Display Data Stem-and-Leaf Graphs (Stemplots), Line Graphs, and Bar Graphs One simple graph, the stem-and-leaf graph or stemplot , comes from the field of exploratory data analysis. It is a good choice when the data sets are small. To create the plot, divide each observation of data into a stem and a leaf. The leaf consists of a final significant digit . For example, 23 has stem two and leaf three. The number 432 has stem 43 and leaf two. Likewise, the number 5,432 has stem 543 and leaf two. The decimal 9.3 has stem nine and leaf three. Write the stems in a vertical line from smallest to largest. Draw a vertical line to the right of the stems. Then write the leaves in increasing order next to their corresponding stem. For Professor Dean's spring pre-calculus class, scores for the first exam were as follows (smallest to largest): 33; 42; 49; 49; 53; 55; 55; 61; 63; 67; 68; 68; 69; 69; 72; 73; 74; 78; 80; 83; 88; 88; 88; 90; 92; 94; 94; 94; 94; 96; 100 Stem-and-Leaf Graph Stem Leaf 3 3 4 2 9 9 5 3 5 5 6 1 3 7 8 8 9 9 7 2 3 4 8 8 0 3 8 8 8 9 0 2 4 4 4 4 6 10 0 The stemplot shows that most scores fell in the 60s, 70s, 80s, and 90s. Eight out of the 31 scores or approximately 26% ( 8 31 ) were in the 90s or 100, a fairly high number of As. Try It For the Park City basketball team, scores for the last 30 games were as follows (smallest to largest): 32; 32; 33; 34; 38; 40; 42; 42; 43; 44; 46; 47; 47; 48; 48; 48; 49; 50; 50; 51; 52; 52; 52; 53; 54; 56; 57; 57; 60; 61 Construct a stem plot for the data. Stem Leaf 3 2 2 3 4 8 4 0 2 2 3 4 6 7 7 8 8 8 9 5 0 0 1 2 2 2 3 4 6 7 7 6 0 1 The stemplot is a quick way to graph data and gives an exact picture of the data. You want to look for an overall pattern and any outliers. An outlier is an observation of data that does not fit the rest of the data. It is sometimes called an extreme value. When you graph an outlier, it will appear not to fit the pattern of the graph. Some outliers are due to mistakes (for example, writing down 50 instead of 500) while others may indicate that something unusual is happening. It takes some background information to explain outliers, so we will cover them in more detail later. The data are the distances (in kilometers) from a home to local supermarkets. Create a stemplot using the data: 1.1; 1.5; 2.3; 2.5; 2.7; 3.2; 3.3; 3.3; 3.5; 3.8; 4.0; 4.2; 4.5; 4.5; 4.7; 4.8; 5.5; 5.6; 6.5; 6.7; 12.3 Do the data seem to have any concentration of values? NOTE The leaves are to the right of the decimal. The value 12.3 may be an outlier. Values appear to concentrate at three and four kilometers. Stem Leaf 1 1 5 2 3 5 7 3 2 3 3 5 8 4 0 2 5 5 7 8 5 5 6 6 5 7 7 8 9 10 11 12 3 Try It The following data show the distances (in miles) from the homes of off-campus statistics students to the college. Create a stem plot using the data and identify any outliers: 0.5; 0.7; 1.1; 1.2; 1.2; 1.3; 1.3; 1.5; 1.5; 1.7; 1.7; 1.8; 1.9; 2.0; 2.2; 2.5; 2.6; 2.8; 2.8; 2.8; 3.5; 3.8; 4.4; 4.8; 4.9; 5.2; 5.5; 5.7; 5.8; 8.0 Stem Leaf 0 5 7 1 1 2 2 3 3 5 5 7 7 8 9 2 0 2 5 6 8 8 8 3 5 8 4 4 8 9 5 2 5 7 8 6 7 8 0 The value 8.0 may be an outlier. Values appear to concentrate at one and two miles. A side-by-side stem-and-leaf plot allows a comparison of the two data sets in two columns. In a side-by-side stem-and-leaf plot, two sets of leaves share the same stem. The leaves are to the left and the right of the stems. and show the ages of presidents at their inauguration and at their death. Construct a side-by-side stem-and-leaf plot using this data. Presidential Ages at Inauguration President Age President Age President Age Washington 57 Lincoln 52 Hoover 54 J. Adams 61 A. Johnson 56 F. Roosevelt 51 Jefferson 57 Grant 46 Truman 60 Madison 57 Hayes 54 Eisenhower 62 Monroe 58 Garfield 49 Kennedy 43 J. Q. Adams 57 Arthur 51 L. Johnson 55 Jackson 61 Cleveland 47 Nixon 56 Van Buren 54 B. Harrison 55 Ford 61 W. H. Harrison 68 Cleveland 55 Carter 52 Tyler 51 McKinley 54 Reagan 69 Polk 49 T. Roosevelt 42 G.H.W. Bush 64 Taylor 64 Taft 51 Clinton 47 Fillmore 50 Wilson 56 G. W. Bush 54 Pierce 48 Harding 55 Obama 47 Buchanan 65 Coolidge 51 Presidential Age at Death President Age President Age President Age Washington 67 Lincoln 56 Hoover 90 J. Adams 90 A. Johnson 66 F. Roosevelt 63 Jefferson 83 Grant 63 Truman 88 Madison 85 Hayes 70 Eisenhower 78 Monroe 73 Garfield 49 Kennedy 46 J. Q. Adams 80 Arthur 56 L. Johnson 64 Jackson 78 Cleveland 71 Nixon 81 Van Buren 79 B. Harrison 67 Ford 93 W. H. Harrison 68 Cleveland 71 Reagan 93 Tyler 71 McKinley 58 Polk 53 T. Roosevelt 60 Taylor 65 Taft 72 Fillmore 74 Wilson 67 Pierce 64 Harding 57 Buchanan 77 Coolidge 60 Ages at Inauguration Ages at Death 9 9 8 7 7 7 6 3 2 4 6 9 8 7 7 7 7 6 6 6 5 5 5 5 4 4 4 4 4 2 2 1 1 1 1 1 0 5 3 6 6 7 7 8 9 8 5 4 4 2 1 1 1 0 6 0 0 3 3 4 4 5 6 7 7 7 8 7 0 1 1 1 2 3 4 7 8 8 9 8 0 1 3 5 8 9 0 0 3 3 The table shows the number of wins and losses the Atlanta Hawks have had in 42 seasons. Create a side-by-side stemand-leaf plot of these wins and losses. Losses Wins Season Losses Wins Season 34 48 1 41 41 22 34 48 2 39 43 23 46 36 3 44 38 24 46 36 4 39 43 25 36 46 5 25 57 26 47 35 6 40 42 27 51 31 7 36 46 28 53 29 8 26 56 29 51 31 9 32 50 30 41 41 10 19 31 31 36 46 11 54 28 32 32 50 12 57 25 33 51 31 13 49 33 34 40 42 14 47 35 35 39 43 15 54 28 36 42 40 16 69 13 37 48 34 17 56 26 38 32 50 18 52 30 39 25 57 19 45 37 40 32 50 20 35 47 41 30 52 21 29 53 42 Another type of graph that is useful for specific data values is a line graph . In the particular line graph shown in , the x -axis (horizontal axis) consists of data values and the y -axis (vertical axis) consists of frequency points . The frequency points are connected using line segments. In a survey, 40 parents were asked how many times per week a teenager must be reminded to do their chores. The results are shown in and in . Number of times teenager is reminded Frequency 0 2 1 5 2 8 3 14 4 7 5 4 Try It In a survey, 40 people were asked how many times per year they had their car in the shop for repairs. The results are shown in . Construct a line graph. Number of times in shop Frequency 0 7 1 10 2 14 3 9 Bar graphs consist of bars that are separated from each other. The bars can be rectangles or they can be rectangular boxes (used in three-dimensional plots), and they can be vertical or horizontal. The bar graph shown in has age groups represented on the x -axis and proportions on the y -axis . The percentage of U.S.-based TikTok users by age is shown in . Construct a bar graph using these data. Age groups Proportion (%) of TikTok users 10–19 32.5% 20–29 29.5% 30–39 16.4% 40–49 13.9% 50+ 7.1% Try It The population in Park City is made up of children, working-age adults, and retirees. shows the three age groups, the number of people in the town from each age group, and the proportion (%) of people in each age group. Construct a bar graph showing the proportions. Age groups Number of people Proportion of population Children 67,059 19% Working-age adults 152,198 43% Retirees 131,662 38% The columns in show the projected data for the year 2030 for the number and percentages of high school graduates by geographic region in the United States. Create a bar graph for these data with the geographic region (qualitative data) on the x -axis and the percentage of high school data (quantitative data) on the y -axis. Region Number of Graduates Percentage of Graduates Northeast 517,720 16.1% Midwest 695,170 21.6% South 1,253,540 39.0% West 749,400 23.3% Try It Park City is broken down into six voting districts. The table shows the percent of the total registered voter population that lives in each district as well as the percent total of the entire population that lives in each district. Construct a bar graph that shows the registered voter population by district. District Registered voter population Overall city population 1 15.5% 19.4% 2 12.2% 15.6% 3 9.8% 9.0% 4 17.4% 18.5% 5 22.8% 20.7% 6 22.3% 16.8% Below is a two-way table showing the types of pets owned by men and women: Dogs Cats Fish Total Men 4 2 2 8 Women 4 6 2 12 Total 8 8 4 20 Given these data, calculate the conditional distributions for the subpopulation of men who own each pet type. Men who own dogs = 4/8 = 0.5 Men who own cats = 2/8 = 0.25 Men who own fish = 2/8 = 0.25 Note: The sum of all of the conditional distributions must equal one. In this case, 0.5 + 0.25 + 0.25 = 1; therefore, the solution "checks". Try It Given the data in , calculate the conditional distributions for the subpopulation of women who own each pet type. Women who own dogs = 4 12 = 0 . 33 Women who own cats = 6 12 = 0 . 5 Women who own fish = 2 12 = 0 . 17 Note: The sum of all of the conditional distributions must equal one. In this case, 0 . 33 + 0 . 5 + 0 . 17 = 1 ; therefore, the solution “checks”. Histograms, Frequency Polygons, and Time Series Graphs For most of the work you do in this book, you will use a histogram to display the data. One advantage of a histogram is that it can readily display large data sets. A rule of thumb is to use a histogram when the data set consists of 100 values or more. A histogram consists of contiguous (adjoining) boxes. It has both a horizontal axis and a vertical axis. The horizontal axis is labeled with what the data represents (for instance, distance from your home to school). The vertical axis is labeled either frequency or relative frequency (or percent frequency or probability). The graph will have the same shape with either label. The histogram (like the stemplot) can give you the shape of the data, the center, and the spread of the data. The relative frequency is equal to the frequency for an observed value of the data divided by the total number of data values in the sample.(Remember, frequency is defined as the number of times an answer occurs.) If: f = frequency n = total number of data values (or the sum of the individual frequencies), and RF = relative frequency, then: RF = f n For example, if three students in an English class of 40 students received from 90% to 100%, then, f = 3, n = 40, and RF = f n = 3 40 = 0.075. 7.5% of the students received 90–100%. 90–100% are quantitative measures. To construct a histogram , first decide how many bars or intervals , also called classes, represent the data. Many histograms consist of five to 15 bars or classes for clarity. The number of bars needs to be chosen. Choose a starting point for the first interval to be less than the smallest data value. A convenient starting point is a lower value carried out to one more decimal place than the value with the most decimal places. For example, if the value with the most decimal places is 6.1 and this is the smallest value, a convenient starting point is 6.05 (6.1 – 0.05 = 6.05). We say that 6.05 has more precision. If the value with the most decimal places is 2.23 and the lowest value is 1.5, a convenient starting point is 1.495 (1.5 – 0.005 = 1.495). If the value with the most decimal places is 3.234 and the lowest value is 1.0, a convenient starting point is 0.9995 (1.0 – 0.0005 = 0.9995). If all the data happen to be integers and the smallest value is two, then a convenient starting point is 1.5 (2 – 0.5 = 1.5). Also, when the starting point and other boundaries are carried to one additional decimal place, no data value will fall on a boundary. The next two examples go into detail about how to construct a histogram using continuous data and how to create a histogram using discrete data. The following data are the heights (in inches to the nearest half inch) of 100 male semiprofessional soccer players. The heights are continuous data, since height is measured. 60; 60.5; 61; 61; 61.5 63.5; 63.5; 63.5 64; 64; 64; 64; 64; 64; 64; 64.5; 64.5; 64.5; 64.5; 64.5; 64.5; 64.5; 64.5 66; 66; 66; 66; 66; 66; 66; 66; 66; 66; 66.5; 66.5; 66.5; 66.5; 66.5; 66.5; 66.5; 66.5; 66.5; 66.5; 66.5; 67; 67; 67; 67; 67; 67; 67; 67; 67; 67; 67; 67; 67.5; 67.5; 67.5; 67.5; 67.5; 67.5; 67.5 68; 68; 69; 69; 69; 69; 69; 69; 69; 69; 69; 69; 69.5; 69.5; 69.5; 69.5; 69.5 70; 70; 70; 70; 70; 70; 70.5; 70.5; 70.5; 71; 71; 71 72; 72; 72; 72.5; 72.5; 73; 73.5 74 The smallest data value is 60. Since the data with the most decimal places has one decimal (for instance, 61.5), we want our starting point to have two decimal places. Since the numbers 0.5, 0.05, 0.005, etc. are convenient numbers, use 0.05 and subtract it from 60, the smallest value, for the convenient starting point. 60 – 0.05 = 59.95 which is more precise than, say, 61.5 by one decimal place. The starting point is, then, 59.95. The largest value is 74, so 74 + 0.05 = 74.05 is the ending value. Next, calculate the width of each bar or class interval. To calculate this width, subtract the starting point from the ending value and divide by the number of bars (you must choose the number of bars you desire). Suppose you choose eight bars. 74.05 - 59.95 = 14.1 14.1 ÷ 8 = 1.76 NOTE We will round up to two and make each bar or class interval two units wide. Rounding up to two is one way to prevent a value from falling on a boundary. Rounding to the next number is often necessary even if it goes against the standard rules of rounding. For this example, using 1.76 as the width would also work. A guideline that is followed by some for the width of a bar or class interval is to take the square root of the number of data values and then round to the nearest whole number, if necessary. For example, if there are 150 values of data, take the square root of 150 and round to 12 bars or intervals. The boundaries are: 59.95 59.95 + 2 = 61.95 61.95 + 2 = 63.95 63.95 + 2 = 65.95 65.95 + 2 = 67.95 67.95 + 2 = 69.95 69.95 + 2 = 71.95 71.95 + 2 = 73.95 73.95 + 2 = 75.95 The heights 60 through 61.5 inches are in the interval 59.95–61.95. The heights that are 63.5 are in the interval 61.95–63.95. The heights that are 64 through 64.5 are in the interval 63.95–65.95. The heights 66 through 67.5 are in the interval 65.95–67.95. The heights 68 through 69.5 are in the interval 67.95–69.95. The heights 70 through 71 are in the interval 69.95–71.95. The heights 72 through 73.5 are in the interval 71.95–73.95. The height 74 is in the interval 73.95–75.95. The following histogram displays the heights on the x -axis and relative frequency on the y -axis. Try It The following data are the shoe sizes of 50 male students. The sizes are continuous data since shoe size is measured. Construct a histogram and calculate the width of each bar or class interval. Suppose you choose six bars. 9; 9; 9.5; 9.5; 10; 10; 10; 10; 10; 10; 10.5; 10.5; 10.5; 10.5; 10.5; 10.5; 10.5; 10.5 11; 11; 11; 11; 11; 11; 11; 11; 11; 11; 11; 11; 11; 11.5; 11.5; 11.5; 11.5; 11.5; 11.5; 11.5 12; 12; 12; 12; 12; 12; 12; 12.5; 12.5; 12.5; 12.5; 14 Smallest value: 9 Largest value: 14 Convenient starting value: 9 – 0.05 = 8.95 Convenient ending value: 14 + 0.05 = 14.05 14.05 − 8.95 6 = 0.85 The calculations suggests using 0.85 as the width of each bar or class interval. You can also use an interval with a width equal to one. Create a histogram for the following data: the number of books bought by 50 part-time college students at ABC College. The number of books is discrete data , since books are counted. 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1 2; 2; 2; 2; 2; 2; 2; 2; 2; 2 3; 3; 3; 3; 3; 3; 3; 3; 3; 3; 3; 3; 3; 3; 3; 3 4; 4; 4; 4; 4; 4 5; 5; 5; 5; 5 6; 6 Eleven students buy one book. Ten students buy two books. Sixteen students buy three books. Six students buy four books. Five students buy five books. Two students buy six books. Because the data are integers, subtract 0.5 from 1, the smallest data value and add 0.5 to 6, the largest data value. Then the starting point is 0.5 and the ending value is 6.5. Next, calculate the width of each bar or class interval. If the data are discrete and there are not too many different values, a width that places the data values in the middle of the bar or class interval is the most convenient. Since the data consist of the numbers 1, 2, 3, 4, 5, 6, and the starting point is 0.5, a width of one places the 1 in the middle of the interval from 0.5 to 1.5, the 2 in the middle of the interval from 1.5 to 2.5, the 3 in the middle of the interval from 2.5 to 3.5, the 4 in the middle of the interval from _______ to _______, the 5 in the middle of the interval from _______ to _______, and the _______ in the middle of the interval from _______ to _______ . 3.5 to 4.5 4.5 to 5.5 6 5.5 to 6.5 Calculate the number of bars as follows: 6.5 - 0.5 = 6 6 ÷ 1 = 6 where 1 is the width of a bar. Therefore, bars = 6. The following histogram displays the number of books on the x -axis and the frequency on the y -axis. Try It The following data are the number of sports played by 50 student athletes. The number of sports is discrete data since sports are counted. 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2 3; 3; 3; 3; 3; 3; 3; 3 20 student athletes play one sport. 22 student athletes play two sports. Eight student athletes play three sports. Fill in the blanks for the following sentence. Since the data consist of the numbers 1, 2, 3, and the starting point is 0.5, a width of one places the 1 in the middle of the interval 0.5 to _____, the 2 in the middle of the interval from _____ to _____, and the 3 in the middle of the interval from _____ to _____. Using this data set, construct a histogram. Number of hours my classmates spent playing video games on weekends 9.95 10 2.25 16.75 0 19.5 22.5 7.5 15 12.75 5.5 11 10 20.75 17.5 23 21.9 24 23.75 18 20 15 22.9 18.8 20.5 Some values in this data set fall on boundaries for the class intervals. A value is counted in a class interval if it falls on the left boundary, but not if it falls on the right boundary. Different researchers may set up histograms for the same data in different ways. There is more than one correct way to set up a histogram. Try It The following data represent the number of employees at various restaurants in New York City. Using this data, create a histogram. 22 35 15 26 40 28 18 20 25 34 39 42 24 22 19 27 22 34 40 20 38 and 28 Use 10–19 as the first interval. Count the money (bills and change) in your pocket or purse. Your instructor will record the amounts. As a class, construct a histogram displaying the data. Discuss how many intervals you think is appropriate. You may want to experiment with the number of intervals. Frequency Polygons Frequency polygons are analogous to line graphs, and just as line graphs make continuous data visually easy to interpret, so too do frequency polygons. To construct a frequency polygon, first examine the data and decide on the number of intervals, or class intervals, to use on the x -axis and y -axis. After choosing the appropriate ranges, begin plotting the data points. After all the points are plotted, draw line segments to connect them. A frequency polygon was constructed from the frequency table below. Frequency distribution for calculus final test scores Lower bound Upper bound Frequency Cumulative frequency 49.5 59.5 5 5 59.5 69.5 10 15 69.5 79.5 30 45 79.5 89.5 40 85 89.5 99.5 15 100 The first label on the x -axis is 44.5. This represents an interval extending from 39.5 to 49.5. Since the lowest test score is 54.5, this interval is used only to allow the graph to touch the x -axis. The point labeled 54.5 represents the next interval, or the first “real” interval from the table, and contains five scores. This reasoning is followed for each of the remaining intervals with the point 104.5 representing the interval from 99.5 to 109.5. Again, this interval contains no data and is only used so that the graph will touch the x -axis. Looking at the graph, we say that this distribution is skewed because one side of the graph does not mirror the other side. Try It Construct a frequency polygon of U.S. Presidents’ ages at inauguration shown in . Age at inauguration Frequency 41.5–46.5 4 46.5–51.5 11 51.5–56.5 14 56.5–61.5 9 61.5–66.5 4 66.5–71.5 2 The first label on the x -axis is 39. This represents an interval extending from 36.5 to 41.5. Since there are no ages less than 41.5, this interval is used only to allow the graph to touch the x -axis. The point labeled 44 represents the next interval, or the first “real” interval from the table, and contains four scores. This reasoning is followed for each of the remaining intervals with the point 74 representing the interval from 71.5 to 76.5. Again, this interval contains no data and is only used so that the graph will touch the x -axis. Looking at the graph, we say that this distribution is skewed because one side of the graph does not mirror the other side. Frequency polygons are useful for comparing distributions. This is achieved by overlaying the frequency polygons drawn for different data sets. We will construct an overlay frequency polygon comparing the scores from with the students’ final numeric grade. Frequency distribution for calculus final test scores Lower bound Upper bound Frequency Cumulative frequency 49.5 59.5 5 5 59.5 69.5 10 15 69.5 79.5 30 45 79.5 89.5 40 85 89.5 99.5 15 100 Frequency distribution for calculus final grades Lower bound Upper bound Frequency Cumulative frequency 49.5 59.5 10 10 59.5 69.5 10 20 69.5 79.5 30 50 79.5 89.5 45 95 89.5 99.5 5 100 Try It We will construct an overlay frequency polygon comparing the scores from with the students’ final test scores in algebra. Frequency Distribution for Algebra Final Test Scores Lower Bound Upper Bound Frequency Cumulative Frequency 49.5 59.5 10 10 59.5 69.5 5 15 69.5 79.5 40 55 79.5 89.5 35 90 89.5 99.5 10 100 Calculus Test Scores v Algebra Test Scores Image Insert Constructing a Time Series Graph Suppose that we want to study the temperature range of a region for an entire month. Every day at noon we note the temperature and write this down in a log. A variety of statistical studies could be done with these data. We could find the mean or the median temperature for the month. We could construct a histogram displaying the number of days that temperatures reach a certain range of values. However, all of these methods ignore a portion of the data that we have collected. One feature of the data that we may want to consider is that of time. Since each date is paired with the temperature reading for the day, we don‘t have to think of the data as being random. We can instead use the times given to impose a chronological order on the data. A graph that recognizes this ordering and displays the changing temperature as the month progresses is called a time series graph. To construct a time series graph, we must look at both pieces of our paired data set . We start with a standard Cartesian coordinate system. The horizontal axis is used to plot the date or time increments, and the vertical axis is used to plot the values of the variable that we are measuring. By doing this, we make each point on the graph correspond to a date and a measured quantity. The points on the graph are typically connected by straight lines in the order in which they occur. The following data shows the Annual Consumer Price Index, each month, for ten years. Construct a time series graph for the Annual Consumer Price Index data only. Year Jan Feb Mar Apr May Jun Jul 1 181.7 183.1 184.2 183.8 183.5 183.7 183.9 2 185.2 186.2 187.4 188.0 189.1 189.7 189.4 3 190.7 191.8 193.3 194.6 194.4 194.5 195.4 4 198.3 198.7 199.8 201.5 202.5 202.9 203.5 5 202.416 203.499 205.352 206.686 207.949 208.352 208.299 6 211.080 211.693 213.528 214.823 216.632 218.815 219.964 7 211.143 212.193 212.709 213.240 213.856 215.693 215.351 8 216.687 216.741 217.631 218.009 218.178 217.965 218.011 9 220.223 221.309 223.467 224.906 225.964 225.722 225.922 10 226.665 227.663 229.392 230.085 229.815 229.478 229.104 Year Aug Sep Oct Nov Dec Annual 1 184.6 185.2 185.0 184.5 184.3 184.0 2 189.5 189.9 190.9 191.0 190.3 188.9 3 196.4 198.8 199.2 197.6 196.8 195.3 4 203.9 202.9 201.8 201.5 201.8 201.6 5 207.917 208.490 208.936 210.177 210.036 207.342 6 219.086 218.783 216.573 212.425 210.228 215.303 7 215.834 215.969 216.177 216.330 215.949 214.537 8 218.312 218.439 218.711 218.803 219.179 218.056 9 226.545 226.889 226.421 226.230 225.672 224.939 10 230.379 231.407 231.317 230.221 229.601 229.594 Try It The following table is a portion of a data set from www.worldbank.org. Use the table to construct a time series graph for CO 2 emissions for the United States. CO 2 emissions Year Ukraine United Kingdom United States 1 352,259 540,640 5,681,664 2 343,121 540,409 5,790,761 3 339,029 541,990 5,826,394 4 327,797 542,045 5,737,615 5 328,357 528,631 5,828,697 6 323,657 522,247 5,656,839 7 272,176 474,579 5,299,563 Uses of a Time Series Graph Time series graphs are important tools in various applications of statistics. When recording values of the same variable over an extended period of time, sometimes it is difficult to discern any trend or pattern. However, once the same data points are displayed graphically, some features jump out. Time series graphs make trends easy to spot. How NOT to Lie with Statistics It is important to remember that the very reason we develop a variety of methods to present data is to develop insights into the subject of what the observations represent. We want to get a "sense" of the data. Are the observations all very much alike or are they spread across a wide range of values, are they bunched at one end of the spectrum or are they distributed evenly and so on. We are trying to get a visual picture of the numerical data. Shortly we will develop formal mathematical measures of the data, but our visual graphical presentation can say much. It can, unfortunately, also say much that is distracting, confusing and simply wrong in terms of the impression the visual leaves. Many years ago Darrell Huff wrote the book How to Lie with Statistics . It has been through 25 plus printings and sold more than one and one-half million copies. His perspective was a harsh one and used many actual examples that were designed to mislead. He wanted to make people aware of such deception, but perhaps more importantly to educate so that others do not make the same errors inadvertently. Again, the goal is to enlighten with visuals that tell the story of the data. Pie charts have a number of common problems when used to convey the message of the data. Too many pieces of the pie overwhelm the reader. More than perhaps five or six categories ought to give an idea of the relative importance of each piece. This is after all the goal of a pie chart, what subset matters most relative to the others. If there are more components than this then perhaps an alternative approach would be better or perhaps some can be consolidated into an "other" category. Pie charts cannot show changes over time, although we see this attempted all too often. In federal, state, and city finance documents pie charts are often presented to show the components of revenue available to the governing body for appropriation: income tax, sales tax motor vehicle taxes and so on. In and of itself this is interesting information and can be nicely done with a pie chart. The error occurs when two years are set side-by-side. Because the total revenues change year to year, but the size of the pie is fixed, no real information is provided and the relative size of each piece of the pie cannot be meaningfully compared. Histograms can be very helpful in understanding the data. Properly presented, they can be a quick visual way to present probabilities of different categories by the simple visual of comparing relative areas in each category. Here the error, purposeful or not, is to vary the width of the categories. This of course makes comparison to the other categories impossible. It does embellish the importance of the category with the expanded width because it has a greater area, inappropriately, and thus visually "says" that that category has a higher probability of occurrence. Time series graphs perhaps are the most abused. A plot of some variable across time should never be presented on axes that change part way across the page either in the vertical or horizontal dimension. Perhaps the time frame is changed from years to months. Perhaps this is to save space or because monthly data was not available for early years. In either case this confounds the presentation and destroys any value of the graph. If this is not done to purposefully confuse the reader, then it certainly is either lazy or sloppy work. Changing the units of measurement of the axis can smooth out a drop or accentuate one. If you want to show large changes, then measure the variable in small units, penny rather than thousands of dollars. And of course to continue the fraud, be sure that the axis does not begin at zero, zero. If it begins at zero, zero, then it becomes apparent that the axis has been manipulated. Perhaps you have a client that is concerned with the volatility of the portfolio you manage. An easy way to present the data is to use long time periods on the time series graph. Use months or better, quarters rather than daily or weekly data. If that doesn't get the volatility down then spread the time axis relative to the rate of return or portfolio valuation axis. If you want to show "quick" dramatic growth, then shrink the time axis. Any positive growth will show visually "high" growth rates. Do note that if the growth is negative then this trick will show the portfolio is collapsing at a dramatic rate. Again, the goal of descriptive statistics is to convey meaningful visuals that tell the story of the data. Purposeful manipulation is fraud and unethical at the worst, but even at its best, making these type of errors will lead to confusion on the part of the analysis. References Doyle, Brandon. "TikTok Statistics – Updated Mar 2023." Wallaroo. March 21, 2023. https://wallaroomedia.com/blog/ social-media/tiktok-statistics/. Table 219.20. Public High School Graduates, by Region, State, and Jurisdiction: Selected Years, 1980-81 through 2030-31.” National Center for Education Statistics. Accessed May 24, 2023. https://nces.ed.gov/programs/digest/d21/tables/dt21_219.20.asp. “Overweight and Obesity: Adult Obesity Facts.” Centers for Disease Control and Prevention. Available online at http://www.cdc.gov/obesity/data/adult.html (accessed September 13, 2013). Data on annual homicides in Detroit, 1961–73, from Gunst & Mason’s book ‘Regression Analysis and its Application’, Marcel Dekker “Timeline: Guide to the U.S. Presidents: Information on every president’s birthplace, political party, term of office, and more.” Scholastic, 2013. Available online at http://www.scholastic.com/teachers/article/timeline-guide-us-presidents (accessed April 3, 2013). “Presidents.” Fact Monster. Pearson Education, 2007. Available online at http://www.factmonster.com/ipka/A0194030.html (accessed April 3, 2013). “Food Security Statistics.” Food and Agriculture Organization of the United Nations. Available online at http://www.fao.org/economic/ess/ess-fs/en/ (accessed April 3, 2013). “Consumer Price Index.” United States Department of Labor: Bureau of Labor Statistics. Available online at http://data.bls.gov/pdq/SurveyOutputServlet (accessed April 3, 2013). “CO2 emissions (kt).” The World Bank, 2013. Available online at http://databank.worldbank.org/data/home.aspx (accessed April 3, 2013). “Births Time Series Data.” General Register Office For Scotland, 2013. Available online at http://www.gro-scotland.gov.uk/statistics/theme/vital-events/births/time-series.html (accessed April 3, 2013). “Demographics: Children under the age of 5 years underweight.” Indexmundi. Available online at http://www.indexmundi.com/g/r.aspx?t=50&v=2224&aml=en (accessed April 3, 2013). Gunst, Richard, Robert Mason. Regression Analysis and Its Application: A Data-Oriented Approach . CRC Press: 1980. “Overweight and Obesity: Adult Obesity Facts.” Centers for Disease Control and Prevention. Available online at http://www.cdc.gov/obesity/data/adult.html (accessed September 13, 2013). Chapter Review A stem-and-leaf plot is a way to plot data and look at the distribution. In a stem-and-leaf plot, all data values within a class are visible. The advantage in a stem-and-leaf plot is that all values are listed, unlike a histogram, which gives classes of data values. A line graph is often used to represent a set of data values in which a quantity varies with time. These graphs are useful for finding trends. That is, finding a general pattern in data sets including temperature, sales, employment, company profit or cost over a period of time. A bar graph is a chart that uses either horizontal or vertical bars to show comparisons among categories. One axis of the chart shows the specific categories being compared, and the other axis represents a discrete value. Some bar graphs present bars clustered in groups of more than one (grouped bar graphs), and others show the bars divided into subparts to show cumulative effect (stacked bar graphs). Bar graphs are especially useful when categorical data is being used. A histogram is a graphic version of a frequency distribution. The graph consists of bars of equal width drawn adjacent to each other. The horizontal scale represents classes of quantitative data values and the vertical scale represents frequencies. The heights of the bars correspond to frequency values. Histograms are typically used for large, continuous, quantitative data sets. A frequency polygon can also be used when graphing large data sets with data points that repeat. The data usually goes on y -axis with the frequency being graphed on the x -axis. Time series graphs can be helpful when looking at large amounts of data for one variable over a period of time. For the next three exercises, use the data to construct a line graph. In a survey, 40 people were asked how many times they visited a store before making a major purchase. The results are shown in . Number of times in store Frequency 1 4 2 10 3 16 4 6 5 4 In a survey, several people were asked how many years it has been since they purchased a mattress. The results are shown in . Years since last purchase Frequency 0 2 1 8 2 13 3 22 4 16 5 9 Several children were asked how many TV shows they watch each day. The results of the survey are shown in . Number of TV shows Frequency 0 12 1 18 2 36 3 7 4 2 The students in Ms. Ramirez’s math class have birthdays in each of the four seasons. shows the four seasons, the number of students who have birthdays in each season, and the percentage (%) of students in each group. Construct a bar graph showing the number of students. Seasons Number of students Proportion of population Spring 8 24% Summer 9 26% Autumn 11 32% Winter 6 18% Using the data from Mrs. Ramirez’s math class supplied in , construct a bar graph showing the percentages. David County has six high schools. Each school sent students to participate in a county-wide science competition. shows the percentage breakdown of competitors from each school, and the percentage of the entire student population of the county that goes to each school. Construct a bar graph that shows the population percentage of competitors from each school. High school Science competition population Overall student population Alabaster 28.9% 8.6% Concordia 7.6% 23.2% Genoa 12.1% 15.0% Mocksville 18.5% 14.3% Tynneson 24.2% 10.1% West End 8.7% 28.8% Use the data from the David County science competition supplied in . Construct a bar graph that shows the county-wide population percentage of students at each school. Sixty-five randomly selected car salespersons were asked the number of cars they generally sell in one week. Fourteen people answered that they generally sell three cars; nineteen generally sell four cars; twelve generally sell five cars; nine generally sell six cars; eleven generally sell seven cars. Complete the table. Data value (# cars) Frequency Relative frequency Cumulative relative frequency What does the frequency column in sum to? Why? 65 What does the relative frequency column in sum to? Why? What is the difference between relative frequency and frequency for each data value in ? The relative frequency shows the proportion of data points that have each value. The frequency tells the number of data points that have each value. What is the difference between cumulative relative frequency and relative frequency for each data value? To construct the histogram for the data in , determine appropriate minimum and maximum x and y values and the scaling. Sketch the histogram. Label the horizontal and vertical axes with words. Include numerical scaling. Answers will vary. One possible histogram is shown: Construct a frequency polygon for the following: Pulse rates for Females Frequency 60–69 12 70–79 14 80–89 11 90–99 1 100–109 1 110–119 0 120–129 1 Actual speed in a 30 MPH zone Frequency 42–45 25 46–49 14 50–53 7 54–57 3 58–61 1 Tar (mg) in nonfiltered cigarettes Frequency 10–13 1 14–17 0 18–21 15 22–25 7 26–29 2 Construct a frequency polygon from the frequency distribution for the 50 highest ranked countries for depth of hunger. Depth of hunger Frequency 230–259 21 260–289 13 290–319 5 320–349 7 350–379 1 380–409 1 410–439 1 Find the midpoint for each class. These will be graphed on the x -axis. The frequency values will be graphed on the y -axis values. Use the two frequency tables to compare the life expectancy of males and females from 20 randomly selected countries. Include an overlayed frequency polygon and discuss the shapes of the distributions, the center, the spread, and any outliers. What can we conclude about the life expectancy of females compared to males? Life expectancy at birth – Females Frequency 49–55 3 56–62 3 63–69 1 70–76 3 77–83 8 84–90 2 Life expectancy at birth – Males Frequency 49–55 3 56–62 3 63–69 1 70–76 1 77–83 7 84–90 5 Construct a times series graph for (a) the number of male births, (b) the number of female births, and (c) the total number of births. Sex/Year 1855 1856 1857 1858 1859 1860 1861 Female 45,545 49,582 50,257 50,324 51,915 51,220 52,403 Male 47,804 52,239 53,158 53,694 54,628 54,409 54,606 Total 93,349 101,821 103,415 104,018 106,543 105,629 107,009 Sex/Year 1862 1863 1864 1865 1866 1867 1868 1869 Female 51,812 53,115 54,959 54,850 55,307 55,527 56,292 55,033 Male 55,257 56,226 57,374 58,220 58,360 58,517 59,222 58,321 Total 107,069 109,341 112,333 113,070 113,667 114,044 115,514 113,354 Sex/Year 1870 1871 1872 1873 1874 1875 Female 56,431 56,099 57,472 58,233 60,109 60,146 Male 58,959 60,029 61,293 61,467 63,602 63,432 Total 115,390 116,128 118,765 119,700 123,711 123,578 The following data sets list full time police per 100,000 citizens along with homicides per 100,000 citizens for a city during the period from 1961 to 1973. Year 1961 1962 1963 1964 1965 1966 1967 Police 260.35 269.8 272.04 272.96 272.51 261.34 268.89 Homicides 8.6 8.9 8.52 8.89 13.07 14.57 21.36 Year 1968 1969 1970 1971 1972 1973 Police 295.99 319.87 341.43 356.59 376.69 390.19 Homicides 28.03 31.49 37.39 46.26 47.24 52.33 Construct a double time series graph using a common x -axis for both sets of data. Which variable increased the fastest? Explain. Did the city's increase in police officers have an impact on the murder rate? Explain. Homework contains the percentages of adults who were never married in the 50 U.S. states and Washington, DC. State Percent (%) State Percent (%) State Percent (%) Alabama 32.2 Kentucky 31.3 North Dakota 27.2 Alaska 24.5 Louisiana 31.0 Ohio 29.2 Arizona 24.3 Maine 26.8 Oklahoma 30.4 Arkansas 30.1 Maryland 27.1 Oregon 26.8 California 24.0 Massachusetts 23.0 Pennsylvania 28.6 Colorado 21.0 Michigan 30.9 Rhode Island 25.5 Connecticut 22.5 Minnesota 24.8 South Carolina 31.5 Delaware 28.0 Mississippi 34.0 South Dakota 27.3 Washington, DC 22.2 Missouri 30.5 Tennessee 30.8 Florida 26.6 Montana 23.0 Texas 31.0 Georgia 29.6 Nebraska 26.9 Utah 22.5 Hawaii 22.7 Nevada 22.4 Vermont 23.2 Idaho 26.5 New Hampshire 25.0 Virginia 26.0 Illinois 28.2 New Jersey 23.8 Washington 25.5 Indiana 29.6 New Mexico 25.1 West Virginia 32.5 Iowa 28.4 New York 23.9 Wisconsin 26.3 Kansas 29.4 North Carolina 27.8 Wyoming 25.1 Use a random number generator to randomly pick eight states. Construct a bar graph of the rates of those eight states. Construct a bar graph for all the states beginning with the letter "A." Construct a bar graph for all the states beginning with the letter "M." Example solution for using the random number generator for the TI-84+ to generate a simple random sample of 8 states. Instructions are as follows. Number the entries in the table 1–51 (Includes Washington, DC; Numbered vertically) Press MATH Arrow over to PRB Press 5:randInt( Enter 51,1,8) Eight numbers are generated (use the right arrow key to scroll through the numbers). The numbers correspond to the numbered states (for this example: {47 21 9 23 51 13 25 4}. If any numbers are repeated, generate a different number by using 5:randInt(51,1)). Here, the states (and Washington DC) are {Arkansas, Washington DC, Idaho, Maryland, Michigan, Mississippi, Virginia, Wyoming}. Corresponding percents are {30.1, 22.2, 26.5, 27.1, 30.9, 34.0, 26.0, 25.1}. Suppose that three book publishers were interested in the number of fiction paperbacks adult consumers purchase per month. Each publisher conducted a survey. In the survey, adult consumers were asked the number of fiction paperbacks they had purchased the previous month. The results are as follows: Publisher A # of books Freq. Rel. freq. 0 10 1 12 2 16 3 12 4 8 5 6 6 2 8 2 Publisher B # of books Freq. Rel. freq. 0 18 1 24 2 24 3 22 4 15 5 10 7 5 9 1 Publisher C # of books Freq. Rel. freq. 0–1 20 2–3 35 4–5 12 6–7 2 8–9 1 Find the relative frequencies for each survey. Write them in the charts. Use the frequency column to construct a histogram for each publisher's survey. For Publishers A and B, make bar widths of one. For Publisher C, make bar widths of two. In complete sentences, give two reasons why the graphs for Publishers A and B are not identical. Would you have expected the graph for Publisher C to look like the other two graphs? Why or why not? Make new histograms for Publisher A and Publisher B. This time, make bar widths of two. Now, compare the graph for Publisher C to the new graphs for Publishers A and B. Are the graphs more similar or more different? Explain your answer. Often, cruise ships conduct all on-board transactions, with the exception of gambling, on a cashless basis. At the end of the cruise, guests pay one bill that covers all onboard transactions. Suppose that 60 single travelers and 70 couples were surveyed as to their on-board bills for a seven-day cruise from Los Angeles to the Mexican Riviera. Following is a summary of the bills for each group. Singles Amount($) Frequency Rel. frequency 51–100 5 101–150 10 151–200 15 201–250 15 251–300 10 301–350 5 Couples Amount($) Frequency Rel. frequency 100–150 5 201–250 5 251–300 5 301–350 5 351–400 10 401–450 10 451–500 10 501–550 10 551–600 5 601–650 5 Fill in the relative frequency for each group. Construct a histogram for the singles group. Scale the x -axis by $50 widths. Use relative frequency on the y -axis. Construct a histogram for the couples group. Scale the x -axis by $50 widths. Use relative frequency on the y -axis. Compare the two graphs: List two similarities between the graphs. List two differences between the graphs. Overall, are the graphs more similar or different? Construct a new graph for the couples by hand. Since each couple is paying for two individuals, instead of scaling the x -axis by $50, scale it by $100. Use relative frequency on the y -axis. Compare the graph for the singles with the new graph for the couples: List two similarities between the graphs. Overall, are the graphs more similar or different? How did scaling the couples graph differently change the way you compared it to the singles graph? Based on the graphs, do you think that individuals spend the same amount, more or less, as singles as they do person by person as a couple? Explain why in one or two complete sentences. Singles Amount($) Frequency Relative frequency 51–100 5 0.08 101–150 10 0.17 151–200 15 0.25 201–250 15 0.25 251–300 10 0.17 301–350 5 0.08 Couples Amount($) Frequency Relative frequency 100–150 5 0.07 201–250 5 0.07 251–300 5 0.07 301–350 5 0.07 351–400 10 0.14 401–450 10 0.14 451–500 10 0.14 501–550 10 0.14 551–600 5 0.07 601–650 5 0.07 See and . In the following histogram data values that fall on the right boundary are counted in the class interval, while values that fall on the left boundary are not counted (with the exception of the first interval where both boundary values are included). In the following histogram, the data values that fall on the right boundary are counted in the class interval, while values that fall on the left boundary are not counted (with the exception of the first interval where values on both boundaries are included). Compare the two graphs: Answers may vary. Possible answers include: Both graphs have a single peak. Both graphs use class intervals with width equal to $50. Answers may vary. Possible answers include: The couples graph has a class interval with no values. It takes almost twice as many class intervals to display the data for couples. Answers may vary. Possible answers include: The graphs are more similar than different because the overall patterns for the graphs are the same. Answers may vary. Compare the graph for the Singles with the new graph for the Couples: Both graphs have a single peak. Both graphs display 6 class intervals. Both graphs show the same general pattern. Answers may vary. Possible answers include: Although the width of the class intervals for couples is double that of the class intervals for singles, the graphs are more similar than they are different. Answers may vary. Possible answers include: You are able to compare the graphs interval by interval. It is easier to compare the overall patterns with the new scale on the Couples graph. Because a couple represents two individuals, the new scale leads to a more accurate comparison. Answers may vary. Possible answers include: Based on the histograms, it seems that spending does not vary much from singles to individuals who are part of a couple. The overall patterns are the same. The range of spending for couples is approximately double the range for individuals. Twenty-five randomly selected students were asked the number of movies they watched the previous week. The results are as follows. # of movies Frequency Relative frequency Cumulative relative frequency 0 5 1 9 2 6 3 4 4 1 Construct a histogram of the data. Complete the columns of the chart. Use the following information to answer the next two exercises: Suppose 111 people who shopped in a special T-shirt store were asked the number of T-shirts they own costing more than $19 each. The percentage of people who own at most three T-shirts costing more than $19 each is approximately: 21 59 41 Cannot be determined c If the data were collected by asking the first 111 people who entered the store, then the type of sampling is: cluster simple random stratified convenience Following are the rates of unmarried adults for the 50 U.S. states and Washington, DC. State Percent (%) State Percent (%) State Percent (%) Alabama 32.2 Kentucky 31.3 North Dakota 27.2 Alaska 24.5 Louisiana 31.0 Ohio 29.2 Arizona 24.3 Maine 26.8 Oklahoma 30.4 Arkansas 30.1 Maryland 27.1 Oregon 26.8 California 24.0 Massachusetts 23.0 Pennsylvania 28.6 Colorado 21.0 Michigan 30.9 Rhode Island 25.5 Connecticut 22.5 Minnesota 24.8 South Carolina 31.5 Delaware 28.0 Mississippi 34.0 South Dakota 27.3 Washington, DC 22.2 Missouri 30.5 Tennessee 30.8 Florida 26.6 Montana 23.0 Texas 31.0 Georgia 29.6 Nebraska 26.9 Utah 22.5 Hawaii 22.7 Nevada 22.4 Vermont 23.2 Idaho 26.5 New Hampshire 25.0 Virginia 26.0 Illinois 28.2 New Jersey 23.8 Washington 25.5 Indiana 29.6 New Mexico 25.1 West Virginia 32.5 Iowa 28.4 New York 23.9 Wisconsin 26.3 Kansas 29.4 North Carolina 27.8 Wyoming 25.1 Construct a bar graph of the unmarried adult rates of your state and the four states closest to your state. Hint: Label the x -axis with the states. Answers will vary. Frequency the number of times a value of the data occurs Histogram a graphical representation in x - y form of the distribution of data in a data set; x represents the data and y represents the frequency, or relative frequency. The graph consists of contiguous rectangles. Relative Frequency the ratio of the number of times a value of the data occurs in the set of all outcomes to the number of all outcomes

Math


Measures of the Location of the Data The common measures of location are quartiles and percentiles . Quartiles divide an ordered data set into four equal parts. The three quartiles of a data set are labeled as Q1, Q2, and Q3. About one-fourth of the data falls on or below the first quartile Q1. About one-half of the data falls on or below the second quartile Q2. About three-fourths of the data falls on or below the third quartile Q3. In the same way, percentiles divide a data set into 100 equal parts. To calculate quartiles and percentiles, the data must be ordered from smallest to largest. Quartiles divide ordered data into quarters. Percentiles divide ordered data into hundredths. To score in the 90 th percentile of an exam does not mean, necessarily, that you received 90% on a test. It means that 90% of test scores are the same or less than your score and 10% of the test scores are the same or greater than your test score. Percentiles are useful for comparing values. For this reason, universities and colleges use percentiles extensively. One instance in which colleges and universities use percentiles is when SAT results are used to determine a minimum testing score that will be used as an acceptance factor. For example, suppose Duke accepts SAT scores at or above the 75 th percentile. That translates into a score of at least 1220. Percentiles are mostly used with very large populations. Therefore, if you were to say that 90% of the test scores are less (and not the same or less) than your score, it would be acceptable because removing one particular data value is not significant. The median is a number that measures the "center" of the data. You can think of the median as the "middle value," but it does not actually have to be one of the observed values. It is a number that separates ordered data into halves. Half the values are the same number or smaller than the median, and half the values are the same number or larger. For example, consider the following data. 1; 11.5; 6; 7.2; 4; 8; 9; 10; 6.8; 8.3; 2; 2; 10; 1 Ordered from smallest to largest: 1; 1; 2; 2; 4; 6; 6.8; 7.2; 8; 8.3; 9; 10; 10; 11.5 Since there are 14 observations, the median is between the seventh value, 6.8, and the eighth value, 7.2. To find the median, add the two values together and divide by two. 6.8 + 7.2 = 14 14 ÷ 2 = 7 The median is seven. Half of the values are smaller than seven and half of the values are larger than seven. Quartiles are numbers that separate the data into quarters. Quartiles may or may not be part of the data. To find the quartiles, first find the median or second quartile. The first quartile, Q 1 , is the middle value of the lower half of the data, and the third quartile, Q 3 , is the middle value, or median, of the upper half of the data. To get the idea, consider the same data set: 1; 1; 2; 2; 4; 6; 6.8; 7.2; 8; 8.3; 9; 10; 10; 11.5 The median or second quartile is seven. The lower half of the data are 1, 1, 2, 2, 4, 6, 6.8. The middle value of the lower half is two. 1; 1; 2; 2; 4; 6; 6.8 The number two, which is part of the data, is the first quartile . One-fourth of the entire sets of values are the same as or less than two and three-fourths of the values are more than two. The upper half of the data is 7.2, 8, 8.3, 9, 10, 10, 11.5. The middle value of the upper half is nine. The third quartile , Q 3, is nine. Three-fourths (75%) of the ordered data set are less than nine. One-fourth (25%) of the ordered data set are greater than nine. The third quartile is part of the data set in this example. The interquartile range is a number that indicates the spread of the middle half or the middle 50% of the data. It is the difference between the third quartile ( Q 3 ) and the first quartile ( Q 1 ). IQR = Q 3 – Q 1 The IQR can help to determine potential outliers . A value is suspected to be a potential outlier if it is less than (1.5)( IQR ) below the first quartile or more than (1.5)( IQR ) above the third quartile . Potential outliers always require further investigation. NOTE A potential outlier is a data point that is significantly different from the other data points. These special data points may be errors or some kind of abnormality or they may be a key to understanding the data. For the following 13 real estate prices, calculate the IQR and determine if any prices are potential outliers. Prices are in dollars. 389,950; 230,500; 158,000; 479,000; 639,000; 114,950; 5,500,000; 387,000; 659,000; 529,000; 575,000; 488,800; 1,095,000 Order the data from smallest to largest. 114,950; 158,000; 230,500; 387,000; 389,950; 479,000; 488,800; 529,000; 575,000; 639,000; 659,000; 1,095,000; 5,500,000 M = 488,800 Q 1 = 230,500 + 387,000 2 = 308,750 Q 3 = 639,000 + 659,000 2 = 649,000 IQR = 649,000 – 308,750 = 340,250 (1.5)( IQR ) = (1.5)(340,250) = 510,375 Q 1 – (1.5)( IQR ) = 308,750 – 510,375 = –201,625 Q 3 + (1.5)( IQR ) = 649,000 + 510,375 = 1,159,375 No house price is less than –201,625. However, 5,500,000 is more than 1,159,375. Therefore, 5,500,000 is a potential outlier . Try It For the following 11 salaries, calculate the IQR and determine if any salaries are outliers. The salaries are in dollars. $33,000 $64,500 $28,000 $54,000 $72,000 $68,500 $69,000 $42,000 $54,000 $120,000 $40,500 Test scores for a college statistics class held during the day are: 99; 56; 78; 55.5; 32; 90; 80; 81; 56; 59; 45; 77; 84.5; 84; 70; 72; 68; 32; 79; 90 Test scores for a college statistics class held during the evening are: 98; 78; 68; 83; 81; 89; 88; 76; 65; 45; 98; 90; 80; 84.5; 85; 79; 78; 98; 90; 79; 81; 25.5 For the two data sets, find the following: The interquartile range. Compare the two interquartile ranges. Any outliers in either set. The five number summary for the day and night classes is Minimum Q 1 Median Q 3 Maximum Day 32 56 74.5 82.5 99 Night 25.5 78 81 89 98 The IQR for the day group is Q 3 – Q 1 = 82.5 – 56 = 26.5 The IQR for the night group is Q 3 – Q 1 = 89 – 78 = 11 The interquartile range (the spread or variability) for the day class is larger than the night class IQR . This suggests more variation will be found in the day class’s class test scores. Day class outliers are found using the IQR times 1.5 rule. So, Q 1 - IQR (1.5) = 56 – 26.5(1.5) = 16.25 Q 3 + IQR (1.5) = 82.5 + 26.5(1.5) = 122.25 Since the minimum and maximum values for the day class are greater than 16.25 and less than 122.25, there are no outliers. Night class outliers are calculated as: Q 1 – IQR (1.5) = 78 – 11(1.5) = 61.5 Q 3 + IQR(1.5) = 89 + 11(1.5) = 105.5 For this class, any test score less than 61.5 is an outlier. Therefore, the scores of 45 and 25.5 are outliers. Since no test score is greater than 105.5, there is no upper end outlier. Try It Find the interquartile range for the following two data sets and compare them. Test Scores for Class A 69; 96; 81; 79; 65; 76; 83; 99; 89; 67; 90; 77; 85; 98; 66; 91; 77; 69; 80; 94 Test Scores for Class B 90; 72; 80; 92; 90; 97; 92; 75; 79; 68; 70; 80; 99; 95; 78; 73; 71; 68; 95; 100 Fifty statistics students were asked how much sleep they get per school night (rounded to the nearest hour). The results were: Amount of sleep per school night (hours) Frequency Relative frequency Cumulative relative frequency 4 2 0.04 0.04 5 5 0.10 0.14 6 7 0.14 0.28 7 12 0.24 0.52 8 14 0.28 0.80 9 7 0.14 0.94 10 3 0.06 1.00 Find the 28 th percentile . Notice the 0.28 in the "cumulative relative frequency" column. Twenty-eight percent of 50 data values is 14 values. There are 14 values less than the 28 th percentile. They include the two 4s, the five 5s, and the seven 6s. The 28 th percentile is between the last six and the first seven. The 28 th percentile is 6.5. Find the median . Look again at the "cumulative relative frequency" column and find 0.52. The median is the 50 th percentile or the second quartile. 50% of 50 is 25. There are 25 values less than the median. They include the two 4s, the five 5s, the seven 6s, and eleven of the 7s. The median or 50 th percentile is between the 25 th , or seven, and 26 th , or seven, values. The median is seven. Find the third quartile . The third quartile is the same as the 75 th percentile. You can "eyeball" this answer. If you look at the "cumulative relative frequency" column, you find 0.52 and 0.80. When you have all the fours, fives, sixes and sevens, you have 52% of the data. When you include all the 8s, you have 80% of the data. The 75 th percentile, then, must be an eight . Another way to look at the problem is to find 75% of 50, which is 37.5, and round up to 38. The third quartile, Q 3 , is the 38 th value, which is an eight. You can check this answer by counting the values. (There are 37 values below the third quartile and 12 values above.) Try it Forty bus drivers were asked how many hours they spend each day running their routes (rounded to the nearest hour). Find the 65 th percentile. Amount of time spent on route (hours) Frequency Relative frequency Cumulative relative frequency 2 12 0.30 0.30 3 14 0.35 0.65 4 10 0.25 0.90 5 4 0.10 1.00 The 65 th percentile is between the last three and the first four. The 65 th percentile is 3.5. A Formula for Finding the k th Percentile If you were to do a little research, you would find several formulas for calculating the k th percentile. Here is one of them. k = the k th percentile. It may or may not be part of the data. i = the index (ranking or position of a data value) n = the total number of data points, or observations Order the data from smallest to largest. Calculate i = k 100 ( n + 1 ) If i is an integer, then the k th percentile is the data value in the i th position in the ordered set of data. If i is not an integer, then round i up and round i down to the nearest integers. Average the two data values in these two positions in the ordered data set. This is easier to understand in an example. Listed are 29 ages for Academy Award winning best actors in order from smallest to largest. 18; 21; 22; 25; 26; 27; 29; 30; 31; 33; 36; 37; 41; 42; 47; 52; 55; 57; 58; 62; 64; 67; 69; 71; 72; 73; 74; 76; 77 Find the 70 th percentile. Find the 83 rd percentile. k = 70 i = the index n = 29 i = k 100 ( n + 1) = ( 70 100 )(29 + 1) = 21. Twenty-one is an integer, and the data value in the 21 st position in the ordered data set is 64. The 70 th percentile is 64 years. k = 83 rd percentile i = the index n = 29 i = k 100 ( n + 1) = ( 83 100 )(29 + 1) = 24.9, which is NOT an integer. Round it down to 24 and up to 25. The age in the 24 th position is 71 and the age in the 25 th position is 72. Average 71 and 72. The 83 rd percentile is 71.5 years. Try It Listed are 29 ages for Academy Award winning best actors in order from smallest to largest. 18; 21; 22; 25; 26; 27; 29; 30; 31; 33; 36; 37; 41; 42; 47; 52; 55; 57; 58; 62; 64; 67; 69; 71; 72; 73; 74; 76; 77 Calculate the 20 th percentile and the 55 th percentile. k = 20. Index = i = k 100 ( n + 1 ) = 20 100 (29 + 1) = 6. The age in the sixth position is 27. The 20 th percentile is 27 years. k = 55. Index = i = k 100 ( n + 1 ) = 55 100 (29 + 1) = 16.5. Round down to 16 and up to 17. The age in the 16 th position is 52 and the age in the 17 th position is 55. The average of 52 and 55 is 53.5. The 55 th percentile is 53.5 years. NOTE You can calculate percentiles using calculators and computers. There are a variety of online calculators. Using : Find the 80 th percentile. Find the 90 th percentile. Find the first quartile. What is another name for the first quartile? Using the data from the frequency table, we have: Notice there are 50 data values in the table, so n = 50. Calculate the index i as follows: i = 80 100 ( 50 + 1 ) = 40 . 8 Since i = 40.8, calculate the mean of the 40 th and 41 st data values. The 40 th data value is 8, the 41 st data value is 9, and the mean of these two data values is 8.5. Thus, the 80 th percentile is 8.5. Calculate the index i as follows: i = 90 100 ( 50 + 1 ) = 45 . 9 Since i = 45.9, calculate the mean of the 45 th and 46 th data values. The 45 th data value is 9, the 46 th data value is 9, and the mean of these two data values is 9. Thus, the 90 th percentile is 9. Another name for the first quartile is the 25 th percentile. Proceed to calculate the 25 th percentile: Calculate the index i as follows: i = 25 100 ( 50 + 1 ) = 12 . 75 Since i = 12.75, calculate the mean of the 12 th and 13 th data values. The 12 th data value is 6, the 13 th data value is 6, and the mean of these two data values is 6. Thus, the first quartile is 6. Try It Refer to the . Find the third quartile. What is another name for the third quartile? Your instructor or a member of the class will ask everyone in class how many sweaters they own. Answer the following questions: How many students were surveyed? What kind of sampling did you do? Construct two different histograms. For each, starting value = _____ ending value = ____. Find the median, first quartile, and third quartile. Construct a table of the data to find the following: the 10 th percentile the 70 th percentile the percent of students who own less than four sweaters A Formula for Finding the Percentile of a Value in a Data Set Order the data from smallest to largest. x = the number of data values counting from the bottom of the data list up to but not including the data value for which you want to find the percentile. y = the number of data values equal to the data value for which you want to find the percentile. n = the total number of data. Calculate x + 0.5 y n (100). Then round to the nearest integer. Listed are 29 ages for Academy Award winning best actors in order from smallest to largest. 18; 21; 22; 25; 26; 27; 29; 30; 31; 33; 36; 37; 41; 42; 47; 52; 55; 57; 58; 62; 64; 67; 69; 71; 72; 73; 74; 76; 77 Find the percentile for 58. Find the percentile for 25. Counting from the bottom of the list, there are 18 data values less than 58. There is one value of 58. x = 18 and y = 1. x + 0.5 y n (100) = 18 + 0.5 ( 1 ) 29 (100) = 63.80. 58 is the 64 th percentile. Counting from the bottom of the list, there are three data values less than 25. There is one value of 25. x = 3 and y = 1. x + 0.5 y n (100) = 3 + 0.5 ( 1 ) 29 (100) = 12.07. Twenty-five is the 12 th percentile. Try It Listed are 30 ages for Academy Award winning best actors in order from smallest to largest. 18; 21; 22; 25; 26; 27; 29; 30; 31, 31; 33; 36; 37; 41; 42; 47; 52; 55; 57; 58; 62; 64; 67; 69; 71; 72; 73; 74; 76; 77 Find the percentiles for 47 and 31. Interpreting Percentiles, Quartiles, and Median A percentile indicates the relative standing of a data value when data are sorted into numerical order from smallest to largest. Percentages of data values are less than or equal to the pth percentile. For example, 15% of data values are less than or equal to the 15 th percentile. Low percentiles always correspond to lower data values. High percentiles always correspond to higher data values. A percentile may or may not correspond to a value judgment about whether it is "good" or "bad." The interpretation of whether a certain percentile is "good" or "bad" depends on the context of the situation to which the data applies. In some situations, a low percentile would be considered "good;" in other contexts a high percentile might be considered "good". In many situations, there is no value judgment that applies. Understanding how to interpret percentiles properly is important not only when describing data, but also when calculating probabilities in later chapters of this text. NOTE When writing the interpretation of a percentile in the context of the given data, the sentence should contain the following information. information about the context of the situation being considered the data value (value of the variable) that represents the percentile the percent of individuals or items with data values below the percentile the percent of individuals or items with data values above the percentile. On a timed math test, the first quartile for time it took to finish the exam was 35 minutes. Interpret the first quartile in the context of this situation. Twenty-five percent of students finished the exam in 35 minutes or less. Seventy-five percent of students finished the exam in 35 minutes or more. A low percentile could be considered good, as finishing more quickly on a timed exam is desirable. (If you take too long, you might not be able to finish.) Try It For the 100-meter dash, the third quartile for times for finishing the race was 11.5 seconds. Interpret the third quartile in the context of the situation. On a 20 question math test, the 70 th percentile for number of correct answers was 16. Interpret the 70 th percentile in the context of this situation. Seventy percent of students answered 16 or fewer questions correctly. Thirty percent of students answered 16 or more questions correctly. A higher percentile could be considered good, as answering more questions correctly is desirable. Try It On a 60 point written assignment, the 80 th percentile for the number of points earned was 49. Interpret the 80 th percentile in the context of this situation. Eighty percent of students earned 49 points or fewer. Twenty percent of students earned 49 or more points. A higher percentile is good because getting more points on an assignment is desirable. At a community college, it was found that the 30 th percentile of credit units that students are enrolled for is seven units. Interpret the 30 th percentile in the context of this situation. Thirty percent of students are enrolled in seven or fewer credit units. Seventy percent of students are enrolled in seven or more credit units. In this example, there is no "good" or "bad" value judgment associated with a higher or lower percentile. Students attend community college for varied reasons and needs, and their course load varies according to their needs. Try It During a season, the 40 th percentile for points scored per player in a game is eight. Interpret the 40 th percentile in the context of this situation. Sharpe Middle School is applying for a grant that will be used to add fitness equipment to the gym. The principal surveyed 15 anonymous students to determine how many minutes a day the students spend exercising. The results from the 15 anonymous students are shown. 0 minutes; 40 minutes; 60 minutes; 30 minutes; 60 minutes 10 minutes; 45 minutes; 30 minutes; 300 minutes; 90 minutes; 30 minutes; 120 minutes; 60 minutes; 0 minutes; 20 minutes Determine the following five values. Min = 0 Q 1 = 20 Med = 40 Q 3 = 60 Max = 300 If you were the principal, would you be justified in purchasing new fitness equipment? Since 75% of the students exercise for 60 minutes or less daily, and since the IQR is 40 minutes (60 – 20 = 40), we know that half of the students surveyed exercise between 20 minutes and 60 minutes daily. This seems a reasonable amount of time spent exercising, so the principal would be justified in purchasing the new equipment. However, the principal needs to be careful. The value 300 appears to be a potential outlier. Q 3 + 1.5( IQR ) = 60 + (1.5)(40) = 120. The value 300 is greater than 120 so it is a potential outlier. If we delete it and calculate the five values, we get the following values: Min = 0 Q 1 = 20 Q 3 = 60 Max = 120 We still have 75% of the students exercising for 60 minutes or less daily and half of the students exercising between 20 and 60 minutes a day. However, 15 students is a small sample and the principal should survey more students to be sure of his survey results. Try It A college statistics instructor is investigating the amount of time students spend working on a final project in the course. The instructor would like students to spend approximately 3 to 4 hours as the typical amount of time to be spent on the project. The instructor collects data from a random sample of 10 students for the number of hours spent working on the final project. The results obtained are as follows: 2 hours; 3 hours; 5 hours; 3 hours; 4 hours; 4 hours; 3 hours; 11 hours; 3 hours; 2 hours. Determine the following five values: Min , Q 1 , Med , Q 3 , Max . Should the instructor modify the final project or leave as is? The five values are obtained as follows: Min = 2 Q 1 = 3 Med = 3 Q 3 = 4 Max = 11 Given that the 3 rd quartile is 4 hours, this indicates that 75% of the students spend 4 hours or less on the project. Also for the interquartile range: IQR = 4 – 3 = 1, we know that half the students spent between 3 and 4 hours on the project. This seems like a reasonable amount of time for students to spend on the final project. However, the value 11 hours appears to be a potential outlier. Q 3 + 1 . 5 ( I Q R ) = 4 + 1 . 5 ( 1 ) = 5 . 5 The value 11 is greater than 5.5, so it is an outlier. If we delete it and recalculate the five values, we get the following values: Min = 2 Q 1 = 2.5 Q 3 = 4 Max = 5 We still have 75% of students spending 4 hours or less on the project. Half of the students spend between 2.5 hours and 4 hours on the project. However, 10 students may be a relatively small sample size. The instructor might consider a larger sample size. References Cauchon, Dennis, Paul Overberg. “Census data shows minorities now a majority of U.S. births.” USA Today, 2012. Available online at http://usatoday30.usatoday.com/news/nation/story/2012-05-17/minority-birthscensus/55029100/1 (accessed April 3, 2013). Data from the United States Department of Commerce: United States Census Bureau. Available online at http://www.census.gov/ (accessed April 3, 2013). “1990 Census.” United States Department of Commerce: United States Census Bureau. Available online at http://www.census.gov/main/www/cen1990.html (accessed April 3, 2013). Data from San Jose Mercury News . Data from Time Magazine ; survey by Yankelovich Partners, Inc. Chapter Review The values that divide a rank-ordered set of data into 100 equal parts are called percentiles. Percentiles are used to compare and interpret data. For example, an observation at the 50 th percentile would be greater than 50 percent of the other observations in the set. Quartiles divide data into quarters. The first quartile ( Q 1 ) is the 25 th percentile,the second quartile ( Q 2 or median) is 50 th percentile, and the third quartile ( Q 3 ) is the 75 th percentile. The interquartile range, or IQR , is the range of the middle 50 percent of the data values. The IQR is found by subtracting Q 1 from Q 3 , and can help determine outliers by using the following two expressions. Q 3 + IQR (1.5) Q 1 – IQR (1.5) Formula Review i = ( k 100 ) ( n + 1 ) where i = the ranking or position of a data value, k = the kth percentile, n = total number of data. Expression for finding the percentile of a data value: ( x + 0.5 y n ) (100) where x = the number of values counting from the bottom of the data list up to but not including the data value for which you want to find the percentile, y = the number of data values equal to the data value for which you want to find the percentile, n = total number of data Listed are 29 ages for Academy Award winning best actors in order from smallest to largest. 18; 21; 22; 25; 26; 27; 29; 30; 31; 33; 36; 37; 41; 42; 47; 52; 55; 57; 58; 62; 64; 67; 69; 71; 72; 73; 74; 76; 77 Find the 40 th percentile. Find the 78 th percentile. The 40 th percentile is 37 years. The 78 th percentile is 70 years. Listed are 32 ages for Academy Award winning best actors in order from smallest to largest. 18; 18; 21; 22; 25; 26; 27; 29; 30; 31; 31; 33; 36; 37; 37; 41; 42; 47; 52; 55; 57; 58; 62; 64; 67; 69; 71; 72; 73; 74; 76; 77 Find the percentile of 37. Find the percentile of 72. Jesse was ranked 37 th in his graduating class of 180 students. At what percentile is Jesse’s ranking? Jesse graduated 37 th out of a class of 180 students. There are 180 – 37 = 143 students ranked below Jesse. There is one rank of 37. x = 143 and y = 1. x + 0.5 y n (100) = 143 + 0.5 ( 1 ) 180 (100) = 79.72. Jesse’s rank of 37 puts him at the 80 th percentile. For runners in a race, a low time means a faster run. The winners in a race have the shortest running times. Is it more desirable to have a finish time with a high or a low percentile when running a race? The 20 th percentile of run times in a particular race is 5.2 minutes. Write a sentence interpreting the 20 th percentile in the context of the situation. A bicyclist in the 90 th percentile of a bicycle race completed the race in 1 hour and 12 minutes. Is he among the fastest or slowest cyclists in the race? Write a sentence interpreting the 90 th percentile in the context of the situation. For runners in a race, a higher speed means a faster run. Is it more desirable to have a speed with a high or a low percentile when running a race? The 40 th percentile of speeds in a particular race is 7.5 miles per hour. Write a sentence interpreting the 40 th percentile in the context of the situation. For runners in a race it is more desirable to have a high percentile for speed. A high percentile means a higher speed which is faster. 40% of runners ran at speeds of 7.5 miles per hour or less (slower). 60% of runners ran at speeds of 7.5 miles per hour or more (faster). On an exam, would it be more desirable to earn a grade with a high or low percentile? Explain. Mina is waiting in line at the Department of Motor Vehicles (DMV). Her wait time of 32 minutes is the 85 th percentile of wait times. Is that good or bad? Write a sentence interpreting the 85 th percentile in the context of this situation. When waiting in line at the DMV, the 85 th percentile would be a long wait time compared to the other people waiting. 85% of people had shorter wait times than Mina. In this context, Mina would prefer a wait time corresponding to a lower percentile. 85% of people at the DMV waited 32 minutes or less. 15% of people at the DMV waited 32 minutes or longer. In a survey collecting data about the salaries earned by recent college graduates, Li found that her salary was in the 78 th percentile. Should Li be pleased or upset by this result? Explain. In a study collecting data about the repair costs of damage to automobiles in a certain type of crash tests, a certain model of car had $1,700 in damage and was in the 90 th percentile. Should the manufacturer and the consumer be pleased or upset by this result? Explain and write a sentence that interprets the 90 th percentile in the context of this problem. The manufacturer and the consumer would be upset. This is a large repair cost for the damages, compared to the other cars in the sample. INTERPRETATION: 90% of the crash tested cars had damage repair costs of $1700 or less; only 10% had damage repair costs of $1700 or more. The University of Wisconsin has two criteria used to set admission standards for students to be admitted to a college in the UW system: Students' GPAs and scores on standardized tests (SATs and ACTs) are entered into a formula that calculates an "admissions index" score. The admissions index score is used to set eligibility standards intended to meet the goal of admitting the top 12% of high school students in the state. In this context, what percentile does the top 12% represent? Students whose GPAs are at or above the 96 th percentile of all students at their high school are eligible (called eligible in the local context), even if they are not in the top 12% of all students in the state. What percentage of students from each high school are "eligible in the local context"? Suppose that you are buying a house. You and your realtor have determined that the most expensive house you can afford is the 34 th percentile. The 34 th percentile of housing prices is $240,000 in the town you want to move to. In this town, can you afford 34% of the houses or 66% of the houses? You can afford 34% of houses. 66% of the houses are too expensive for your budget. INTERPRETATION: 34% of houses cost $240,000 or less. 66% of houses cost $240,000 or more. Use the following information to answer the next six exercises. Sixty-five randomly selected car salespersons were asked the number of cars they generally sell in one week. Fourteen people answered that they generally sell three cars; nineteen generally sell four cars; twelve generally sell five cars; nine generally sell six cars; eleven generally sell seven cars. First quartile = _______ Second quartile = median = 50 th percentile = _______ 4 Third quartile = _______ Interquartile range ( IQR ) = _____ – _____ = _____ 6 – 4 = 2 10 th percentile = _______ 70 th percentile = _______ 6 Homework The median age for Black people in the United States currently is 30.9 years; for White people in the United States it is 42.3 years. Based upon this information, give two reasons why the median age for Black people could be lower than the median age for White people. Does the lower median age for Black people necessarily mean that Black people die younger than White people? Why or why not? How might it be possible for Black people and White people to die at approximately the same age, but for the median age for White people to be higher? Six hundred adult Americans were asked by telephone poll, "What do you think constitutes a middle-class income?" The results are in . Also, include left endpoint, but not the right endpoint. Salary ($) Relative frequency < 20,000 0.02 20,000–25,000 0.09 25,000–30,000 0.19 30,000–40,000 0.26 40,000–50,000 0.18 50,000–75,000 0.17 75,000–99,999 0.02 100,000+ 0.01 What percentage of the survey answered "not sure"? What percentage think that middle-class is from $25,000 to $50,000? Construct a histogram of the data. Should all bars have the same width, based on the data? Why or why not? How should the <20,000 and the 100,000+ intervals be handled? Why? Find the 40 th and 80 th percentiles Construct a bar graph of the data 1 – (0.02+0.09+0.19+0.26+0.18+0.17+0.02+0.01) = 0.06 0.19+0.26+0.18 = 0.63 Answers may vary. 40 th percentile will fall between 30,000 and 40,000 80 th percentile will fall between 50,000 and 75,000 Answers may vary. Interquartile Range or IQR , is the range of the middle 50 percent of the data values; the IQR is found by subtracting the first quartile from the third quartile. Outlier an observation that does not fit the rest of the data Percentile a number that divides ordered data into hundredths; percentiles may or may not be part of the data. The median of the data is the second quartile and the 50 th percentile. The first and third quartiles are the 25 th and the 75 th percentiles, respectively. Quartiles the numbers that separate the data into quarters; quartiles may or may not be part of the data. The second quartile is the median of the data.

Math


Measures of the Center of the Data The "center" of a data set is also a way of describing location. The two most widely used measures of the "center" of the data are the mean (average) and the median . To calculate the mean weight of 50 people, add the 50 weights together and divide by 50. Technically this is the arithmetic mean. We will discuss the geometric mean later. To find the median weight of the 50 people, order the data and find the number that splits the data into two equal parts meaning an equal number of observations on each side. The weight of 25 people are below this weight and 25 people are heavier than this weight. The median is generally a better measure of the center when there are extreme values or outliers because it is not affected by the precise numerical values of the outliers. The mean is the most common measure of the center. NOTE The words “mean” and “average” are often used interchangeably. The substitution of one word for the other is common practice. The technical term is “arithmetic mean” and “average” is technically a center location. Formally, the arithmetic mean is called the first moment of the distribution by mathematicians. However, in practice among non-statisticians, “average" is commonly accepted for “arithmetic mean.” When each value in the data set is not unique, the mean can be calculated by multiplying each distinct value by its frequency and then dividing the sum by the total number of data values. The letter used to represent the sample mean is an x with a bar over it (pronounced “ x bar”): x – . The Greek letter μ (pronounced "mew") represents the population mean . One of the requirements for the sample mean to be a good estimate of the population mean is for the sample taken to be truly random. To see that both ways of calculating the mean are the same, consider the sample: 1; 1; 1; 2; 2; 3; 4; 4; 4; 4; 4 x – = 1 + 1 + 1 + 2 + 2 + 3 + 4 + 4 + 4 + 4 + 4 11 = 2.7 x ¯ = 3 ( 1 ) + 2 ( 2 ) + 1 ( 3 ) + 5 ( 4 ) 11 = 2.7 In the second calculation, the frequencies are 3, 2, 1, and 5. You can quickly find the location of the median by using the expression n + 1 2 . The letter n is the total number of data values in the sample. If n is an odd number, the median is the middle value of the ordered data (ordered smallest to largest). If n is an even number, the median is equal to the two middle values added together and divided by two after the data has been ordered. For example, if the total number of data values is 97, then n + 1 2 = 97 + 1 2 = 49. The median is the 49 th value in the ordered data. If the total number of data values is 100, then n + 1 2 = 100 + 1 2 = 50.5. The median occurs midway between the 50 th and 51 st values. The location of the median and the value of the median are not the same. The upper case letter M is often used to represent the median. The next example illustrates the location of the median and the value of the median. A hospital administrator keeps track of the ages (in years) of patients visiting the emergency room over a one-week period (data are sorted from smallest to largest): 3; 4; 8; 8; 10; 11; 12; 13; 14; 15; 15; 16; 16; 17; 17; 18; 21; 22; 22; 24; 24; 25; 26; 26; 27; 27; 29; 29; 31; 32; 33; 33; 34; 34; 35; 37; 40; 44; 44; 47; Calculate the mean and the median. The calculation for the mean is: x – = [ 3 + 4 + ( 8 ) ( 2 ) + 10 + 11 + 12 + 13 + 14 + ( 15 ) ( 2 ) + ( 16 ) ( 2 ) + ... + 35 + 37 + 40 + ( 44 ) ( 2 ) + 47 ] 40 = 23.6 To find the median, M , first use the formula for the location. The location is: n + 1 2 = 40 + 1 2 = 20.5 Starting at the smallest value, the median is located between the 20 th and 21 st values (the two 24s): 3; 4; 8; 8; 10; 11; 12; 13; 14; 15; 15; 16; 16; 17; 17; 18; 21; 22; 22; 24; 24; 25; 26; 26; 27; 27; 29; 29; 31; 32; 33; 33; 34; 34; 35; 37; 40; 44; 44; 47; M = 24 + 24 2 = 24 Try It The following data show the number of months patients typically wait on a transplant list before getting surgery. The data are ordered from smallest to largest. Calculate the mean and median. 3 4 5 7 7 7 7 8 8 9 9 10 10 10 10 10 11 12 12 13 14 14 15 15 17 17 18 19 19 19 21 21 22 22 23 24 24 24 24 Suppose that in a small town of 50 people, one person earns $5,000,000 per year and the other 49 each earn $30,000. Which is the better measure of the "center": the mean or the median? x ¯ = 5 , 000 , 000 + 49 ( 30 , 000 ) 50 = 129,400 M = 30,000 (There are 49 people who earn $30,000 and one person who earns $5,000,000.) The median is a better measure of the "center" than the mean because 49 of the values are 30,000 and one is 5,000,000. The 5,000,000 is an outlier. The 30,000 gives us a better sense of the middle of the data. Try It In a sample of 60 households, one house is worth $2,500,000. Twenty-nine houses are worth $280,000, and all the others are worth $315,000. Which is the better measure of the “center”: the mean or the median? Another measure of the center is the mode. The mode is the most frequent value. There can be more than one mode in a data set as long as those values have the same frequency and that frequency is the highest. A data set with two modes is called bimodal. Statistics exam scores for 20 students are as follows: 50 53 59 59 63 63 72 72 72 72 72 76 78 81 83 84 84 84 90 93 Find the mode. The most frequent score is 72, which occurs five times. Mode = 72. Try It The number of books checked out from the library from 25 students are as follows: 0 0 0 1 2 3 3 4 4 5 5 7 7 7 7 8 8 8 9 10 10 11 11 12 12 Find the mode. Five real estate exam scores are 430, 430, 480, 480, 495. The data set is bimodal because the scores 430 and 480 each occur twice. When is the mode the best measure of the "center"? Consider a weight loss program that advertises a mean weight loss of six pounds the first week of the program. The mode might indicate that most people lose two pounds the first week, making the program less appealing. NOTE The mode can be calculated for qualitative data as well as for quantitative data. For example, if the data set is: red, red, red, green, green, yellow, purple, black, blue, the mode is red. Try It Five credit scores are 680, 680, 700, 720, 720. The data set is bimodal because the scores 680 and 720 each occur twice. Consider the annual earnings of workers at a factory. The mode is $25,000 and occurs 150 times out of 301. The median is $50,000 and the mean is $47,500. What would be the best measure of the “center”? Calculating the Arithmetic Mean of Grouped Frequency Tables When only grouped data is available, you do not know the individual data values (we only know intervals and interval frequencies); therefore, you cannot compute an exact mean for the data set. What we must do is estimate the actual mean by calculating the mean of a frequency table. A frequency table is a data representation in which grouped data is displayed along with the corresponding frequencies. To calculate the mean from a grouped frequency table we can apply the basic definition of mean: mean = data sum number of data values We simply need to modify the definition to fit within the restrictions of a frequency table. Since we do not know the individual data values we can instead find the midpoint of each interval. The midpoint is lower boundary + upper boundary 2 . We can now modify the mean definition to be Mean of Frequency Table = ∑ f m ∑ f where f = the frequency of the interval and m = the midpoint of the interval. A frequency table displaying professor Blount’s last statistic test is shown. Find the best estimate of the class mean. Grade interval Number of students 50–56.5 1 56.5–62.5 0 62.5–68.5 4 68.5–74.5 4 74.5–80.5 2 80.5–86.5 3 86.5–92.5 4 92.5–98.5 1 Find the midpoints for all intervals Grade interval Midpoint 50–56.5 53.25 56.5–62.5 59.5 62.5–68.5 65.5 68.5–74.5 71.5 74.5–80.5 77.5 80.5–86.5 83.5 86.5–92.5 89.5 92.5–98.5 95.5 Calculate the sum of the product of each interval frequency and midpoint. ∑  f m 53.25 ( 1 ) + 59.5 ( 0 ) + 65.5 ( 4 ) + 71.5 ( 4 ) + 77.5 ( 2 ) + 83.5 ( 3 ) + 89.5 ( 4 ) + 95.5 ( 1 ) = 1460.25 μ = ∑ f m ∑ f = 1460.25 19 = 76.86 Try It A researcher conducted a study on the effect that playing video games has on memory recall. As part of the study, they compiled the following data: Hours teenagers spend on video games Number of teenagers 0–3.5 3 3.5–7.5 7 7.5–11.5 12 11.5–15.5 7 15.5–19.5 9 What is the best estimate for the mean number of hours spent playing video games? Find the midpoint of each interval, multiply by the corresponding number of teenagers, add the results and then divide by the total number of teenagers The midpoints are 1.75, 5.5, 9.5, 13.5,17.5. Mean = (1.75)(3) + (5.5)(7) + (9.5)(12) + (13.5)(7) + (17.5)(9) = 409.75/38 = 10.78 References Data from The World Bank, available online at http://www.worldbank.org (accessed April 3, 2013). “Demographics: Obesity – adult prevalence rate.” Indexmundi. Available online at http://www.indexmundi.com/g/r.aspx?t=50&v=2228&l=en (accessed April 3, 2013). Chapter Review The mean and the median can be calculated to help you find the "center" of a data set. The mean is the best estimate for the actual data set, but the median is the best measurement when a data set contains several outliers or extreme values. The mode will tell you the most frequently occurring datum (or data) in your data set. The mean, median, and mode are extremely helpful when you need to analyze your data, but if your data set consists of ranges which lack specific values, the mean may seem impossible to calculate. However, the mean can be approximated if you add the lower boundary with the upper boundary and divide by two to find the midpoint of each interval. Multiply each midpoint by the number of values found in the corresponding range. Divide the sum of these values by the total number of data values in the set. Formula Review μ = ∑ f m ∑ f Where f = interval frequencies and m = interval midpoints. The arithmetic mean for a sample (denoted by x ¯ ) is x ¯ = Sum of all values in the sample Number of values in the sample The arithmetic mean for a population (denoted by μ ) is μ = Sum of all values in the population Number of values in the population Find the mean for the following frequency tables. Grade Frequency 49.5–59.5 2 59.5–69.5 3 69.5–79.5 8 79.5–89.5 12 89.5–99.5 5 Daily low temperature Frequency 49.5–59.5 53 59.5–69.5 32 69.5–79.5 15 79.5–89.5 1 89.5–99.5 0 Points per game Frequency 49.5–59.5 14 59.5–69.5 32 69.5–79.5 15 79.5–89.5 23 89.5–99.5 2 Use the following information to answer the next three exercises: The following data show the lengths of boats moored in a marina. The data are ordered from smallest to largest: 16 17 19 20 20 21 23 24 25 25 25 26 26 27 27 27 28 29 30 32 33 33 34 35 37 39 40 Calculate the mean. Mean: 16 + 17 + 19 + 20 + 20 + 21 + 23 + 24 + 25 + 25 + 25 + 26 + 26 + 27 + 27 + 27 + 28 + 29 + 30 + 32 + 33 + 33 + 34 + 35 + 37 + 39 + 40 = 738; 738 27 = 27.33 Identify the median. Identify the mode. The most frequent lengths are 25 and 27, which occur three times. Mode = 25, 27 Use the following information to answer the next three exercises: Sixty-five randomly selected car salespersons were asked the number of cars they generally sell in one week. Fourteen people answered that they generally sell three cars; nineteen generally sell four cars; twelve generally sell five cars; nine generally sell six cars; eleven generally sell seven cars. Calculate the following: sample mean = x ¯ = _______ median = _______ 4 mode = _______ Homework The countries with the highest rates of obesity in the world have obesity rates that range from 11.4% to 74.6%. This data is summarized in the following table. Percent of population Number of countries 11.4–20.45 29 20.45–29.45 13 29.45–38.45 4 38.45–47.45 0 47.45–56.45 2 56.45–65.45 1 65.45–74.45 0 74.45–83.45 1 What is the best estimate of the average obesity percentage for these countries? The United States has an average obesity rate of 33.9%. Is this rate above average or below? How does the United States compare to other countries? gives the percent of children under five considered to be underweight. What is the best estimate for the mean percentage of underweight children? Percent of underweight children Number of countries 16–21.45 23 21.45–26.9 4 26.9–32.35 9 32.35–37.8 7 37.8–43.25 6 43.25–48.7 1 The mean percentage, x – = 1328.65 50 = 26.57 Bringing It Together Javier and Ercilia are supervisors at a shopping mall. Each was given the task of estimating the mean distance that shoppers live from the mall. They each randomly surveyed 100 shoppers. The samples yielded the following information. Javier Ercilia x ¯ 6.0 miles 6.0 miles s 4.0 miles 7.0 miles How can you determine which survey was correct ? Explain what the difference in the results of the surveys implies about the data. If the two histograms depict the distribution of values for each supervisor, which one depicts Ercilia's sample? How do you know? Use the following information to answer the next three exercises : We are interested in the number of years students in a particular elementary statistics class have lived in California. The information in the following table is from the entire section. Number of years Frequency Number of years Frequency 7 1 22 1 14 3 23 1 15 1 26 1 18 1 40 2 19 4 42 2 20 3 Total = 20 What is the IQR ? 8 11 15 35 a What is the mode? 19 19.5 14 and 20 22.65 Is this a sample or the entire population? sample entire population neither b Frequency Table a data representation in which grouped data is displayed along with the corresponding frequencies Mean (arithmetic) a number that measures the central tendency of the data; a common name for mean is 'average.' The term 'mean' is a shortened form of 'arithmetic mean.' By definition, the mean for a sample (denoted by x ¯ ) is x ¯ = Sum of all values in the sample Number of values in the sample , and the mean for a population (denoted by μ ) is μ = Sum of all values in the population Number of values in the population . Mean (geometric) a measure of central tendency that provides a measure of average geometric growth over multiple time periods. Median a number that separates ordered data into halves; half the values are the same number or smaller than the median and half the values are the same number or larger than the median. The median may or may not be part of the data. Midpoint the mean of an interval in a frequency table Mode the value that appears most frequently in a set of data

Math


Sigma Notation and Calculating the Arithmetic Mean Formula for Population Mean μ = 1 N ∑ i = 1 N x i Formula for Sample Mean x – = 1 n ∑ i = 1 n x i This unit is here to remind you of material that you once studied and said at the time “I am sure that I will never need this!” Here are the formulas for a population mean and the sample mean. The Greek letter μ is the symbol for the population mean and x ¯ is the symbol for the sample mean. Both formulas have a mathematical symbol that tells us how to make the calculations. It is called Sigma notation because the symbol is the Greek capital letter sigma: Σ. Like all mathematical symbols it tells us what to do: just as the plus sign tells us to add and the x tells us to multiply. These are called mathematical operators. The Σ symbol tells us to add a specific list of numbers. Let’s say we have a sample of animals from the local animal shelter and we are interested in their average age. If we list each value, or observation, in a column, you can give each one an index number. The first number will be number 1 and the second number 2 and so on. Animal Age 1 9 2 1 3 8.5 4 10.5 5 10 6 8.5 7 12 8 8 9 1 10 9.5 Each observation represents a particular animal in the sample. Purr is animal number one and is a 9 year old cat, Toto is animal number 2 and is a 1 year old puppy and so on. To calculate the mean we are told by the formula to add up all these numbers, ages in this case, and then divide the sum by 10, the total number of animals in the sample. Animal number one, the cat Purr, is designated as X 1 , animal number 2, Toto, is designated as X 2 and so on through Dundee who is animal number 10 and is designated as X 10 . The i in the formula tells us which of the observations to add together. In this case it is X 1 through X 10 which is all of them. We know which ones to add by the indexing notation, the i = 1 and the n or capital N for the population. For this example the indexing notation would be i = 1 and because it is a sample we use a small n on the top of the Σ which would be 10. The standard deviation requires the same mathematical operator and so it would be helpful to recall this knowledge from your past. The sum of the ages is found to be 78 and dividing by 10 gives us the sample mean age as 7.8 years. A group of 10 children are on a scavenger hunt to find different color rocks. The results are shown in the below. The column on the right shows the number of colors of rocks each child has. What is the mean number of rocks? Child Rock colors 1 5 2 5 3 6 4 2 5 4 6 3 7 7 8 2 9 1 10 10 A group of children are measured to determine the average height of the group. The results are in below. What is the mean height of the group to the nearest hundredth of an inch? Child Height in inches Adam 45.21 Betina 39.45 Chen 43.78 Donna 48.76 Edhas 37.39 Fran 39.90 George 45.56 Heather 46.24 39.48 in. A person compares prices for five automobiles. The results are in . What is the mean price of the cars the person has considered? Price $20,987 $22,008 $19,998 $23,433 $21,444 $21,574 A customer protection service has obtained 8 bags of candy that are supposed to contain 16 ounces of candy each. The candy is weighed to determine if the average weight is at least the claimed 16 ounces. The results are in given in . What is the mean weight of a bag of candy in the sample? Weight in ounces 15.65 16.09 16.01 15.99 16.02 16.00 15.98 16.08 15.98 ounces A teacher records grades for a class of 70, 72, 79, 81, 82, 82, 83, 90, and 95. What is the mean of these grades? 81.56 A family is polled to see the mean of the number of hours per day the television set is on. The results, starting with Sunday, are 6, 3, 2, 3, 1, 3, and 7 hours. What is the average number of hours the family had the television set on to the nearest whole number? 4 hours A city received the following rainfall for a recent year. What is the mean number of inches of rainfall the city received monthly, to the nearest hundredth of an inch? Use . Month Rainfall in inches January 2.21 February 3.12 March 4.11 April 2.09 May 0.99 June 1.08 July 2.99 August 0.08 September 0.52 October 1.89 November 2.00 December 3.06 2.01 inches A football team scored the following points in its first 8 games of the new season. Starting at game 1 and in order the scores are 14, 14, 24, 21, 7, 0, 38, and 28. What is the mean number of points the team scored in these eight games? 18.25 Homework A sample of 10 prices is chosen from a population of 100 similar items. The values obtained from the sample, and the values for the population, are given in and respectively. Is the mean of the sample within $1 of the population mean? What is the difference in the sample and population means? Prices of the sample $21 $23 $21 $24 $22 $22 $25 $21 $20 $24 Prices of the population Frequency $20 20 $21 35 $22 15 $23 10 $24 18 $25 2 Yes The sample is 0.5 higher. A standardized test is given to ten people at the beginning of the school year with the results given in below. At the end of the year the same people were again tested. What is the average improvement? Does it matter if the means are subtracted, or if the individual values are subtracted? Student Beginning score Ending score 1 1100 1120 2 980 1030 3 1200 1208 4 998 1000 5 893 948 6 1015 1030 7 1217 1224 8 1232 1245 9 967 988 10 988 997 20 No A small class of 7 students has a mean grade of 82 on a test. If six of the grades are 80, 82,86, 90, 90, and 95, what is the other grade? 51 A class of 20 students has a mean grade of 80 on a test. Nineteen of the students has a mean grade between 79 and 82, inclusive. What is the lowest possible grade of the other student? What is the highest possible grade of the other student? 42 99 If the mean of 20 prices is $10.39, and 5 of the items with a mean of $10.99 are sampled, what is the mean of the other 15 prices? $10.19

Math