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 Q 1 , Q 2 , and Q 3 . About one-fourth of the data falls on or below the first quartile Q 1 . About one-half of the data falls on or below the second quartile Q 2 . About three-fourths of the data falls on or below the third quartile Q 3 . 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 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 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. 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. 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. 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. 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 $240,000 or more. Use the following information to answer the next six exercises. Sixty-five randomly selected car salespersons were asked the number of cars they generally sell in one week. Fourteen people answered that they generally sell three cars; nineteen generally sell four cars; twelve generally sell five cars; nine generally sell six cars; eleven generally sell seven cars. First quartile = _______ Second quartile = median = 50 th percentile = _______ 4 Third quartile = _______ Interquartile range ( IQR ) = _____ – _____ = _____ 6 – 4 = 2 10 th percentile = _______ 70 th percentile = _______ 6 Homework The median age for Black people in the U.S. is 30.9 years; for U.S. White people it is 42.3 years. Based upon this information, give two reasons why the median age for Black people could be lower than the median age for White people. Does the lower median age for Black people necessarily mean that Black people die younger than White people? Why or why not? How might it be possible for Black people and White people to die at approximately the same age, but for the median age for White people to be higher? Six hundred adult Americans were asked by telephone poll, "What do you think constitutes a middle-class income?" The results are in . Also, include left endpoint, but not the right endpoint. Salary ($) 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. Given the following box plot: which quarter has the smallest spread of data? What is that spread? which quarter has the largest spread of data? What is that spread? find the interquartile range ( IQR ). are there more data in the interval 5–10 or in the interval 10–13? How do you know this? which interval has the fewest data in it? How do you know this? 0–2 2–4 10–12 12–13 need more information The following box plot shows the U.S. population for a certain year. Are there fewer or more children (age 17 and under) than senior citizens (age 65 and over)? How do you know? 12.6% are age 65 and over. Approximately what percentage of the population are working age adults (above age 17 to age 65)? more children; the left whisker shows that 25% of the population are children 17 and younger. The right whisker shows that 25% of the population are adults 50 and older, so adults 65 and over represent less than 25%. 62.4% 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


Box Plots Box plots (also called box-and-whisker plots or box-whisker plots ) give a good graphical image of the concentration of the data. They also show how far the extreme values are from most of the data. A box plot is constructed from five values: the minimum value, the first quartile, the median, the third quartile, and the maximum value. We use these values to compare how close other data values are to them. To construct a box plot, use a horizontal or vertical number line and a rectangular box. The smallest and largest data values label the endpoints of the axis. The first quartile marks one end of the box and the third quartile marks the other end of the box. Approximately the middle 50 percent of the data fall inside the box. The "whiskers" extend from the ends of the box to the smallest and largest data values. The median or second quartile can be between the first and third quartiles, or it can be one, or the other, or both. The box plot gives a good, quick picture of the data. NOTE You may encounter box-and-whisker plots that have dots marking outlier values. In those cases, the whiskers are not extending to the minimum and maximum values. Consider, again, this dataset. 1 1 2 2 4 6 6.8 7.2 8 8.3 9 10 10 11.5 The first quartile is two, the median is seven, and the third quartile is nine. The smallest value is one, and the largest value is 11.5. The following image shows the constructed box plot. NOTE See the calculator instructions on the TI web site or in the appendix. The two whiskers extend from the first quartile to the smallest value and from the third quartile to the largest value. The median is shown with a dashed line. NOTE It is important to start a box plot with a scaled number line . Otherwise the box plot may not be useful. The following data are the heights of 40 students in a statistics class. 59 60 61 62 62 63 63 64 64 64 65 65 65 65 65 65 65 65 65 66 66 67 67 68 68 69 70 70 70 70 70 71 71 72 72 73 74 74 75 77 Construct a box plot with the following properties; the calculator intructions for the minimum and maximum values as well as the quartiles follow the example. Minimum value = 59 Maximum value = 77 Q 1: First quartile = 64.5 Q 2: Second quartile or median= 66 Q 3: Third quartile = 70 Each quarter has approximately 25% of the data. The spreads of the four quarters are 64.5 – 59 = 5.5 (first quarter), 66 – 64.5 = 1.5 (second quarter), 70 – 66 = 4 (third quarter), and 77 – 70 = 7 (fourth quarter). So, the second quarter has the smallest spread and the fourth quarter has the largest spread. Range = maximum value – the minimum value = 77 – 59 = 18 Interquartile Range: IQR = Q 3 – Q 1 = 70 – 64.5 = 5.5. The interval 59–65 has more than 25% of the data so it has more data in it than the interval 66 through 70 which has 25% of the data. The middle 50% (middle half) of the data has a range of 5.5 inches. To find the minimum, maximum, and quartiles: Enter data into the list editor (Pres STAT 1:EDIT ). If you need to clear the list, arrow up to the name L1 , press CLEAR, and then arrow down. Put the data values into the list L1 . Press STAT and arrow to CALC. Press 1:1-VarStats . Enter L1 . Press ENTER. Use the down and up arrow keys to scroll. Smallest value = 59. Largest value = 77. Q 1 : First quartile = 64.5. Q 2 : Second quartile or median = 66. Q 3 : Third quartile = 70. To construct the box plot: Press 4:Plotsoff . Press ENTER. Arrow down and then use the right arrow key to go to the fifth picture, which is the box plot. Press ENTER. Arrow down to Xlist: Press 2nd 1 for L1 Arrow down to Freq: Press ALPHA . Press 1. Press Zoom. Press 9: ZoomStat . Press TRACE, and use the arrow keys to examine the box plot. Try It The following data are the number of pages in 40 books on a shelf. Construct a box plot using a graphing calculator, and state the interquartile range. 136 140 178 190 205 215 217 218 232 234 240 255 270 275 290 301 303 315 317 318 326 333 343 349 360 369 377 388 391 392 398 400 402 405 408 422 429 450 475 512 For some sets of data, some of the largest value, smallest value, first quartile, median, and third quartile may be the same. For instance, you might have a data set in which the median and the third quartile are the same. In this case, the diagram would not have a dotted line inside the box displaying the median. The right side of the box would display both the third quartile and the median. For example, if the smallest value and the first quartile were both one, the median and the third quartile were both five, and the largest value was seven, the box plot would look like: In this case, at least 25% of the values are equal to one. Twenty-five percent of the values are between one and five, inclusive. At least 25% of the values are equal to five. The top 25% of the values fall between five and seven, inclusive. 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 Find the smallest and largest values, the median, and the first and third quartile for the day class. Find the smallest and largest values, the median, and the first and third quartile for the night class. For each data set, what percentage of the data is between the smallest value and the first quartile? the first quartile and the median? the median and the third quartile? the third quartile and the largest value? What percentage of the data is between the first quartile and the largest value? Create a box plot for each set of data. Use one number line for both box plots. Which box plot has the widest spread for the middle 50% of the data (the data between the first and third quartiles)? What does this mean for that set of data in comparison to the other set of data? Min = 32 Q 1 = 56 M = 74.5 Q 3 = 82.5 Max = 99 Min = 25.5 Q 1 = 78 M = 81 Q 3 = 89 Max = 98 Day class: There are six data values ranging from 32 to 56: 30%. There are six data values ranging from 56 to 74.5: 30%. There are five data values ranging from 74.5 to 82.5: 25%. There are five data values ranging from 82.5 to 99: 25%. There are 16 data values between the first quartile, 56, and the largest value, 99: 75%. Night class: The first data set has the wider spread for the middle 50% of the data. The IQR for the first data set is greater than the IQR for the second set. This means that there is more variability in the middle 50% of the first data set. Try It The following data set shows the heights in inches for the boys in a class of 40 students. 66; 66; 67; 67; 68; 68; 68; 68; 68; 69; 69; 69; 70; 71; 72; 72; 72; 73; 73; 74 The following data set shows the heights in inches for the girls in a class of 40 students. 61; 61; 62; 62; 63; 63; 63; 65; 65; 65; 66; 66; 66; 67; 68; 68; 68; 69; 69; 69 Construct a box plot using a graphing calculator for each data set, and state which box plot has the wider spread for the middle 50% of the data. Graph a box-and-whisker plot for the data values shown. 10 10 10 15 35 75 90 95 100 175 420 490 515 515 790 The five numbers used to create a box-and-whisker plot are: Min: 10 Q 1 : 15 Med: 95 Q 3 : 490 Max: 790 The following graph shows the box-and-whisker plot. Try It Follow the steps you used to graph a box-and-whisker plot for the data values shown. 0 5 5 15 30 30 45 50 50 60 75 110 140 240 330 References Data from West Magazine . Chapter Review Box plots are a type of graph that can help visually organize data. To graph a box plot the following data points must be calculated: the minimum value, the first quartile, the median, the third quartile, and the maximum value. Once the box plot is graphed, you can display and compare distributions of data. Use the following information to answer the next two 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. Construct a box plot below. Use a ruler to measure and scale accurately. Looking at your box plot, does it appear that the data are concentrated together, spread out evenly, or concentrated in some areas, but not in others? How can you tell? More than 25% of salespersons sell four cars in a typical week. You can see this concentration in the box plot because the first quartile is equal to the median. The top 25% and the bottom 25% are spread out evenly; the whiskers have the same length. Homework In a survey of 20-year-olds in China, Germany, and the United States, people were asked the number of foreign countries they had visited in their lifetime. The following box plots display the results. In complete sentences, describe what the shape of each box plot implies about the distribution of the data collected. Have more U.S. citizens or more Germans surveyed been to over eight foreign countries? Compare the three box plots. What do they imply about the foreign travel of 20-year-old residents of the three countries when compared to each other? Given the following box plot, answer the questions. Think of an example (in words) where the data might fit into the above box plot. In 2–5 sentences, write down the example. What does it mean to have the first and second quartiles so close together, while the second to third quartiles are far apart? Answers will vary. Possible answer: State University conducted a survey to see how involved its students are in community service. The box plot shows the number of community service hours logged by participants over the past year. Because the first and second quartiles are close, the data in this quarter is very similar. There is not much variation in the values. The data in the third quarter is much more variable, or spread out. This is clear because the second quartile is so far away from the third quartile. Given the following box plots, answer the questions. In complete sentences, explain why each statement is false. Data 1 has more data values above two than Data 2 has above two. The data sets cannot have the same mode. For Data 1 , there are more data values below four than there are above four. For which group, Data 1 or Data 2, is the value of “7” more likely to be an outlier? Explain why in complete sentences. A survey was conducted of 130 purchasers of new BMW 3 series cars, 130 purchasers of new BMW 5 series cars, and 130 purchasers of new BMW 7 series cars. In it, people were asked the age they were when they purchased their car. The following box plots display the results. In complete sentences, describe what the shape of each box plot implies about the distribution of the data collected for that car series. Which group is most likely to have an outlier? Explain how you determined that. Compare the three box plots. What do they imply about the age of purchasing a BMW from the series when compared to each other? Look at the BMW 5 series. Which quarter has the smallest spread of data? What is the spread? Look at the BMW 5 series. Which quarter has the largest spread of data? What is the spread? Look at the BMW 5 series. Estimate the interquartile range (IQR). Look at the BMW 5 series. Are there more data in the interval 31 to 38 or in the interval 45 to 55? How do you know this? Look at the BMW 5 series. Which interval has the fewest data in it? How do you know this? 31–35 38–41 41–64 Each box plot is spread out more in the greater values. Each plot is skewed to the right, so the ages of the top 50% of buyers are more variable than the ages of the lower 50%. The BMW 3 series is most likely to have an outlier. It has the longest whisker. Comparing the median ages, younger people tend to buy the BMW 3 series, while older people tend to buy the BMW 7 series. However, this is not a rule, because there is so much variability in each data set. The second quarter has the smallest spread. There seems to be only a three-year difference between the first quartile and the median. The third quarter has the largest spread. There seems to be approximately a 14-year difference between the median and the third quartile. IQR ~ 17 years There is not enough information to tell. Each interval lies within a quarter, so we cannot tell exactly where the data in that quarter is concentrated. The interval from 31 to 35 years has the fewest data values. Twenty-five percent of the values fall in the interval 38 to 41, and 25% fall between 41 and 64. Since 25% of values fall between 31 and 38, we know that fewer than 25% fall between 31 and 35. Twenty-five randomly selected students were asked the number of movies they watched the previous week. The results are as follows: # of movies Frequency 0 5 1 9 2 6 3 4 4 1 Construct a box plot of the data. Bringing It Together A certain small town in the United states has a population of 27,873 people. Their ages are as follows: Age Group Percent of Community 0–17 18.9 18–24 8.0 25–34 22.8 35–44 15.0 45–54 13.1 55–64 11.9 65+ 10.3 Construct a histogram of the age distribution for this small town. The bars will not be the same width for this example. Why not? What impact does this have on the reliability of the graph? What percentage of the community is under age 35? Which box plot most resembles the information above? For graph, answers may vary. 49.7% of the community is under the age of 35. Based on the information in the table, graph (a) most closely represents the data. Box plot a graph that gives a quick picture of the middle 50% of the data First Quartile the value that is the median of the of the lower half of the ordered data set Frequency Polygon looks like a line graph but uses intervals to display ranges of large amounts of data Interval also called a class interval; an interval represents a range of data and is used when displaying large data sets Paired Data Set two data sets that have a one to one relationship so that: both data sets are the same size, and each data point in one data set is matched with exactly one point from the other set. Skewed used to describe data that is not symmetrical; when the right side of a graph looks “chopped off” compared the left side, we say it is “skewed to the left.” When the left side of the graph looks “chopped off” compared to the right side, we say the data is “skewed to the right.” Alternatively: when the lower values of the data are more spread out, we say the data are skewed to the left. When the greater values are more spread out, the data are skewed to the right.

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. To find the median weight of the 50 people, order the data and find the number that splits the data into two equal parts. 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. 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 To find the mean and the median: Clear list L1. Pres STAT 4:ClrList . Enter 2nd 1 for list L1. Press ENTER. Enter data into the list editor. Press STAT 1:EDIT . Put the data values into list L1. Press STAT and arrow to CALC. Press 1:1-VarStats. Press 2nd 1 for L1 and then ENTER. Press the down and up arrow keys to scroll. x ¯ = 23.6, M = 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. Statistical software will easily calculate the mean, the median, and the mode. Some graphing calculators can also make these calculations. In the real world, people make these calculations using software. 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”? The Law of Large Numbers and the Mean The Law of Large Numbers says that if you take samples of larger and larger size from any population, then the mean x ¯ of the sample is very likely to get closer and closer to µ . This is discussed in more detail later in the text. Sampling Distributions and Statistic of a Sampling Distribution You can think of a sampling distribution as a relative frequency distribution with a great many samples. (See Sampling and Data for a review of relative frequency). Suppose thirty randomly selected students were asked the number of movies they watched the previous week. The results are in the relative frequency table shown below. # of movies Relative Frequency 0 5 30 1 15 30 2 6 30 3 3 30 4 1 30 If you let the number of samples get very large (say, 300 million or more), the relative frequency table becomes a relative frequency distribution . A statistic is a number calculated from a sample. Statistic examples include the mean, the median and the mode as well as others. The sample mean x ¯ is an example of a statistic which estimates the population mean μ . Calculating the 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? 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. 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? If the two box plots 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 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 . 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


Skewness and the Mean, Median, and Mode Consider the following data set. 4; 5; 6; 6; 6; 7; 7; 7; 7; 7; 7; 8; 8; 8; 9; 10 This data set can be represented by following histogram. Each interval has width one, and each value is located in the middle of an interval. The histogram (shown in ) displays a symmetrical distribution of data. A distribution is symmetrical if a vertical line can be drawn at some point in the histogram such that the shape to the left and the right of the vertical line are mirror images of each other. The mean, the median, and the mode are each seven for these data. In a perfectly symmetrical distribution, the mean and the median are the same. This example has one mode (unimodal), and the mode is the same as the mean and median. In a symmetrical distribution that has two modes (bimodal), the two modes would be different from the mean and median. The histogram for the data: 4 5 6 6 6 7 7 7 7 8 (shown in ) is not symmetrical. The right-hand side seems "chopped off" compared to the left side. A distribution of this type is called skewed to the left because it is pulled out to the left. The mean is 6.3, the median is 6.5, and the mode is seven. Notice that the mean is less than the median, and they are both less than the mode. The mean and the median both reflect the skewing, but the mean reflects it more so. The histogram for the data: 6 7 7 7 7 8 8 8 9 10 , is also not symmetrical. It is skewed to the right . The mean is 7.7, the median is 7.5, and the mode is seven. Of the three statistics, the mean is the largest, while the mode is the smallest . Again, the mean reflects the skewing the most. The mean is affected by outliers that do not influence the median. Therefore, when the distribution of data is skewed to the left, the mean is often less than the median. When the distribution is skewed to the right, the mean is often greater than the median. In symmetric distributions, we expect the mean and median to be approximately equal in value. This is an important connection between the shape of the distribution and the relationship of the mean and median. It is not, however, true for every data set. The most common exceptions occur in sets of discrete data. Skewness and symmetry become important when we discuss probability distributions in later chapters. Statistics are used to compare and sometimes identify authors. The following lists shows a simple random sample that compares the letter counts for three authors. Terry: 7; 9; 3; 3; 3; 4; 1; 3; 2; 2 Davis: 3; 3; 3; 4; 1; 4; 3; 2; 3; 1 Maris: 2; 3; 4; 4; 4; 6; 6; 6; 8; 3 Make a dot plot for the three authors and compare the shapes. Calculate the mean for each. Calculate the median for each. Describe any pattern you notice between the shape and the measures of center. Terry’s distribution has a right (positive) skew. Davis’ distribution has a left (negative) skew Maris’ distribution is symmetrically shaped. Terry’s mean is 3.7, Davis’ mean is 2.7, Maris’ mean is 4.6. Terry’s median is three, Davis’ median is three. Maris’ median is four. It appears that the median is always closest to the high point (the mode), while the mean tends to be farther out on the tail. In a symmetrical distribution, the mean and the median are both centrally located close to the high point of the distribution. Try It Discuss the mean, median, and mode for each of the following problems. Is there a pattern between the shape and measure of the center? a. b. The Ages Former U.S Presidents Died 4 6 9 5 3 6 7 7 7 8 6 0 0 3 3 4 4 5 6 7 7 7 8 7 0 1 1 2 3 4 7 8 8 9 8 0 1 3 5 8 9 0 0 3 3 Key: 8|0 means 80. c. Chapter Review Looking at the distribution of data can reveal a lot about the relationship between the mean, the median, and the mode. There are three types of distributions . A left (or negative) skewed distribution has a shape like . A right (or positive) skewed distribution has a shape like . A symmetrical distribution looks like . Use the following information to answer the next three exercises: State whether the data are symmetrical, skewed to the left, or skewed to the right. 1 1 1 2 2 2 2 3 3 3 3 3 3 3 3 4 4 4 5 5 The data are symmetrical. The median is 3 and the mean is 2.85. They are close, and the mode lies close to the middle of the data, so the data are symmetrical. 16 17 19 22 22 22 22 22 23 87 87 87 87 87 88 89 89 90 91 The data are skewed right. The median is 87.5 and the mean is 88.2. Even though they are close, the mode lies to the left of the middle of the data, and there are many more instances of 87 than any other number, so the data are skewed right. When the data are skewed left, what is the typical relationship between the mean and median? When the data are symmetrical, what is the typical relationship between the mean and median? When the data are symmetrical, the mean and median are close or the same. What word describes a distribution that has two modes? Describe the shape of this distribution. The distribution is skewed right because it looks pulled out to the right. Describe the relationship between the mode and the median of this distribution. Describe the relationship between the mean and the median of this distribution. The mean is 4.1 and is slightly greater than the median, which is four. Describe the shape of this distribution. Describe the relationship between the mode and the median of this distribution. The mode and the median are the same. In this case, they are both five. Are the mean and the median the exact same in this distribution? Why or why not? Describe the shape of this distribution. The distribution is skewed left because it looks pulled out to the left. Describe the relationship between the mode and the median of this distribution. Describe the relationship between the mean and the median of this distribution. The mean and the median are both six. The mean and median for the data are the same. 3 4 5 5 6 6 6 6 7 7 7 7 7 7 7 Is the data perfectly symmetrical? Why or why not? Which is the greatest, the mean, the mode, or the median of the data set? 11 11 12 12 12 12 13 15 17 22 22 22 The mode is 12, the median is 12.5, and the mean is 15.1. The mean is the largest. Which is the least, the mean, the mode, and the median of the data set? 56 56 56 58 59 60 62 64 64 65 67 Of the three measures, which tends to reflect skewing the most, the mean, the mode, or the median? Why? The mean tends to reflect skewing the most because it is affected the most by outliers. In a perfectly symmetrical distribution, when would the mode be different from the mean and median? Homework The median age of the U.S. population in 1980 was 30.0 years. In 1991, the median age was 33.1 years. What does it mean for the median age to rise? Give two reasons why the median age could rise. For the median age to rise, is the actual number of children less in 1991 than it was in 1980? Why or why not?

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