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useful for all sorts of math problems. estigaaaaationtiontiontiontion — Transforming Functions estigestig estig pter 8 Invvvvvestig pter 8 In ChaChaChaChaChapter 8 In pter 8 In pter 8 In 435435435435435 436436436436436436436436436436 Appendixes Glossary................................................... 438 Formula Sh... |
points ccccclosed inter losed inter losed inter losed inter losure pre pre pre pre proper dition and dition and umber ad umber ad eal-n eal-n ties (of r r r r real-n ties (of ties (of oper oper losur losur ccccclosur operties (of dition and umber addition and eal-number ad dition and umber ad eal-n ties (of oper losur ... |
of tions a system of equations with equations pendent system of equa tions pendent system of equa infinitely many possible solutions discriminant discriminant discriminant for a quadratic equation ax2 + bx + c = 0, the discriminant discriminant discriminant is b2 – 4ac distrib distrib distributiutiutiutiutivvvvve pr e... |
f common factor of two or more other expressions; all other common factors will also be factors of the GCF ouping symbols gggggrrrrrouping symbols ouping symbols ouping symbols symbols that show the order in which ouping symbols mathematical operations should be carried out — such as parentheses and brackets I tion) t... |
same variables, e ter and where each variable is raised to the same power in every term and and tion in tion in linear equa linear equa and y an equation that can be written in the tion in x and linear equation in and tion in linear equa linear equa form Ax + By = C (where A and B are not both zero) M monomial monomia... |
polynomial actorizationtiontiontiontion a factorization of a number where each actoriza actoriza prime f prime f prime factoriza actoriza prime f prime f factor is a prime number umber umber prime n prime n umber a number that can only be divided by itself and 1 prime number prime n umber prime n oduct oduct prprprprp... |
multiplicative inverse of an expression eciprocal ocal ecipr rrrrrelaelaelaelaelationtiontiontiontion any set of ordered pairs (the first number or expression in each pair can be thought of as the relation’s “input,” the second as the relation’s “output”) an equationtiontiontiontion an equation’s roots are its solutio... |
numbers 0, 1, 2, 3,...; umber umber hole n hole n wwwwwhole n hole number umber hole n the set of all whole numbers is denoted W X -intercececececeptptptptpt the x-coordinate of a point where a graph meets -inter -inter x-inter -inter the x-axis Y -intercececececeptptptptpt the y-coordinate of a point where a graph me... |
ggggrrrrrouping symbols ouping symbols ouping symbols ouping symbols — working from the innermost grouping ouping symbols symbols to the outermost. Exponents Exponents Exponents. 2. Exponents Exponents visions tions and dididididivisions visions tions tions Multiplica Multiplica 3. Multiplica visions, working from left... |
c| < m means c – m < x < c + m Special Products of Binomials Point-slope form: y – y1 = m(x – x1) Slope-intercept form: y = mx + b Standard form of a linear equation: Ax + By = C Two lines with slopes m1 and m2 are: parallel if m1 = m2 perpendicular if m1 × m2 = –1 Quadratics Basic form of a quadratic equation: ax2 + ... |
x + c If (x – a)(x – b) = 0, then x = a or x = b. ⎞ ⎟⎟⎟⎟ ⎠ FFFFFororororormmmmmulasulasulasulasulas 441441441441441 Index A abscissa 164 absolute values 19, 21, 128, equations 128-132 inequalities 158-160 addends 14, 21 addition 14, 15, 21, 27, 28 of fractions 52, 53 of polynomials 266, 267, 272 property of equality 76... |
, 288, 388 common denominator 403, 406-409 dependent systems of linear equations 239, 240 difference 24 difference of two squares 298, 325, 326 discriminant 372-378 disjunctions 156, 159, 160 distance 103-106 distributive laws 31 distributive property 34, 35, 67, 68, 70, 71, 276, 281 use in combining like terms 67 use ... |
313-317, 318-320, 322, 323, 325-330 polynomials using long division 291, 292 prime factorization 53, 54, 81 quadratic expressions and equations 294, 295, 310-320, 325-328, 333-336, 338-341 third-degree polynomials 322, 323 factors 14, 22, 302, 303 common factors 288, 289, 302, 303, 305-308, 313, 314, 322, 323, 329, 33... |
omial 266 order of operations 27-29, 399, 400 ordered pairs 164, 175, 179, 420, 421, 424 ordinate 164 origin 164 P parabolas 362, 363 parallel lines 197-199, 209 parentheses 148 as grouping symbols 27 percent mixture problems 108-111, 113, 114, 116-119, 255, 256 perfect square trinomials 327, 328, 342-345, 251-253, 353... |
, 284, 286 rules of the number system 14-22, 24-35 run 189 S second-degree polynomials 318-320 sequences of integers 95-97 sets 2-6, 420-422, 424-426, 430-433 sign of a number 19 signs of coordinates in different quadrants 167 simplifying algebraic expressions 13, 67, 68, 70, 71, 73-75, 147, 148, 150-153, 263, 264, 266... |
341, 351-353, 355-357 speed 103-106 square roots 40-42 properties of 43, 44 square root method for solving quadratics 338-341 stacking method for multiplying polynomials 278, 279 W whole numbers 5 word problems 89, 90, 92, 93, 95-97, 99-101, 103-106, 108-111, 113, 114, 116-119, 121, 122, 124-126, 150-153, 249-251, 253,... |
andtwofromgroupB)andwereexcludedfromthestatisticalanalysissincetheyrequestedtheremovaloftheneedles.OnepatientfromgroupAdidnotgiveherconsenttotheimplantofthesemi-permanentneedles.IngroupA,themeannumberofFig.1Theappropriatearea(M)versustheinappropriatearea(S)usedinthetreatmentofmigraineattacksS174NeurolSci(2011)32(Suppl1... |
of the data are understood. For instance, it is always important to be sure we know what each variable means and the units of measurement. Descriptions of the loan50 variables are given in Figure 1.4. loan amount 7500 25000 14500... 3000 interest rate 7.34 9.43 6.08... 7.96 term 36 60 36... 36 grade A B A... A state M... |
represents the county data set, which is shown as a data matrix in Figure 1.5. These data come from the US Census, with much of the data coming from the US Census Bureau’s American Community Survey (ACS). Unlike the Decennial Census, which takes place every 10 years and attempts to collect basic demographic data from ... |
-negative numbers (0, 1, 2,...). For this reason, the population variable is said to be discrete since it can only take numerical values with jumps. On the other hand, the unemployment rate variable is said to be continuous. The variable state can take up to 51 values after accounting for Washington, DC: AL, AK,..., an... |
it represents a count, num migraines is a discrete numerical variable. all variablesnumericalcategoricalcontinuousdiscretenominal(unordered categorical)ordinal(ordered categorical) 20 CHAPTER 1. DATA COLLECTION 1.2.3 Relationships between variables Many analyses are motivated by a researcher looking for a relationship... |
and vice-versa. Homeownership Rate020406080100020406080100lPercent of Units in Multi−Unit Structures 1.2. DATA BASICS 21 Figure 1.9: A scatterplot showing pop change against median hh income. Owsley County of Kentucky, is highlighted, which lost 3.63% of its population from 2010 to 2017 and had median household income... |
data in a table, where the rows correspond to individuals or cases and the columns correspond to the variables, the values of which are recorded for each individual. • Variables can be numerical (measured on a numerical scale) or categorical (taking on levels, such as low/medium/high). Numerical variables can be conti... |
to 10. On average, the participants in the Buteyko group experienced a significant reduction in asthma symptoms and an improvement in quality of life.12 (a) Identify the main research question of the study. (b) Who are the subjects in this study, and how many are included? (c) What are the variables in the study? Ident... |
LECTION 1.6 Stealers, study components. In a study of the relationship between socio-economic class and unethical behavior, 129 University of California undergraduates at Berkeley were asked to identify themselves as having low or high social-class by comparing themselves to others with the most (least) money, most (le... |
from each species in the data set.15 (a) How many cases were included in the data? (b) How many numerical variables are included in the data? Indicate what they are, and if they are continuous or discrete. (c) How many categorical variables are included in the data, and what are they? List the corresponding levels (ca... |
a) List the variables used in creating this visualization. (b) Indicate whether each variable in the study is numerical or categorical. If numerical, identify as contin- uous or discrete. If categorical, indicate if the variable is ordinal. 1.12 UN Votes. The visualization below shows voting patterns the United States,... |
in patients with severe heart disease? Each research question refers to a target population. In the first question, the target population is all swordfish in the Atlantic ocean, and each fish represents a case. Often times, it is too expensive to collect data for every case in a population. Instead, a sample is taken. A ... |
�sh will be the statistic. Two statistics we will study are the mean (also called the average) and proportion. When we are discussing a population, we label the mean as µ (the Greek letter, mu), while we label the sample mean as ¯x (read as x-bar ). When we are discussing a proportion in the context of a population, we... |
swordfish, so the average mercury concentration in swordfish must be dangerously high. 2. I met two students who took more than 7 years to graduate from Duke, so it must take longer to graduate at Duke than at many other colleges. 3. My friend’s dad had a heart attack and died after they gave him a new heart disease dru... |
the two is actually causal, even if there is an association identified between the two variables. We use these labels only to keep track of which variable we suspect affects the other. In many cases, the relationship is complex or unknown. It may be unclear whether variable A explains variable B or whether variable B ex... |
into one of two groups: the first group receives a placebo (fake treatment) and the second group receives the drug. See the case study in Section 1.1 for another example of an experiment, though that study did not employ a placebo. EXAMPLE 1.14 Suppose that a researcher is interested in the average tip customers at a p... |
pollution exposure during gestation was calculated for each birth. (a) Identify the population of interest and the sample in this study. (b) Comment on whether or not the results of the study can be generalized to the population, and if the findings of the study can be used to establish causal relationships. 1.14 Cheat... |
presented with a jar of individually wrapped candies and informed that the candies were for children in a nearby laboratory, but that they could take some if they wanted. After completing some unrelated tasks, participants reported the number of candies they had taken. It was found that those who were identified as upp... |
conclude eating chocolate improves health outcomes? • How do opinion polls work? How do research organizations collect the data, and what types of bias should we look out for? Learning objectives 1. Identify possible confounding factors in a study and explain, in context, how they could con- found. 2. Distinguish amon... |
more likely to use sunscreen, and people who are out in the sun all day are more likely to get skin cancer. Research shows us the development of skin cancer is due to the sun exposure. The variables of sunscreen usage and sun exposure are confounded, and without this research, we would have no way of knowing which one... |
, there is no guarantee that all confounding variables can be examined or measured. In the same way, the county data set is an observational study with confounding variables, and its data cannot be used to make causal conclusions. sun exposureuse sunscreenskin cancer? 1.4. OBSERVATIONAL STUDIES AND SAMPLING STRATEGIES ... |
to be included in the sample. Why pick a sample randomly? Why not just pick a sample by hand? Consider the following scenario. 24Answers will vary. Population density may be important. If a county is very dense, then this may require a larger fraction of residents to live in multi-unit structures. Additionally, the hi... |
population has an equal chance of being included and there is no implied connection between the cases in the sample. A common downfall is a convenience sample, where individuals who are easily accessible are more likely to be included in the sample. For instance, if a political survey is done by stopping people walkin... |
response bias is often subtle, one must pay careful attention to how questions were asked when attempting to draw conclusions from the data. EXAMPLE 1.21 Suppose a high school student wants to investigate the student body’s opinions on the food in the cafeteria. Let’s assume that she manages to survey every student in... |
last section. Here we provide a more technical treatment of this method and introduce four new random sampling methods: systematic, stratified, cluster, and multistage.27 Figure 1.14 provides a graphical representation of simple versus systematic sampling while Figure 1.15 provides a graphical representation of stratifi... |
sample. Such a sample would be impossible from a systematic sample. Property 2 of simple random samples does not hold for other types of random samples. 26The researcher should be concerned about non-response bias, and sampling more people will not eliminate this issue. The same type of people that did not respond to ... |
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● 40 CHAPTER 1. DATA COLLECTION Sometimes there is a variable that is known to be associated with the quantity we want to estimate. In this case, a stratified random sample might be selected. Stratified sampling is a divide-a... |
each team. Next, let’s consider a sampling technique that randomly selects groups of people. Cluster sampling is much like simple random sampling, but instead of randomly selecting individuals, we randomly select groups or clusters. Unlike stratified sampling, cluster sampling is most helpful when there is a lot of cas... |
In the top panel, stratified sampling was used: cases were grouped into strata, and then simple random sampling was employed within each stratum. In the middle panel, cluster sampling was used, where data were binned into nine cluster and three clusters were randomly selected. In the bottom panel, multistage sampling w... |
employed? Describe how to collect such a sample to get a sample size of 60. Because grade level affects the likelihood of having a part-time job, we should take a stratified random sample. To do this, we can take a simple random sample of 15 students from each grade. This will give us equal representation from each grad... |
and volunteer sam- ples, which likely introduce bias. Instead, use a random sampling method. • Generalizations from a sample can be made to a population only if the sample is random. Furthermore, the generalization can be made only to the population from which the sample was randomly selected, not to a larger or differ... |
fferent teaching assistants. The professor wants to conduct a survey about how satisfied the students are with the course, and he believes that the lab section a student is in might affect the student’s overall satisfaction with the course. (a) What type of study is this? (b) Suggest a sampling strategy for carrying out t... |
old in one cluster, 19 years old in one cluster, etc.), then randomly sample three clusters and survey all students in those clusters. 1.24 Random digit dialing. The Gallup Poll uses a procedure called random digit dialing, which creates phone numbers based on a list of all area codes in America in conjunction with th... |
ask them what their family size is, will this be a good measure of household size? Or will our average be biased? If so, will it overestimate or underestimate the true value? 1.27 Sampling strategies. A statistics student who is curious about the relationship between the amount of time students spend on social network... |
have shown symptoms of sleep disorders.” A friend of yours who read the article says, “The study shows that sleep disorders lead to bullying in school children.” Is this statement justified? If not, how best can you describe the conclusion that can be drawn from this study? 30Justin Hepler and Dolores Albarrac´ın. “Att... |
levels for each factor and the total number of treatments. 1.5.1 Reducing bias in human experiments In the last section we investigated observational studies and sampling strategies. While these are effective tools for answering certain research questions, often times researchers want to measure the effect of a treatmen... |
want patients to know which group they are in. When researchers keep the patients uninformed about their treatment, the study is said to be blind or single-blind. But there is one problem: if a patient doesn’t receive a treatment, she will know she is in the control group. The solution to this problem is to give fake ... |
or make the same. A researcher can directly control the appearance of the treatment, the time of day it is taken, etc. She cannot directly control variables such as gender or age. To control for these other types of variables, she might consider blocking, which is described in Section 1.5.3. Randomization. Researchers... |
treatments, one of which may be a placebo, and 300 subjects. To carry out a completely randomized design, we could randomly assign each subject a unique number from 1 to 300, then subjects with numbers 1-100 would get treatment 1, subjects 101-200 would get treatment 2, and subjects 201- 300 would get treatment 3. Not... |
to influence the response variable whether or not a smoker will quit. A reasonable variable, therefore, would be the number of years that the smoker has been smoking. The subjects could be separated into three blocks based on number of years of smoking and each block randomly divided into the two treatment groups. 1.5.... |
variables that affect the response can be distributed unevenly among the treatment groups, thus biasing the experiment in one direction. In a matched pairs A third type of design, known as matched pairs addresses this problem. experiment, pairs of people are matched on as many variables as possible, so that the compari... |
soft punk, loud dance, loud classical, loud punk. Each combination is a treatment. Therefore, this experiment will have 2 factors and 6 treatments. To replicate each treatment 10 times, one would need to play the game 60 times. GUIDED PRACTICE 1.32 A researcher wants to compare the effectiveness of four different drugs.... |
that the treatment groups differ on some characteristic due to chance alone (i.e. in spite of the randomization). • An ideal experiment is randomized, controlled, and double-blind. • A completely randomized experiment involves randomly assigning the subjects to the different treatment groups. To do this, first number the... |
aff and students at a university. A quarter of the patients were assigned a placebo, and the rest were evenly divided between 1g Vitamin C, 3g Vitamin C, or 3g Vitamin C plus additives to be taken at onset of a cold for the following two days. All tablets had identical appearance and packaging. The nurses who handed the... |
. A researcher is interested in the effects of exercise on mental health and he proposes the following study: Use stratified random sampling to ensure representative proportions of 18-30, 31-40 and 41- 55 year olds from the population. Next, randomly assign half the subjects from each age group to exercise twice a week, ... |
is used in an experiment to separate subjects into blocks and then compare responses within those blocks. All subjects in a block are used in the experiment, not just a sample of them. • Random sampling vs Random assignment. Random sampling refers to sampling a subset of a population for the purpose of inference to th... |
control group. One group was given 25 grams of chia seeds twice a day, and the other was given a placebo. The subjects volunteered to be a part of the study. After 12 weeks, the scientists found no significant difference between the groups in appetite or weight loss.40 (a) What type of study is this? (b) What are the ex... |
85% said “no”. Based on these results, the school officials conclude that a great majority of the parents have no difficulty spending time with their kids after school. (b) A survey is conducted on a simple random sample of 1,000 women who recently gave birth, asking them about whether or not they smoked during pregnancy.... |
A newspaper article reporting on the study states, “The results of this study provide proof that giving young adults fresh fruits and vegetables to eat can have psychological benefits, even over a brief period of time.” How would you suggest revising this statement so that it can be supported by the study? 41Tamlin S C... |
Driver’s race Black Hispanic White Black Hispanic White · · · No. of stops per year 266 1008 6322 1169 9453 10826 · · · 16 27 1157 % of stopped cars searched 0.08 0.05 0.02 0.05 0.04 0.02 · · · 0.24 0.04 0.03 drivers arrested 0.02 0.02 0.01 0.01 0.01 0.01 · · · 0.10 0.03 0.03 (a) What variables were collected on each ... |
2.2 Numerical summaries and box plots 2.3 Normal distribution 2.4 Considering categorical data 2.5 Case study: malaria vaccine (special topic) 58 After collecting data, the next stage in the investigative process is to describe and summarize the data. In this chapter, we will look at ways to summarize numerical and ca... |
shown in Figure 2.1, illustrating the relationship between the number of line breaks (line breaks) and number of characters (num char) in emails for the email50 data set. In any scatterplot, each point represents a single case. Since there are 50 cases in email50, there are 50 points in Figure 2.1. EXAMPLE 2.1 A scatt... |
the trend. There is also a negative association, as higher rates of poverty tend to be associated with lower median household income. GUIDED PRACTICE 2.4 What do scatterplots reveal about the data, and how are they useful?1 1Answers may vary. Scatterplots are helpful in quickly spotting associations relating variables... |
the vertical axis and water consumption on the horizontal axis, then we would create a ∩ shape. 0%10%20%30%40%50%$0$20k$40k$60k$80k$100k$120kPoverty Rate (Percent)Median Household Income 62 CHAPTER 2. SUMMARIZING DATA Rather than look at the data as a list of numbers, which makes the distribution difficult to discern, w... |
1244 1 | 67 2 | 2 2 | 5556679 3 | 3 | 4 | 23 4 | 5 | 5 | 6 | 4 Legend: 1 | 2 = 12,000 Figure 2.5: A split stem-and-leaf. 3There are a lot of numbers on the first row because there are a lot of values in the data set less than 10 thousand. 2.1. EXAMINING NUMERICAL DATA 63 GUIDED PRACTICE 2.7 What is the smallest number i... |
etry in the distribution. 2. Identifying data collection or entry errors. For instance, we re-examined the email pur- ported to have 64 thousand characters to ensure this value was accurate. 3. Providing insight into interesting properties of the data. 4The smallest number is less than 1 thousand, and the largest is 64... |
shown in Figure 2.6. Characters (in thousands) Count 0-5 5-10 10-15 15-20 20-25 25-30 19 12 6 2 3 5 · · · · · · 55-60 60-65 0 1 Figure 2.7: The counts for the binned num char data. Figure 2.8: A histogram of num char. This histogram uses bins or class intervals of width 5. Explore this histogram and dozens of histogra... |
emails is evenly distribution on this interval, then we can estimate that approximately 19/5 ≈ 4 emails fell in the range between 0 and 1 thousand. EXAMPLE 2.15 What percent of the emails had 10 thousand or more characters? From the first example, we know that 31 emails had fewer than 10 thousand characters. Since ther... |
INING NUMERICAL DATA 67 EXAMPLE 2.18 Approximately 25 of the emails had fewer than how many characters? This time we are given a cumulative frequency, so we start at 25 on the vertical axis and trace it across to see which bin it hits. It hits the 5-10 thousand bin, so 25 of the emails had fewer than a value somewhere ... |
skewed. GUIDED PRACTICE 2.20 Take a look at the dot plot in Figure 2.6. Can you see the skew in the data? Is it easier to see the skew in the frequency histogram, the dot plot, or the stem-and-leaf plot?11 9The shape will remain exactly the same. Changing from frequency to relative frequency involves dividing all the ... |
2 pets but that a smaller number of households will have 3, 4, 5, or more pets, so there will be greater density over the small numbers, suggesting the distribution will have a long right tail and be right skewed. 13Another definition of mode, which is not typically used in statistics, is the value with the most occurr... |
the smaller numbers), it is right skewed. • Distributions may be unimodal, bimodal, or multimodal. • Two graphs that are useful for showing the distribution of a small number of observations are the stem-and-leaf plot and dot plot. These graphs are ideal for displaying data from small samples because they show the exa... |
number of wins. In order to win, the team has to have scored more points (runs) than their opponent in any given game. As such, number of runs is often a good proxy for the success of the team. The table below shows number of runs, home runs, and batting averages for a random sample of 10 teams in the 2014 Major Leagu... |
19 Apple Jacks 20 Dora the Explorer Fiber % 3.3% 3.4% 7.1% 7.4% 7.4% 9.4% 10.0% 10.0% 10.7% 11.1% (a) Create a stem and leaf plot of the distribution of the fiber content of these cereals. (b) Create a dot plot of the fiber content of these cereals. (c) Create a histogram and a relative frequency histogram of the fiber c... |
collected on life spans (in years) and gestation lengths (in days) for 62 mammals. A scatterplot of life span versus length of gestation is shown below.18 (a) What type of an association is apparent between life span and length of gestation? (b) What type of an association would you expect to see if the axes of the pl... |
�ect changing units has on each of the summary quantities. 7. Use quartiles, percentiles, and Z-scores to measure the relative position of a data point within the data set. 8. Compare the distribution of a numerical variable using dot plots / histograms with the same scale, back-to-back stem-and-leaf plots, or parallel... |
that were received in January and March. We could compute a mean for this population in the same way as the sample mean, however, the population mean has a special label: µ. The symbol µ is the Greek letter mu and represents the average of all observations in the population. Sometimes a subscript, such as x, is used t... |
2 to the number of characters in the second email (7.0, in thousands), and xi corresponds to the number of characters in the ith email in the data set. 20The sample size was n = 50. 2.2. NUMERICAL SUMMARIES AND BOX PLOTS 75 The median provides another measure of center. The median splits an ordered data set in half. Th... |
3540455055606570MeanMedian 76 CHAPTER 2. SUMMARIZING DATA 2.2.2 Standard deviation as a measure of spread The U.S. Census Bureau reported that in 2019, the median family income was $80,944 and the mean family income was $108,587. Is a family income of $60,000 far from the mean or somewhat close to the mean? In order to... |
(−11.0)2 + · · · + 4.22 50 − 1 102.01 + 21.16 + 121.00 + · · · + 17.64 49 = 172.44 We divide by n − 1, rather than dividing by n, when computing the variance; you need not worry about this mathematical nuance for the material in this textbook. Notice that squaring the deviations does two things. First, it makes large ... |
income of $60,000 far from the mean or relatively close to the mean? Because $60,000 is less that one standard deviation from the mean, it is relatively close to the mean. If the value were more than 2 standard deviations away from the mean, we would consider it far from the mean. In the next section, we encounter a b... |
with about 65,000 characters. In this chapter we use standard deviation as a descriptive statistic to describe the variability in a given data set. In Chapter 4 we will use standard deviation to assess how close a sample mean is likely to be to the population mean. 23Figure 2.14 shows three distributions that look qui... |
. If an observation is equal to the mean, then the Z-score is 0. EXAMPLE 2.34 Head lengths of brushtail possums have a mean of 92.6 mm and standard deviation 3.6 mm. Compute the Z-scores for possums with head lengths of 95.4 mm and 85.8 mm. For x1 = 95.4 mm: For x2 = 85.8 mm: Z1 = x1 − µ σ = 95.4 − 92.6 3.6 = 0.78 Z2 =... |
June in LA. 2.2.4 Box plots and quartiles A box plot summarizes a data set using five summary statistics while also plotting unusual observations, called outliers. Figure 2.15 provides a box plot of the num char variable from the email50 data set. The five summary statistics used in a box plot are known as the five-numbe... |
in 50 emails. The median (6,890) splits the data into the bottom 50% and the top 50%. Explore dozens of boxplots with histograms using American Community Survey data on Tableau Public. INTERQUARTILE RANGE (IQR) The IQR is the length of the box in a box plot. It is computed as where Q1 and Q3 are the 25th and 75th perc... |
DATA EXAMPLE 2.38 Compare the box plot to the graphs previously discussed: stem-and-leaf plot, dot plot, frequency and relative frequency histogram. What can we learn more easily from a box plot? What can we learn more easily from the other graphs? It is easier to immediately identify the quartiles from a box plot. Th... |
than -41, so there are no outliers on the low end. The observation at 80 is greater than 77, so 80 is an outlier on the high end. 27False. Since Q1 is the 25th percentile and the median is the 50th percentile, 25% of the data fall between Q1 and the median. Similarly, 25% of the data fall between Q2 and the median. Th... |
following information. down arrow to see all of the summary statistics. It will be necessary to hit the ¯x Σx Σx2 Sx σx Mean Sum of all the data values Sum of all the squared data values Sample standard deviation Population standard deviation Sample size or # of data points n minX Minimum Q1 Med Median maxX Maximum Fi... |
, then hit 2). 2. Use the arrow buttons to navigate to the list you would like to delete. 3. Select (F6) to see more options. 4. Select DEL-A (F4) and then F1 to confirm. GUIDED PRACTICE 2.42 Enter the following 10 data points into a calculator. Find the summary statistics and make a box plot of the data.28 5, 8, 1, 19,... |
observed? What would happen to these summary statistics if the observation at 64,401 had been even larger, say 150,000? These scenarios are plotted alongside the original data in Figure 2.16, and sample statistics are computed under each scenario in Figure 2.17. Figure 2.16: Dot plots of the original character count d... |
values. The mean and standard deviation are much more affected by changes in extreme observations. EXAMPLE 2.47 The median and IQR do not change much under the three scenarios in Figure 2.17. Why might this be the case? Since there are no large gaps between observations around the three quartiles, adding, deleting, or ... |
0.5, the mean was multiplied by -0.5. Multiplying all of the values by a negative flipped the sign of numbers, which affects the location of the center, but not the spread. Multiplying all of the values by -0.5 multiplied the standard deviation by +0.5 since the standard deviation cannot be negative. EXAMPLE 2.51 Again, ... |
temperatures shown in both Celsius and Fahrenheit. ADDING SHIFTS THE VALUES, MULTIPLYING STRETCHES OR CONTRACTS THEM Adding a constant to every value in a data set shifts the mean but does not affect the standard deviation. Multiplying the values in a data set by a constant will change the mean and the standard deviati... |
Figure 2.19 to give a better sense of some of the raw median income data. 35New IQR = 9 5 Q3 + 32 − 9 5 Q1 + 32 = 9 5 (Q3 − Q1) = 9 5 × (old IQR). 204060801000204060CelsiusFahrenheitFrequencyTemperature 90 CHAPTER 2. SUMMARIZING DATA Median Income for 150 Counties, in $1000s Population Gain 43.6 51.8 63.3 34.1 46.3 40... |
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