Chapter 1. Chapter 1: Looking at Data—Distributions

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Looking at Data—Distributions

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Introduction

Statistics is the science of learning from data. Data are numerical or qualitative descriptions of the objects that we want to study. In this chapter, we will master the art of examining data.

We begin in Section 1.1 with some basic ideas about data. We will learn about the different types of data that are collected and how data sets are organized.

Section 1.2 starts our process of learning from data by looking at graphs. These visual displays give us a picture of the overall patterns in a set of data. We have excellent software tools that help us make these graphs. However, it takes a little experience and a lot of judgment to study the graphs carefully and to explain what they tell us about our data.

Section 1.3 continues our process of learning from data by computing numerical summaries. These sets of numbers describe key characteristics of the patterns that we saw in our graphical summaries.

The final section in this chapter helps us make the transition from data summaries to statistical models that are used to draw conclusions and to make predictions. Specifically, we learn about using density curves to describe a set of data and are introduced to the Normal distributions. These distributions can be used to describe many sets of data that we will encounter. They also play a fundamental role in many of the methods of statistical analysis.

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A statistical analysis starts with a set of data. We construct a set of data by first deciding what cases, or units, we want to study. For each case, we record information about characteristics that we call variables.

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Some variables, like the type of restaurant, the name of the restaurant, and the item simply place coupons into categories. The regular price and discount price columns have numerical values for which we can do arithmetic. It makes sense to give an average of the regular prices, but it does not make sense to give an “average” type of restaurant. We can, however, do arithmetic to compare the regular prices classified by type of restaurant.

An appropriate label for your cases should be chosen carefully. In our food coupon example, a natural choice of a label would be the name of the restaurant. However, if there are two or more coupons available for a particular restaurant, or if a restaurant is a chain with different discounts offered at different locations, then the name of the restaurant would not uniquely label each of the coupons. In the restaurant discount coupon file, the first variable, ID, is a unique label for each coupon.

The display in Figure 1.1 is from an Excel spreadsheetspreadsheet. Spreadsheets are very useful for doing the kind of simple computations that you will do in Exercise 1.2. You can type in a formula and have the same computation performed for each row.

Note that the names we have chosen for the variables in our spreadsheet do not have spaces. For example, instead of “Restaurant Name” for the name of the restaurant, we simply use Name. In some statistical software packages, however, spaces are not allowed in variable names. For this reason, when creating spreadsheets for eventual use with statistical software, it is best to avoid spaces in variable names. Another convention is to use an underscore (_) where you would normally use a space. For our data set, we could have used Regular_Price and Discount_Price for the two price variables.

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Another important part of the description of any quantitative variable is its unit of measurement. For both RegPrice and DiscPrice, the unit of measurementunit of measurement is clearly dollars. In other settings, it may not be as obvious. For example, if we were measuring heights of children, we might choose to use either inches or centimeters. The units of measurement are an important part of the description of a quantitative variable.

Key characteristics of a data set

In practice, any set of data is accompanied by background information that helps us understand the data. When you plan a statistical study or explore data from someone else’s work, ask yourself the following questions:

In our example of statistics class data, the possible values for the grade variable are A, B, C, D, and F. When computing grade point averages, many colleges and universities translate these letter grades into numbers using A = 4, B = 3, C = 2, D = 1, and F = 0. The transformed variable with numeric values is considered to be quantitative because we can average the numerical values across different courses to obtain a grade point average.

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Sometimes, experts argue about numerical scales such as this. They ask whether or not the difference between an A and a B is the same as the difference between a D and an F. Similarly, many questionnaires ask people to respond on a 1 to 5 scale, with 1 representing strongly agree, 2 representing agree, etc. Again we could ask whether or not the five possible values for this scale are equally spaced in some sense. From a practical point of view, the averages that can be computed when we convert categorical scales such as these to numerical values frequently provide a very useful way to summarize data.

Often, the variables in a statistical study are easy to understand: height in centimeters, study time in minutes, and so on. But each area of work also has its own special variables. A psychologist uses the Minnesota Multiphasic Personality Inventory (MMPI), and a physical fitness expert measures “VO2 max” (the volume of oxygen consumed per minute while exercising at your maximum capacity). Both of these variables are measured with special instrumentsinstrument. VO2 max is measured by exercising while breathing into a mouthpiece connected to an apparatus that measures oxygen consumed. Scores on the MMPI are based on a long questionnaire, which is also called an instrument.

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Part of mastering your field of work is learning what variables are important and how they are best measured. Because details of particular measurements usually require knowledge of the particular field of study, we will say little about them.

Be sure that each variable really does measure what you want it to. A poor choice of variables can lead to misleading conclusions. Often, for example, the raterate at which something occurs is a more meaningful measure than a simple count of occurrences.

In Example 1.6, when we computed the graduation rate, we used the total number of students to adjust the number of graduates. We constructed a new variable by dividing the number of graduates by the total number of students. Computing a rate is just one of several ways of adjusting one variable to create anotheradjusting one variable to create another. We often divide one variable by another to compute a more meaningful variable to study. Example 1.20 (page 20) is another type of adjustment.

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Exercises 1.5 and 1.6 illustrate an important point about presenting the results of your statistical calculations. Always consider how to best communicate your results to a general audience. For example, the numbers produced by your calculator or by statistical software frequently contain more digits that are needed. Be sure that you do not include extra information generated by software that will distract from a clear explanation of what you have found.

For Exercises 1.1 and 1.2, see page 3; for Exercise 1.3, see page 5; for Exercise 1.4, see page 5; and for Exercises 1.5 and 1.6, see page 6.

Question 1.7

1. (a) What are the cases?

2. (b) Identify the variables and their possible values.

3. (c) Classify each variable as categorical or quantitative. Be sure to include at least one of each.

4. (d) Was a label used? Explain your answer.

5. (e) Summarize the key characteristics of your data set.

Question 1.8

1. (a) What are the cases?

2. (b) Identify the variables and their possible values.

3. (c) Classify each variable as categorical or quantitative. Be sure to include at least one of each.

4. (d) Use a label and explain how you chose it.

5. (e) Summarize the key characteristics of your data set.

Question 1.9

1. (a) What are the cases for this data set?

2. (b) Describe each type of information as a label, a quantitative variable, or a categorical variable.

3. (c) Set up a spreadsheet that could be used to record the data. Give appropriate column headings and five sample cases.

Question 1.10

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Question 1.11

1. (a) Classify each of these variables as categorical or quantitative and give reasons for your answers.

2. (b) For each variable give the possible values.

Question 1.12

Question 1.13

Question 1.14

1. (a) What are the cases for this set of data?

2. (b) Is there a label variable? If yes, what is it?

3. (c) Identify each variable as categorical or quantitative.

4. (d) Explain how you might use each of the quantitative variables to explain something about the states.

5. (e) Consider a variable computed as the number of students in each state who attend college in the state divided by the total number of students from the state who attend college. Explain how you would use this variable to explain something about the states.

Question 1.15

Statistical tools and ideas help us examine data to describe their main features. This examination is called exploratory data analysisexploratory data analysis. Like an explorer crossing unknown lands, we want first to simply describe what we see. Here are two basic strategies that help us organize our exploration of a set of data:

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We follow these principles in organizing our learning. This chapter presents methods for describing a single variable. We will study relationships among several variables in Chapter 2. Within each chapter, we will begin with graphical displays, then add numerical summaries for a more complete description.

Categorical variables: Bar graphs and pie charts

The values of a categorical variable are labels for the categories, such as “yes” and “no.” The distribution of a categorical variabledistribution of a categorical variable lists the categories and gives either the countcount or the percentpercent of cases that fall in each category. An alternative to the percent is the proportionproportion, the count divided by the sum of the counts. Note that the percent is simply the proportion times 100.

Note that the last value of the variable resource is “Other,” which includes all other online resources that were given as selection options. For data sets that have a large number of values for a categorical variable, we often create a category such as this that includes categories that have relatively small counts or percents. Careful judgment is needed when doing this. You don’t want to cover up some important piece of information contained in the data by combining data in this way.

The use of graphical methods allows us to see this information and other characteristics of the data easily. We now examine two types of graphs.

The categories in a bar graph can be put in any order. In Figure 1.2, we ordered the resources based on their preference percents. For other data sets, an alphabetical ordering or some other arrangement might produce a more useful graphical display.

You should always consider the best way to order the values of the categorical variable in a bar graph. Choose an ordering that will be useful to you. If you have difficulty, ask a friend if your choice communicates what you expect. Note that a bar graph using counts will look the same as a bar graph using percents. A pie chart naturally uses percents.

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To make a pie chart, you must include all the categories that make up a whole. A category such as “Other” in this example can be used, but the sum of the percents for all the categories should be 100%. This constraint makes bar graphs more flexible.

Quantitative variables: Stemplots and histograms

A stemplot (also called a stem-and-leaf plot) gives a quick picture of the shape of a distribution while including the actual numerical values in the graph. Stemplots work best for small numbers of observations that are all greater than 0.

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When you wish to compare two related distributions, a back-to-back stemplotback-to-back stemplot with common stems is useful. The leaves on each side are ordered out from the common stem.

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There are two modifications of the basic stemplot that can be helpful in different situations. You can double the number of stems in a plot by splitting each stemsplitting stems into two: one with leaves 0 to 4 and the other with leaves 5 through 9. When the observed values have many digits, it is often best to trimtrimming the numbers by removing the last digit or digits before making a stemplot. If you are using software, you can round the data, which is what was done for the data given in Example 1.11.

You must use your judgment in deciding whether to split stems and whether to trim or round, though statistical software will often make these choices for you. Remember that the purpose of a stemplot is to display the shape of a distribution. If there are many stems with no leaves or only one leaf, trimming will reduce the number of stems. Let’s take a look at the effect of splitting the stems for our SCF data.

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Histograms

Stemplots display the actual values of the observations. This feature makes stemplots awkward for large data sets. Moreover, the picture presented by a stemplot divides the observations into groups (stems) determined by the number system rather than by judgment.

Histograms do not have these limitations. A histogramhistogram breaks the range of values of a variable into classes and displays only the count or percent of the observations that fall into each class. You can choose any convenient number of classes, but you should choose classes of equal width.

Making a histogram by hand requires more work than a stemplot. Histograms do not display the actual values observed. For these reasons, we prefer stemplots for small data sets.

The construction of a histogram is best shown by example. Most statistical software packages will make a histogram for you.

Large sets of data are often reported in the form of frequency tables when it is not practical to publish the individual observations. In addition to the frequency (count) for each class, we may be interested in the fraction or percent of the observations that fall in each class. A histogram of percents looks just like a frequency histogram such as Figure 1.7. Simply relabel the vertical scale to read in percents. Use histograms of percents for comparing several distributions that have different numbers of observations.

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Our eyes respond to the area of the bars in a histogram. Because the classes are all the same width, area is determined by height and all classes are fairly represented. There is no one right choice of the classes in a histogram. Too few classes will give a “skyscraper” graph, with all values in a few classes with tall bars. Too many will produce a “pancake” graph, with most classes having one or no observations. Neither choice will give a good picture of the shape of the distribution. You must use your judgment in choosing classes to display the shape. Statistical software will choose the classes for you. The software’s choice is often a good one, but you can change it if you want.

You should be aware that the appearance of a histogram can change when you change the classes. The histogram function in the One-Variable Statistical Calculator applet on the text website allows you to change the number of classes by dragging with the mouse, so that it is easy to see how the choice of classes affects the histogram.

Although histograms resemble bar graphs, their details and uses are distinct. A histogram shows the distribution of counts or percents among the values of a single variable. A bar graph compares the counts or percents of different items. The horizontal axis of a bar graph need not have any measurement scale but simply identifies the items being compared.

Draw bar graphs with blank space between the bars to separate the items being compared. Draw histograms with no space, to indicate that all values of the variable are covered. Some spreadsheet programs, which are not primarily intended for statistics, will draw histograms as if they were bar graphs, with space between the bars. Often, you can tell the software to eliminate the space to produce a proper histogram.

Data analysis in action: Don’t hang up on me

Many businesses operate call centers to serve customers who want to place an order or make an inquiry. Customers want their requests handled thoroughly. Businesses want to treat customers well, but they also want to avoid wasted time on the phone. They therefore monitor the length of calls and encourage their representatives to keep calls short.

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We started our study of the customer service center data by examining a few cases, the ones displayed in Table 1.2. It would be very difficult to examine all 31,492 cases in this way. How can we do this? Let’s try a histogram.

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It turned out that the bank penalized representatives whose average call length was too long—so some representatives just hung up on customers to bring their average length down. Neither the customers nor the bank were happy about this. The bank changed its policy, and later data showed that calls under 10 seconds had almost disappeared.

The extreme values of a distribution are in the tailstails of the distribution. The high values are in the upper, or right, tail and the low values are in the lower, or left, tail. The overall pattern in Figure 1.8 is made up of the many moderate call lengths and the long right tail of more lengthy calls. The striking deviation from the overall pattern is the surprising number of very short calls in the left tail.

Our examination of the call center data illustrates some important principles:

Examining distributions

Making a statistical graph is not an end in itself. The purpose of the graph is to help us understand the data. After you make a graph, always ask, “What do I see?” Once you have displayed a distribution, you can see its important features as follows.

In Section 1.3, we will learn how to describe center and spread numerically. For now, we can describe the center of a distribution by its midpoint, the value with roughly half the observations taking smaller values and half taking larger values. We can describe the spread of a distribution by giving the smallest and largest values. Stemplots and histograms display the shape of a distribution in the same way. Just imagine a stemplot turned on its side so that the larger values lie to the right.

Some things to look for in describing shape are

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Some variables commonly have distributions with predictable shapes. Many biological measurements on specimens from the same species and sex—lengths of bird bills, heights of young women—have symmetric distributions. Money amounts, on the other hand, usually have right-skewed distributions. There are many moderately priced houses, for example, but the few very expensive mansions give the distribution of house prices a strong right-skew.

Dealing with outliers

In data sets smaller than the service call data, you can spot outliers by looking for observations that stand apart (either high or low) from the overall pattern of a histogram or stemplot. Identifying outliers is a matter for judgment. Look for points that are clearly apart from the body of the data, not just the most extreme observations in a distribution. You should search for an explanation for any outlier. Sometimes outliers point to errors made in recording the data. In other cases, the outlying observation may be caused by equipment failure or other unusual circumstances.

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The state of California is an outlier in the previous example because it has a very large number of undergraduate students. California has the largest population of all the states, so we might expect it to have a large number of college students. Let’s look at these data in a different way.

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In Example 1.19, we looked at the distribution of the number of undergraduate students, while in Example 1.20, we adjusted these data by expressing the counts as number per 1000 people in each state. Which way is correct? The answer depends upon why you are examining the data.

If you are interested in marketing a product to undergraduate students, the unadjusted numbers would be of interest because you want to reach the most people. On the other hand, if you are interested in comparing states with respect to how well they provide opportunities for higher education to their residents, the population-adjusted values would be more suitable. Always think about why you are doing a statistical analysis, and this will guide you in choosing an appropriate analytic strategy.

Here is an example with a different kind of outlier.

Time plots

Whenever data are collected over time, it is a good idea to plot the observations in time order. Displays of the distribution of a variable that ignore time order, such as stemplots and histograms, can be misleading when there is systematic change over time.

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The data described in the preceding example are based on a subset of the subjects in a study of 248 subjects. The researchers were particularly concerned about subjects whose levels were deficient, defined as a serum vitamin D level of less than 50 nmol/l. They found that there was a 3.8-fold higher deficiency rate in February–March than in August–September: 91.2% versus 24.3%. To ensure that individuals from this population have adequate levels of vitamin D, some form of supplementation is needed, particularly during certain times of the year.

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For Exercise 1.16, see page 11; for Exercise 1.17, see page 12; for Exercises 1.18 and 1.19, see page 14; for Exercise 1.20, see page 16; for Exercises 1.21 and 1.22, see page 16; for Exercise 1.23, see page 19; and for Exercise 1.24, see page 21.

Question 1.25

1. (a) Use a bar graph to describe the numbers of users in these countries.

2. (b) Do you think that the United States is an outlier in this data set? Explain your answer.

3. (c) Describe the major features of your graph in a short paragraph.

Question 1.26

1. (a) Use a bar graph to describe the increase in users in these countries.

2. (b) Describe the major features of your graph in a short paragraph.

3. (c) Do you think a stemplot would be a better graphical display for these data? Give reasons for your answer.

4. (d) Write a short paragraph about possible business opportunities suggested by the data you described in this exercise and the previous one.

Question 1.27

1. (a) Make a bar graph of these data.

2. (b) Give a short summary of how the number of passengers varied with class.

3. (c) If you made a bar graph of the percent of passengers in each class, would the general features of the graph differ from the one you made in part (a)? Explain your answer.

Question 1.28

1. (a) Make a pie chart to display the data.

2. (b) Compare the pie chart with the bar graph. Which do you prefer? Give reasons for your answer.

Question 1.29

Question 1.30

1. (a) Make a stemplot of the data.

2. (b) Describe the pattern of the distribution.

3. (c) Are there any outliers? If yes, describe them and explain why you have declared them to be outliers.

4. (d) Describe the shape, center, and spread of the distribution.

Question 1.31

Question 1.32

1. (a) Look at the table and describe how the energy consumption varies from month to month.

2. (b) Make a time plot of the data and describe the patterns.

3. (c) Suppose you wanted to communicate information about the month-to-month variation in energy consumption. Which would be more effective, the table of the data or the graph? Give reasons for your answer.

Question 1.33

1. (a) Analyze these data using the questions in the previous exercise as a guide.

2. (b) Compare the patterns across the two years. Describe any similarities and differences.

Question 1.34

Question 1.35

Question 1.36

1. (a) Add the weights. The sum is not exactly equal to the value of 250.9 million tons given in the table. Why?

2. (b) Make a bar graph of the percents. The graph gives a clearer picture of the main contributors to garbage if you order the bars from tallest to shortest.

3. (c) Also make a pie chart of the percents. Comparing the two graphs, notice that it is easier to see the small differences among “Food scraps,” “Plastics,” and “Yard trimmings” in the bar graph.

Question 1.37

1. (a) Describe these data with a bar graph.

2. (b) Describe these data with a pie chart.

3. (c) Which graphical summary do you prefer. Give reasons for your answer.

Question 1.38

Question 1.39

1. (a) How many variables does this data set contain? Which are categorical variables and which are quantitative variables?

2. (b) Make a stemplot of the distribution of GPA, after rounding to the nearest tenth of a point.

3. (c) Describe the shape, center, and spread of the GPA distribution. Identify any suspected outliers from the overall pattern.

4. (d) Make a back-to-back stemplot of the rounded GPAs for female and male students. Write a brief comparison of the two distributions.

Question 1.40

Question 1.41

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Question 1.42

We can begin our data exploration with graphs, but numerical summaries make our analysis more specific. For categorical variables, numerical summaries are the counts or percents that we use to construct pie charts or bar graphs. In this section, we focus on numerical summaries for quantitative variables. A brief description of the distribution of a quantitative variable should include its shape and numbers describing its center and spread. We describe the shape of a distribution based on inspection of a histogram or a stemplot. Now we will learn specific ways to use numbers to measure the center and spread of a distribution. We can calculate these numerical measures for any quantitative variable. But to interpret measures of center and spread, and to choose among the several measures we will learn, you must think about the shape of the distribution and the meaning of the data. The numbers, like graphs, are aids to understanding, not “the answer” in themselves.

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Measuring center: The mean

Numerical description of a distribution begins with a measure of its center or average. The two common measures of center are the mean and the median. The mean is the “average value” and the median is the “middle value.” These are two different ideas for “center,” and the two measures behave differently. We need precise recipes for the mean and the median.

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The Σ (capital Greek sigma) in the formula for the mean is short for “add them all up.” The bar over the x indicates the mean of all the x-values. Pronounce the mean $\overline{x}$ as “x-bar.” This notation is so common that writers who are discussing data use $\overline{x}$, $\overline{y}$, etc., without additional explanation. The subscripts on the observations $x_i$ are a way of keeping the n observations separate.

The value of the mean will not necessarily be equal to the value of one of the observations in the data set. Our example of time to start a business illustrates this fact.

In practice, you can key the data into your calculator and hit the Mean key. You don’t have to actually add and divide. But you should know that this is what the calculator is doing.

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Exercise 1.43 illustrates an important weakness of the mean as a measure of center: the mean is sensitive to the influence of a few extreme observations. These may be outliers, but a skewed distribution that has no outliers will also pull the mean toward its long tail. Because the mean cannot resist the influence of extreme observations, we say that it is not a resistant measureresistant measure of center.

A measure that is resistant does more than limit the influence of outliers. Its value does not respond strongly to changes in a few observations, no matter how large those changes may be. The mean fails this requirement because we can make the mean as large as we wish by making a large enough increase in just one observation. A resistant measure is sometimes called a robust measurerobust measure.

Measuring center: The median

We used the midpoint of a distribution as an informal measure of center in Section 1.2. The median is the formal version of the midpoint, with a specific rule for calculation.

Note that the formula $\left(n+1\right)/2$ does not give the median, just the location of the median in the ordered list. Medians require little arithmetic, so they are easy to find by hand for small sets of data. Arranging even a moderate number of observations in order is tedious, however, so that finding the median by hand for larger sets of data is unpleasant. Even simple calculators have an $\overline{x}$ button, but you will need computer software or a graphing calculator to automate finding the median.

Note that you can use the stemplot in Figure 1.13 (page 28) directly to compute the median. In the stemplot the cases are already ordered and you simply need to count from the top or the bottom to the desired location.

Mean versus median

Exercises 1.43 and 1.45 illustrate an important difference between the mean and the median. Suriname is an outlier. It pulls the mean time to start a business up from 16 days to 24 days. The median increased slightly, from 11.5 days to 12 days.

The median is more resistant than the mean. If the largest start time in the data set was 1200 days, the median for all 25 countries would still be 12 days. The largest observation just counts as one observation above the center, no matter how far above the center it lies. The mean uses the actual value of each observation and so will chase a single large observation upward.

The best way to compare the response of the mean and median to extreme observations is to use an interactive applet that allows you to place points on a line and then drag them with your computer’s mouse. Exercises 1.83, 1.84, and 1.85 use the Mean and Median applet on the website for this text to compare the mean and the median.

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The median and mean are the most common measures of the center of a distribution. The mean and median of a symmetric distribution are close together. If the distribution is exactly symmetric, the mean and median are exactly the same. In a skewed distribution, the mean is farther out in the long tail than is the median.

The endowment for a college or university is money set aside and invested. The income from the endowment is usually used to support various programs. The distribution of the sizes of the endowments of colleges and universities is strongly skewed to the right. Most institutions have modest endowments, but a few are very wealthy. The median endowment of colleges and universities in a recent year was $93 million—but the mean endowment was $498 million.²¹ The few wealthy institutions pull the mean up but do not affect the median. Don’t confuse the “average” value of a variable (the mean) with its “typical” value, which we might describe by the median.

We can now give a better answer to the question of how to deal with outliers in data. First, look at the data to identify outliers and investigate their causes. You can then correct outliers if they are wrongly recorded, delete them for good reason, or otherwise give them individual attention. The outlier in Example 1.21 (page 21) can be dropped from the data once we discover that it is an error. If you have no clear reason to drop outliers, you may want to use resistant measures in your analysis, so that outliers have little influence over your conclusions. The choice is often a matter for judgment.

Measuring spread: The quartiles

A measure of center alone can be misleading. Two countries with the same median family income are very different if one has extremes of wealth and poverty and the other has little variation among families. A drug manufactured with the correct mean concentration of active ingredient is dangerous if some batches are much too high and others much too low.

We are interested in the spread or variability of incomes and drug potencies as well as their centers. The simplest useful numerical description of a distribution consists of both a measure of center and a measure of spread.

We can describe the spread or variability of a distribution by giving several percentiles. The median divides the data in two; half of the observations are above the median and half are below the median. We could call the median the 50th percentile. The upper quartilequartile is the median of the upper half of the data. Similarly, the lower quartile is the median of the lower half of the data. With the median, the quartiles divide the data into four equal parts; 25% of the data are in each part.

We can do a similar calculation for any percent. The pth percentilepercentile of a distribution is the value that has p percent of the observations fall at or below it. To calculate a percentile, arrange the observations in increasing order and count up the required percent from the bottom of the list.

Our definition of percentiles is a bit inexact because there is not always a value with exactly p percent of the data at or below it. We will be content to take the nearest observation for most percentiles, but the quartiles are important enough to require an exact rule.

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Here is an example.

Notice that the quartiles are resistant. For example, $Q_3$ would have the same value if the highest start time was 530 days rather than 53 days.

Be careful when several observations take the same numerical value. Write down all the observations and apply the rules just as if they all had distinct values.

There are several rules for calculating quartiles, which often give slightly different values. The differences are generally small. For describing data, just report the values that your software gives.

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The five-number summary and boxplots

In Section 1.2, we used the smallest and largest observations to indicate the spread of a distribution. These single observations tell us little about the distribution as a whole, but they give information about the tails of the distribution that is missing if we know only $Q_1$, M, and $Q_3$. To get a quick summary of both center and spread, use all five numbers.

The five-number summary leads to another visual representation of a distribution, the boxplot.

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The lines extending to the smallest and largest observations are sometimes called whiskerswhiskers, and boxplots are sometimes called box-and-whisker plotsbox-and-whisker plots. Software provides many varieties of boxplots, some of which use different choices for the placement of the whiskers.

When you look at a boxplot, first locate the median, which marks the center of the distribution. Then look at the spread. The quartiles show the spread of the middle half of the data, and the extremes (the smallest and largest observations) show the spread of the entire data set.

The 1.5 × IQR rule for suspected outliers

If we look at the data in Table 1.2 (page 17), we can spot a clear outlier, a call lasting 2631 seconds, more than twice the length of any other call. How can we describe the spread of this distribution? The smallest and largest observations are extremes that do not describe the spread of the majority of the data. The distance between the quartiles (the range of the center half of the data) is a more resistant measure of spread than the range. This distance is called the interquartile range.

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The quartiles and the IQR are not affected by changes in either tail of the distribution. They are resistant, therefore, because changes in a few data points have no further effect once these points move outside the quartiles.

However, no single numerical measure of spread, such as IQR, is very useful for describing skewed distributions. The two sides of a skewed distribution have different spreads, so one number can’t summarize them. We can often detect skewness from the five-number summary by comparing how far the first quartile and the minimum are from the median (left tail) with how far the third quartile and the maximum are from the median (right tail). The interquartile range is mainly used as the basis for a rule of thumb for identifying suspected outliers.

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Two variations on the basic boxplot can be very useful. The first, called a modified boxplotmodified boxplot, uses the 1.5 × IQR rule. The lines that extend out from the quartiles are terminated in whiskers that are 1.5 × IQR in length. Points beyond the whiskers are plotted individually and are classified as outliers according to the 1.5 × IQR rule.

The other variation is to use two or more boxplots in the same graph to compare groups measured on the same variable. These are called side-by-side boxplotsside-by-side boxplots. The following example illustrates these two variations.

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Measuring spread: The standard deviation

The five-number summary is not the most common numerical description of a distribution. That distinction belongs to the combination of the mean to measure center and the standard deviation to measure spread, or variability. The standard deviation measures spread by looking at how far the observations are from their mean.

The idea behind the variance and the standard deviation as measures of spread is as follows: The deviations $x_i-\overline{x}$ display the spread of the values x_i about their mean $\overline{x}$. Some of these deviations will be positive and some negative because some of the observations fall on each side of the mean. In fact, the sum of the deviations of the observations from their mean will always be zero. Squaring the deviations makes the negative deviations positive so that observations far from the mean in either direction have large positive squared deviations. The variance is the average squared deviation. Therefore, s² and s will be large if the observations are widely spread about their mean and small if the observations are all close to the mean.

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Exercise 1.80 asks you to calculate the seven deviations from Example 1.32, square them, and find s² and s directly from the deviations. Working one or two short examples by hand helps you understand how the standard deviation is obtained. In practice, you will use either software or a calculator that will find s.

The idea of the variance is straightforward: it is the average of the squares of the deviations of the observations from their mean. The details we have just presented, however, raise some questions.

Why do we square the deviations?

Why do we emphasize the standard deviation rather than the variance?

Why do we average by dividing by n − 1 rather than n in calculating the variance?

Properties of the standard deviation

Here are the basic properties of the standard deviation s as a measure of spread.

The use of squared deviations renders s even more sensitive than $\overline{x}$ to a few extreme observations. For example, when we add Suriname to our sample of 24 countries for the analysis of the time to start a business (Exercise 1.43, page 29, and Exercise 1.45, page 31), we increase the standard deviation from 14.2 to 40.8! Distributions with outliers and strongly skewed distributions have standard deviations that do not give much helpful information about such distributions.

Choosing measures of center and spread

How do we choose between the five-number summary and $\overline{x}$ and s to describe the center and spread of a distribution? Because the two sides of a strongly skewed distribution have different spreads, no single number such as s describes the spread well. The five-number summary, with its two quartiles and two extremes, does a better job.

Remember that a graph gives the best overall picture of a distribution. Numerical measures of center and spread report specific facts about a distribution, but they do not describe its shape. Numerical summaries do not disclose the presence of multiple modes or gaps, for example. Always plot your data.

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Changing the unit of measurement

The same variable can be recorded in different units of measurement. Americans commonly record distances in miles and temperatures in degrees Fahrenheit, while the rest of the world measures distances in kilometers and temperatures in degrees Celsius. Fortunately, it is easy to convert numerical descriptions of a distribution from one unit of measurement to another. This is true because a change in the measurement unit is a linear transformation of the measurements.

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Linear transformations do not change the shape of a distribution. If measurements on a variable x have a right-skewed distribution, any new variable x_new obtained by a linear transformation x_new = a + bx (for b > 0) will also have a right-skewed distribution. If the distribution of x is symmetric and unimodal, the distribution of x_new remains symmetric and unimodal.

Although a linear transformation preserves the basic shape of a distribution, the center and spread will change. Because linear changes of measurement scale are common, we must be aware of their effect on numerical descriptive measures of center and spread. Fortunately, the changes follow a simple pattern.

Here is a summary of the rules for linear transformations:

In Example 1.35, when we converted from score to points, we described the transformation as dividing by 4. The multiplication part of the summary of the effect of a linear transformation applies to this case because division by 4 is the same as multiplication by 0.25. Similarly, the second part of the summary applies to subtraction as well as addition because subtraction is simply the addition of a negative number.

The measures of spread IQR and s do not change when we add the same number a to all the observations because adding a constant changes the location of the distribution but leaves the spread unaltered. You can find the effect of a linear transformation x_new = a + bx by combining these rules. For example, if x has mean $\overline{x}$, the transformed variable x_new has mean a + b$\overline{x}$.

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For Exercises 1.43 and 1.44, see page 29; for Exercises 1.45 to 1.47, see page 31; for Exercise 1.48, see page 33; for Exercises 1.49 and 1.50, see page 34; for Exercise 1.51, see page 35; for Exercise 1.52, see page 37; for Exercise 1.53, see page 39; for Exercise 1.54, see page 40; for Exercise 1.55, see page 40; and for Exercise 1.56, see page 45.

Question 1.57

1. (a) Compute the mean for these data.

2. (b) Compute the median for these data.

3. (c) Which measure do you prefer for describing the center of this distribution? Explain your answer. (You may include a graphical summary as part of your explanation.)

Question 1.58

1. (a) Compute the mean for these data.

2. (b) Compute the median for these data.

3. (c) Which measure do you prefer for describing the center of this distribution? Explain your answer. (You may include a graphical summary as part of your explanation.)

Question 1.59

1. (a) Compute the standard deviation for these data.

2. (b) Compute the quartiles for these data.

3. (c) Give the five-number summary and explain the meaning of each of the five numbers.

4. (d) Which numerical summaries do you prefer for describing the distribution, the mean, and the standard deviation of the five-number summary? Explain your answer. (You may include a graphical summary as part of your explanation.)

Question 1.60

1. (a) Compute the standard deviation for these data.

2. (b) Compute the quartiles for these data.

3. (c) Give the five-number summary and explain the meaning of each of the five numbers.

4. (d) Which numerical summaries do you prefer for describing the distribution, the mean, and the standard deviation of the five-number summary? Explain your answer. (You may include a graphical summary as part of your explanation.)

Question 1.61

1. (a) Make a histogram and use it to describe the distribution of potassium absorption.

2. (b) Make a boxplot and use it to describe the distribution of potassium absorption.

3. (c) Compare the stemplot, the histogram, and the boxplot as graphical summaries of this distribution. Which do you prefer? Give reasons for your answer.

Question 1.62

1. (a) Make a histogram and use it to describe the distribution of potassium absorption.

2. (b) Make a boxplot and use it to describe the distribution of potassium absorption.

3. (c) Compare the stemplot, the histogram, and the boxplot as graphical summaries of this distribution. Which do you prefer? Give reasons for your answer.

Question 1.63

Question 1.64

1. (a) Summarize what you see in the boxplots and compare it with what you saw in the stemplots.

2. (b) For comparing these two distributions, do you prefer back-to-back stemplots or side-by-side boxplots? Give reasons for your answer.

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Question 1.65

1. (a) Compute the mean for these data.

2. (b) Compute the median for these data.

3. (c) Which measure do you prefer for describing the center of this distribution? Explain your answer. (You may include a graphical summary as part of your explanation.)

Question 1.66

1. (a) Compute the standard deviation for these data.

2. (b) Compute the quartiles for these data.

3. (c) Which measure do you prefer for describing the spread of this distribution? Explain your answer. (You may include a graphical summary as part of your explanation.)

Question 1.67

1. (a) Find the IQR for these data.

2. (b) Use the 1.5 × IQR rule to identify and name any outliers.

3. (c) Make a boxplot for these data and describe the distribution using only the information in the boxplot.

4. (d) Make a modified boxplot for these data and describe the distribution using only the information in the boxplot.

5. (e) Make a stemplot for these data.

6. (f) Compare the boxplot, the modified boxplot, and the stemplot. Evaluate the advantages and disadvantages of each graphical summary for describing the distribution of the double stout data.

Question 1.68

1. (a) Find the mean reflectance for these smolts.

2. (b) Find the median reflectance for these smolts.

3. (c) Do you prefer the mean or the median as a measure of center for these data? Give reasons for your preference.

Question 1.69

1. (a) Find the standard deviation of the reflectance for these smolts.

2. (b) Find the quartiles of the reflectance for these smolts.

3. (c) Do you prefer the standard deviation or the quartiles as a measure of spread for these data? Give reasons for your preference.

Question 1.70

1. (a) Find the IQR for the smolt data.

2. (b) Use the 1.5 × IQR rule to identify any outliers.

3. (c) Make a boxplot for the smolt data and describe the distribution using only the information in the boxplot.

4. (d) Make a modified boxplot for these data and describe the distribution using only the information in the boxplot.

5. (e) Make a stemplot for these data.

6. (f) Compare the boxplot, the modified boxplot, and the stemplot. Evaluate the advantages and disadvantages of each graphical summary for describing the distribution of the smolt reflectance data.

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Question 1.71

1. (a) Summarize the data graphically and numerically. Give reasons for the methods you chose to use in your summaries.

2. (b) Do you think that your numerical summaries do an effective job of describing these data? Why or why not?

3. (c) There appear to be two distinct clusters of weights for these potatoes. Divide the sample into two subsamples based on the clustering. Give the mean and standard deviation for each subsample. Do you think that this way of summarizing these data is better than a numerical summary that uses all the data as a single sample? Give a reason for your answer.

Question 1.72

1. (a) Use graphical and numerical summaries of your choice to describe the data. Give reasons for your choice.

2. (b) The data set contains an outlier. Explain why this particular beer is unusual.

3. (c) For the outlier, give a short description of how you think this particular beer should be marketed.

Question 1.73

1. (a) Calculate the mean with and without the outlier. Do the same for the median. Explain how these values change when the outliers is excluded.

2. (b) Calculate the standard deviation with and without the outlier. Do the same for the quartiles. Explain how these values change when the outlier is excluded.

3. (c) Write a short paragraph summarizing what you have learned in this exercise.

Question 1.74

1. (a) Analyze the data and summarize the distribution of calories for these 159 brands of beer.

2. (b) In the previous exercise, you identified one brand of beer as an outlier. To what extent is this brand an outlier in the distribution of calories? Explain your answer.

3. (c) Does the distribution of calories suggest marketing strategies for this brand of beer? Describe some marketing strategies.

Question 1.75

Question 1.76

Question 1.77

Question 1.78

Question 1.79

Question 1.80

Question 1.81

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Question 1.82

1. (a) Find $\overline{x}$ and s for the IQ data. In large populations, IQ scores are standardized to have mean 100 and standard deviation 15. In what way does the distribution of IQ among these students differ from the overall population?

2. (b) Find the median IQ score. It is, as we expect, close to the mean.

3. (c) Find the mean and median for the GPA data. The two measures of center differ a bit. What feature of the data (see your stemplot in Exercise 1.39 or make a new stemplot) explains the difference?

Question 1.83

Question 1.84

1. (a) Pull the single rightmost observation out to the right. (Place the cursor on the point, hold down a mouse button, and drag the point.) How does the mean behave? How does the median behave? Explain briefly why each measure acts as it does.

2. (b) Now drag the rightmost point to the left as far as you can. What happens to the mean? What happens to the median as you drag this point past the other two (watch carefully)?

Question 1.85

1. (a) Add one additional observation without changing the median. Where is your new point?

2. (b) Use the applet to convince yourself that when you add yet another observation (there are now nine in all), the median does not change no matter where you put the seventh point. Explain why this must be true.

Question 1.86

1. (a) Verify that the mean is 15 and find the standard deviation for the 10 cases for which x is not missing.

2. (b) Create a new data set with 20 cases by setting the values for the 10 missing cases to 15. Compute the mean and standard deviation for this data set.

3. (c) Summarize what you have learned about the possible effects of this type of imputation on the mean and the standard deviation.

Question 1.87

1. (a) Choose four numbers that have the smallest possible standard deviation.

2. (b) Choose four numbers that have the largest possible standard deviation.

3. (c) Is more than one choice possible in either part (a) or part (b)? Explain.

Question 1.88

1. (a) Find the five-number summary for these data.

2. (b) Make a boxplot.

3. (c) Make a histogram.

4. (d) Write a short summary of the major features of this distribution. Do you prefer the boxplot or the histogram for these data?

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Question 1.89

Question 1.90

Question 1.91

Question 1.92

We now have a kit of graphical and numerical tools for describing distributions. What is more, we have a clear strategy for exploring data on a single quantitative variable:

Technology has expanded the set of graphs that we can choose for Step 1. It is possible, though painful, to make histograms by hand. Using software, clever algorithms can describe a distribution in a way that is not feasible by hand, by fitting a smooth curve to the data in addition to or instead of a histogram. The curves used are called density curvesdensity curves. Before we examine density curves in detail, here is an example of what software can do.

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A smooth density curve is an idealization that gives the overall pattern of the data but ignores minor irregularities. We first discuss density curves in general and then focus on a special class of density curves, the bell-shaped Normal curves.

Density curves

One way to think of a density curve is as a smooth approximation to the irregular bars of a histogram. Figure 1.21 shows a histogram of the scores of all 947 seventh-grade students in Gary, Indiana, on the vocabulary part of the Iowa Test of Basic Skills. Scores of many students on this national test have a very regular distribution. The histogram is symmetric, and both tails fall off quite smoothly from a single center peak. There are no large gaps or obvious outliers. The curve drawn through the tops of the histogram bars in Figure 1.21 is a good description of the overall pattern of the data.

The density curve in Figure 1.21 is a Normal curve. Density curves, like distributions, come in many shapes. Figure 1.22 shows two density curves, a symmetric Normal density curve and a right-skewed curve.

We will discuss Normal density curves in detail in this section because of the important role that they play in statistics. There are, however, many applications where the use of other families of density curves are essential.

A density curve of an appropriate shape is often an adequate description of the overall pattern of a distribution. Outliers, which are deviations from the overall pattern, are not described by the curve.

Measuring center and spread for density curves

Our measures of center and spread apply to density curves as well as to actual sets of observations, but only some of these measures are easily seen from the curve. A mode of a distribution described by a density curve is a peak point of the curve, the location where the curve is highest. Because areas under a density curve represent proportions of the observations, the median is the point with half the total area on each side. You can roughly locate the quartiles by dividing the area under the curve into quarters as accurately as possible by eye. The IQR is the distance between the first and third quartiles. There are mathematical ways of calculating areas under curves. These allow us to locate the median and quartiles exactly on any density curve.

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What about the mean and standard deviation? The mean of a set of observations is their arithmetic average. If we think of the observations as weights strung out along a thin rod, the mean is the point at which the rod would balance. This fact is also true of density curves. The mean is the point at which the curve would balance if it were made out of solid material. Figure 1.23 illustrates this interpretation of the mean.

A symmetric curve, such as the Normal curve in Figure 1.22(a), balances at its center of symmetry. Half the area under a symmetric curve lies on either side of its center, so this is also the median.

For a right-skewed curve, such as those shown in Figures 1.22(b) and 1.23, the small area in the long right tail tips the curve more than the same area near the center. The mean (the balance point), therefore, lies to the right of the median. It is hard to locate the balance point by eye on a skewed curve. There are mathematical ways of calculating the mean for any density curve, so we are able to mark the mean as well as the median in Figure 1.22(b). The standard deviation can also be calculated mathematically, but it can’t be located by eye on most density curves.

A density curve is an idealized description of a distribution of data. For example, the density curve in Figure 1.21 is exactly symmetric, but the histogram of vocabulary scores is only approximately symmetric. We therefore need to distinguish between the mean and standard deviation of the density curve and the numbers $\overline{x}$ and s computed from the actual observations. The usual notation for the mean of an idealized distribution is μmean μ (the Greek letter mu). We write the standard deviation of a density curve as σstandard deviation σ (the Greek letter sigma). In Chapter 5, we refer to $\overline{x}$ and s as statistics associated with a sample and to μ and σ as parameters associated with a population.

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Normal distributions

One particularly important class of density curves has already appeared in Figures 1.21 and 1.22(a). These density curves are symmetric, unimodal, and bell-shaped. They are called Normal curvesNormal curves, and they describe Normal distributionsNormal distributions. All Normal distributions have the same overall shape.

The exact density curve for a particular Normal distribution is specified by giving the distribution’s mean μ and its standard deviation σ. The mean is located at the center of the symmetric curve and is the same as the median. Changing μ without changing σ moves the Normal curve along the horizontal axis without changing its spread.

The standard deviation σ controls the spread of a Normal curve. Figure 1.24 shows two Normal curves with different values of σ. The curve with the larger standard deviation is more spread out.

The standard deviation σ is the natural measure of spread for Normal distributions. Not only do μ and σ completely determine the shape of a Normal curve, but we can locate σ by eye on the curve. Here’s how. As we move out in either direction from the center μ, the curve changes from falling ever more steeply

to falling ever less steeply

The points at which this change of curvature takes place are located at distance σ on either side of the mean m. You can feel the change as you run your finger along a Normal curve, and so find the standard deviation. Remember that m and σ alone do not specify the shape of most distributions, and that the shape of density curves in general does not reveal σ. These are special properties of Normal distributions.

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There are other symmetric bell-shaped density curves that are not Normal. The Normal density curves are specified by a particular equation. The height of the density curve at any point x is given by

$\frac{1}{\sigma \sqrt{2\pi }}{e}^{-\frac{1}{2}{\left(\frac{x-\mu }{\sigma }\right)}^2}$

We will not make direct use of this fact, although it is the basis of mathematical work with Normal distributions. Notice that the equation of the curve is completely determined by the mean μ and the standard deviation σ.

Why are the Normal distributions important in statistics? Here are three reasons.

However, even though many sets of data follow a Normal distribution, many do not. Most income distributions, for example, are skewed to the right and so are not Normal. Non-Normal data, like nonnormal people, not only are common but are also sometimes more interesting than their Normal counterparts.

The 68–95–99.7 rule

Although there are many Normal curves, they all have common properties. Here is one of the most important.

Figure 1.25 illustrates the 68–95–99.7 rule. By remembering these three numbers, you can think about Normal distributions without constantly making detailed calculations.

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Because we will mention Normal distributions often, a short notation is helpful. We abbreviate the Normal distribution with mean μ and standard deviation σ as N(μ, σ)N(μ, σ). For example, the distribution of young women’s heights is N(64.5, 2.5).

Standardizing observations

As the 68–95–99.7 rule suggests, all Normal distributions share many properties. In fact, all Normal distributions are the same if we measure in units of size σ about the mean μ as center. Changing to these units is called standardizing. To standardize a value, subtract the mean of the distribution and then divide by the standard deviation.

A z-score tells us how many standard deviations the original observation falls away from the mean, and in which direction. Observations larger than the mean are positive when standardized, and observations smaller than the mean are negative.

To compare scores based on different measures, z-scores can be very useful. For example, see Exercise 1.124 (page 73), where you are asked to compare an SAT score with an ACT score.

We need a way to write variables, such as “height” in Example 1.38, that follow a theoretical distribution such as a Normal distribution. We use capital letters near the end of the alphabet for such variables. If X is the height of a young woman, we can then shorten “the height of a young woman is less than 68 inches” to “X < 68.” We will use lowercase x to stand for any specific value of the variable X.

We often standardize observations from symmetric distributions to express them in a common scale. We might, for example, compare the heights of two children of different ages by calculating their z-scores. The standardized heights tell us where each child stands in the distribution for his or her age group.

Standardizing is a linear transformation that transforms the data into the standard scale of z-scores. We know that a linear transformation does not change the shape of a distribution, and that the mean and standard deviation change in a simple manner. In particular, the standardized values for any distribution always have mean 0 and standard deviation 1.

If the variable we standardize has a Normal distribution, standardizing does more than give a common scale. It makes all Normal distributions into a single distribution, and this distribution is still Normal. Standardizing a variable that has any Normal distribution produces a new variable that has the standard Normal distribution.

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Normal distribution calculations

Areas under a Normal curve represent proportions of observations from that Normal distribution. There is no formula for areas under a Normal curve. Calculations use either software that calculates areas or a table of areas. The table and most software calculate one kind of area: cumulative proportionscumulative proportion. A cumulative proportion is the proportion of observations in a distribution that lie at or below a given value. When the distribution is given by a density curve, the cumulative proportion is the area under the curve to the left of a given value. Figure 1.27 shows the idea more clearly than words do.

The key to calculating Normal proportions is to match the area you want with areas that represent cumulative proportions. Then get areas for cumulative proportions either from software or (with an extra step) from a table. The following examples show the method in pictures.

There is no area under a smooth curve that is exactly over the point 800. Consequently, the area to the right of 800 (the proportion of scores > 800) is the same as the area at or to the right of this point (the proportion of scores ≥ 800). The actual data may contain a student who scored exactly 800 on the SAT. That the proportion of scores exactly equal to 800 is 0 for a Normal distribution is a consequence of the idealized smoothing of Normal distributions for data.

How do we find the numerical values of the areas in Examples 1.40 and 1.41? If you use software, just plug in mean 1010 and standard deviation 225. Then ask for the cumulative proportions for 800 and for 620. (Your software will probably refer to these as “cumulative probabilities.” We will learn in Chapter 4 why the language of probability fits.) Sketches of the areas that you want similar to the ones in Examples 1.40 and 1.41 are very helpful in making sure that you are doing the correct calculations.

You can use the Normal Curve applet on the text website to find Normal proportions. The applet is more flexible than most software—it will find any Normal proportion, not just cumulative proportions. The applet is an excellent way to understand Normal curves. But, because of the limitations of web browsers, the applet is not as accurate as statistical software.

If you are not using software, you can find cumulative proportions for Normal curves from a table. That requires an extra step, as we now explain.

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Using the standard Normal table

The extra step in finding cumulative proportions from a table is that we must first standardize to express the problem in the standard scale of z-scores. This allows us to get by with just one table, a table of standard Normal cumulative proportions. Table A in the back of the book gives standard Normal probabilities. The picture at the top of the table reminds us that the entries are cumulative proportions, areas under the curve to the left of a value z.

Now that you see how Table A works, let’s redo the NCAA Examples 1.40 and 1.41 using the table.

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The area from the table in Example 1.43 (0.8238) is slightly less accurate than the area from software in Example 1.40 (0.8247) because we must round z to two places when we use Table A. The difference is rarely important in practice.

Sometimes we encounter a value of z more extreme than those appearing in Table A. For example, the area to the left of z = −4 is not given in the table. The z-values in Table A leave only area 0.0002 in each tail unaccounted for. For practical purposes, we can act as if there is zero area outside the range of Table A.

Inverse Normal calculations

Examples 1.40 to 1.44 illustrate the use of Normal distributions to find the proportion of observations in a given event, such as “SAT score between 620 and 800.” We may instead want to find the observed value corresponding to a given proportion.

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Statistical software will do this directly. Without software, use Table A backward, finding the desired proportion in the body of the table and then reading the corresponding z from the left column and top row.

Normal quantile plots

The Normal distributions provide good descriptions of some distributions of real data, such as the Iowa Test vocabulary scores. The distributions of some other common variables are usually skewed and therefore distinctly non-Normal. Examples include economic variables such as personal income and gross sales of business firms, the survival times of cancer patients after treatment, and the service lifetime of mechanical or electronic components. While experience can suggest whether or not a Normal distribution is plausible in a particular case, it is risky to assume that a distribution is Normal without actually inspecting the data.

A histogram or stemplot can reveal distinctly non-Normal features of a distribution, such as outliers, pronounced skewness, or gaps and clusters. If the stemplot or histogram appears roughly symmetric and unimodal, however, we need a more sensitive way to judge the adequacy of a Normal model. The most useful tool for assessing Normality is another graph, the Normal quantile plotNormal quantile plot.

Here is the basic idea of a Normal quantile plot. The graphs produced by software use more sophisticated versions of this idea. It is not practical to make Normal quantile plots by hand.

Any Normal distribution produces a straight line on the plot because standardizing turns any Normal distribution into a standard Normal distribution. Standardizing is a linear transformation that can change the slope and intercept of the line in our plot but cannot turn a line into a curved pattern.

Figures 1.30 and 1.31 are Normal quantile plots for data we have met earlier. The data x are plotted vertically against the corresponding standard Normal z-score plotted horizontally. The z-score scale generally extends from −3 to 3 because almost all of a standard Normal curve lies between these values. These figures show how Normal quantile plots behave.

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Real data often show some departure from the theoretical Normal model. When you examine a Normal quantile plot, look for shapes that show clear departures from Normality. Don’t overreact to minor wiggles in the plot. When we discuss statistical methods that are based on the Normal model, we are interested in whether or not the data are sufficiently Normal for these procedures to work properly. We are not concerned about minor deviations from Normality. Many common methods work well as long as the data are approximately Normal and outliers are not present.

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Many distributions that we have met have a single peak, or mode. The distribution described in Example 1.48 has three modes and is called a trimodal distributiontrimodal distribution. A distribution that has two modes is called a bimodal distributionbimodal distribution.

The previous example reminds of a continuing theme for data analysis. We looked at a histogram and a density estimate and saw something interesting. This led us to speculation. Additional data on the type and location of the seats may explain more about the prices than we see in Figure 1.32.

For Exercises 1.93 and 1.94, see page 59; for Exercises 1.95 and 1.96, see page 60; for Exercises 1.97 and 1.98, see page 64; and for Exercises 1.99 and 1.100, see page 66.

Question 1.101

1. (a) Sketch a symmetric distribution that is not Normal. Mark the location of the mean and the median.

2. (b) Sketch a distribution that is skewed to the left. Mark the location of the mean and the median.

Question 1.102

1. (a) Sketch a Normal curve that has mean 30 and standard deviation 8.

2. (b) On the same x axis, sketch a Normal curve that has mean 30 and standard deviation 12.

3. (c) How does the Normal curve change when the standard deviation is varied but the mean stays the same?

Question 1.103

1. (a) Sketch a Normal curve that has mean 30 and standard deviation 8.

2. (b) On the same x axis, sketch a Normal curve that has mean 40 and standard deviation 8.

3. (c) How does the Normal curve change when the mean is varied but the standard deviation stays the same?

Question 1.104

1. (a) Sketch this Normal distribution.

2. (b) Make a table that includes values of the scores corresponding to plus or minus one, two, and three standard deviations from the mean. Mark these points on your sketch along with the mean.

3. (c) Apply the 68–95–99.7 rule to this distribution. Give the ranges of reading score values that are within one, two, and three standard deviations of the mean.

Question 1.105

Question 1.106

Question 1.107

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Question 1.108

Question 1.109

Question 1.110

1. (a) The mean number of words spoken per day by the women was 14,297 with a standard deviation of 6441. Use the 68–95–99.7 rule to describe this distribution.

2. (b) Do you think that applying the rule in this situation is reasonable? Explain your answer.

3. (c) The men averaged 14,060 words per day with a standard deviation of 9056. Answer the questions in parts (a) and (b) for the men.

4. (d) Do you think that the data support the conventional wisdom? Explain your answer. Note that in Section 7.2 we will learn formal statistical methods to answer this type of question.

Question 1.111

1. (a) Answer the questions from the previous exercise for the Mexican study.

2. (b) The means for both men and women are higher for the Mexican study than for the U.S. study. What conclusions can you draw from this observation?

Question 1.112