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 as “x-bar.” This notation is so common that writers who are discussing data use , , etc., without additional explanation. The subscripts on the observations 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 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 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 percentile percentile 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, 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 , M, and . 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
box-and-whisker plots, and boxplots are sometimes called box-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 display the spread of the values x_i about their mean . 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 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 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 , the transformed variable x_new has mean a + b.