Statistical tools and ideas help us examine data to describe their main features. This examination is called exploratory 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:

We follow these principles in organizing our learning. The rest of this chapter presents methods for describing a single variable. We study relationships among two or more variables in Chapter 2. Within each chapter, we 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 variable lists the categories and gives either the count or the percent of cases that fall in each category.

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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 will allow 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.

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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.

We use graphical displays to help us learn things from data. Here is another example.

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We have discussed two tools to make a graphical summary for these data—pie charts and bar charts. Let's consider possible uses of the data to help us to choose a useful graph. Which cost centers are generating large costs? Notice that the display of the data in Figure 1.4 is organized to help us answer this question. The cost centers are ordered by the annual cost, largest to smallest. The data display also gives the annual cost as a percent of the total. We see that parts and materials have an annual cost of $1,325,000, which is 31% of $4,250,500, the total cost.

The last column in the display gives the cumulative percent which is the sum of the percents for the cost center in the given row and all above it. We see that the three largest cost centers—parts and materials, manufacturing equipment, and salaries—account for 66% of the total annual costs.

A bar graph whose categories are ordered from most frequent to least frequent is called a Pareto chart.⁵ Pareto charts are frequently used in quality control settings. There, the purpose is often to identify common types of defects in a manufactured product. Deciding upon strategies for corrective action can then be based on what would be most effective. Chapter 12 gives more examples of settings where Pareto charts are used.

Let’s use a Pareto chart to look at our cost analysis data.

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Bar graphs, pie charts, and Pareto charts can help you see characteristics of a distribution quickly. We now examine quantitative variables, where graphs are essential tools.

Quantitative variables: Histograms

Quantitative variables often take many values. A graph of the distribution is clearer if nearby values are grouped together. The most common graph of the distribution of a single quantitative variable is a histogram.

Our data set contains 2895 cases. The two variables in the data set are the date of the auction and the interest rate. To learn something about T-bill interest rates, we begin with a 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 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.

Our eyes respond to the area of the bars in a histogram.⁸ 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. Always use your judgment in choosing classes to display the shape. Statistics software will choose the classes for you, but there are usually options for changing them.

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. The next example illustrates a situation where the wrong choice of classes will cause you to miss a very important characteristic of a data set.

We started our study of the customer service center data by examining a few cases, the ones displayed in Table 1.1. It would be very difficult to examine all 31,492 cases in this way. We need a better method. Let’s try a histogram.

The choice of the classes is an important part of making a histogram. Let’s look at the customer service center call lengths again.

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If we let software choose the classes, we would miss one of the most important features of the data, the calls of very short duration. We were alerted to this unexpected characteristic of the data by our examination of the 80 cases displayed in Table 1.1. Beware of letting statistical software do your thinking for you. Example 1.15 illustrates the danger of doing this. To do an effective analysis of data, we often need to look at data in more than one way. For histograms, looking at several choices of classes will lead us to a good choice.

Quantitative variables: Stemplots

Histograms are not the only graphical display of distributions of quantitative variables. For small data sets, a stemplot is quicker to make and presents more detailed information. It is sometimes referred to as a back-of-the-envelope technique. Popularized by the statistician John Tukey, it was designed to give a quick and informative look at the distribution of a quantitative variable. A stemplot was originally designed to be made by hand, although many statistical software packages include this capability.

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You can choose the classes in a histogram. The classes (the stems) of a stem-plot are given to you. When the observed values have many digits, it is often best to round the numbers to just a few digits before making a stemplot, as we did in Example 1.16.

You can also split stems to double the number of stems when all the leaves would otherwise fall on just a few stems. Each stem then appears twice. Leaves 0 to 4 go on the upper stem, and leaves 5 to 9 go on the lower stem. Rounding and splitting stems are matters for judgment, like choosing the classes in a histogram. Stemplots work well for small sets of data. When there are more than 100 observations, a histogram is almost always a better choice.

Stemplots can also be used to compare two distributions. This type of plot is called a back-to-back stemplot. We put the leaves for one group to the right of the stem and the leaves for the other group on the left. Here is an example.

Special considerations apply for very large data sets. It is often useful to take a sample and examine it in detail as a first step. This is what we did in Example 1.16. Sampling can be done in many different ways. A company with a very large number of customer records, for example, might look at those from a particular region or country for an initial analysis.

Interpreting histograms and stemplots

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.

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We learn how to describe center and spread numerically in Section 1.3. 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.

When you describe a distribution, concentrate on the main features. Look for major peaks, not for minor ups and downs in the bars of the histogram. Look for clear outliers, not just for the smallest and largest observations. Look for rough symmetry or clear skewness.

The overall shape of a distribution is important information about a variable. Some types of data regularly produce distributions that are symmetric or skewed. For example, data on the diameters of ball bearings produced by a manufacturing process tend to be symmetric. Data on incomes (whether of individuals, companies, or nations) are usually strongly skewed to the right. There are many moderate incomes, some large incomes, and a few very large incomes. Do remember that many distributions have shapes that are neither symmetric nor skewed. Some data show other patterns. Scores on an exam, for example, may have a cluster near the top of the scale if many students did well. Or they may show two distinct peaks if a tough problem divided the class into those who did and didn—t solve it. Use your eyes and describe what you see.

Time plots

Many variables are measured at intervals over time. We might, for example, measure the cost of raw materials for a manufacturing process each month or the price of a stock at the end of each day. In these examples, our main interest is change over time. To display change over time, make a time plot.

More details about how to analyze data that vary over time are given in Chapter 13. For now, we examine how a time plot can reveal some additional important information about T-bill interest rates.

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In Example 1.12 (page 12) we examined the distribution of T-bill interest rates for the period December 12, 1958, to May 30, 2014. The histogram in Figure 1.6 showed us the shape of the distribution. By looking at the time plot in Figure 1.12, we now see that there is more to this data set than is revealed by the histogram. This scenario illustrates the types of steps used in an effective statistical analysis of data. We are rarely able to completely plan our analysis in advance, set up the appropriate steps to be taken, and then click on the appropriate buttons in a software package to obtain useful results. An effective analysis requires that we proceed in an organized way, use a variety of analytical tools as we proceed, and exercise careful judgment at each step in the process.