C h a p t e r   4

Summarizing Univariate Data: The Distribution Platform

The Situation at INCU

The Distribution Platform

Summarizing Quantitative Variables

Graphic Analysis Panel

Quantiles Panel

Moments Panel

Additional Moments

Summary Measures for Qualitative Variables

Frequencies Panel

Summarizing by a Qualitative Variable

Summary

Chapter Glossary

Questions and Problems

Notes

In the last chapter, we discussed visual presentations of data as tables or charts. In this chapter, we want to explore simple descriptive measures of data. These descriptive measures will consider only one variable at a time—hence, the “Univariate” in the title of this chapter. We will also introduce additional charts in this chapter that depend on these descriptive measures. The material in this chapter, besides providing the tools to more completely describe a set of data, also provides the basis for our discussion in the following three chapters, which present the foundations for statistical inference.

The default for results of the distribution platform is to display three output panes. The first pane, labeled with the name of the column, displays graphic output for the column; the second panel labeled “Quantiles” displays various quantiles. The third panel labeled “Moments” displays summary measures for that column. The outline icon allows you to close and open different output segments. The top outline icon opens and closes all of the output for that column. The drop-down menu at the top of the output allows you to specify different display options. Figure 4.4 shows the first level menu items that are displayed by clicking the symbol.

The first menu item Display Options leads to submenus. The submenu items are shown in Figure 4.5. The first two items in this submenu function like the outline icons to open and close the “Quantiles” and “Moments” portions of the output. The last menu item (Horizontal Layout) changes the layout of the display. In the default layout, the output panels are arranged vertically. Selecting the Horizontal Layout option will arrange the panels horizontally. This option operates as a toggle switch. After you switch to horizontal layout, the menu item will read Vertical Layout, and selecting it will arrange the panels vertically. The third option (More Moments) adds additional summary statistics to the Moments panel. We will look at this additional output in a moment. First, we want to discuss the three output panels in more detail and describe some of the additional options in Figure 4.5.

However, we can get additional details on the histogram by selecting additional options in JMP. The Histogram Options menu item leads to a submenu shown in Figure 4.6.

The first option determines whether the histogram panel is displayed or not. If you uncheck this option, the histogram will no longer be displayed, but the box plot will still be there. The second option changes the histogram into a shadowgram as shown in Figure 4.7. The shadowgram displays the same basic information as a histogram and is similar to what is usually called a frequency polygon, which is a line chart version of the histogram.

The third submenu option changes the orientation of the histogram from vertical to horizontal. Again, this is a toggle switch so that if it is selected again, it returns the histogram to a vertical orientation. The next option (Std Error Bars) will be discussed in Chapter 6 when we discuss confidence intervals. The next option (Set Bin Width) is used to see the width of the intervals used. As can be seen in Figure 4.3, the current interval width is 10. If you select the Set Bin Width option for Age and change the 10 to 5, you will see the histogram shown in Figure 4.8(a). You can experiment with different interval widths until you get the pattern that makes the most sense to you.

The next three options relate to the other axis of the histogram, which is currently unlabeled. The first of these (Count Axis) puts the frequency on the unlabeled axis. The second option (Prob Axis) puts the relative frequency on that axis. Recall from Chapter 3 that the relative frequency is the actual frequency for the interval divided by the number of observations. Figure 4.8(b) illustrates the graph with the frequency count on the axis. With this axis in place, we can finally see approximately what the frequencies are in each interval of the graph. The third option for the axis (Density Axis) will not be discussed in this text.

The final two options add clarity to the graph by labeling the bars of the graph with either the actual frequency for that interval (Show Counts) or the relative frequency (Show Percents). Figure 4.9 shows both options in a horizontal orientation.

The stem-and-leaf diagram for age was very straightforward since the original values were all two-digit numbers. But what happens if we have values such as 23, 413, or 31.42? In these cases, JMP must make decisions about what is represented by the stem and what the leaf means. In some situations, JMP may round off values. In this case, some information may be lost because of the rounding, but each observation is still represented in the graph, even if the value is rounded. In other situations, JMP may drop fractional values to make the graph clearer. What this means is that JMP will make intelligent choices about the significant digits in the data and will tell you at the bottom of the stem and leaf diagram what the values represent.

The primary advantage of interactivity comes in comparing the distribution of two different columns of data. Notice that in Figure 4.11 not only are the data connected to the histogram for age but also connected to the histogram for tenure. This allows us to see, for example, that all of the individuals making between $40,000 and $50,000 have tenure of 15 years or less. This interactivity between graphs allows a deeper exploration of the relationship between variables.

The quantiles panel shows the extreme quantiles that are the minimum and maximum. It also displays the first and third quartiles along with the median. In addition, it shows the quantiles close to the minimum and maximum values. This is because values close to the extremes of the distribution can sometimes be of special significance in statistics. For our purposes, the most important quantiles are the minimum and maximum values, the two quartiles, and the median. For age, the results show that the median age is 45 and that 50% of the customers are between 34.25 years and 54 years old. The youngest is 4 and the oldest is 76. The difference between the upper and lower quartiles (in this case, 54 – 34.25 = 15.75) is called the interquartile range.

For example, for the Age variable in Figure 4.12, the box is defined by the quartiles 34.25 and 54. The line near the middle of the box represents the median value—in this case 45. The “diamond” portion of the plot represents the mean value (discussed later in this chapter) along with a confidence interval. We will not focus on this part of the diagram nor the red bracket in the figure that represents the “shortest half” of the values, which is the densest portion of the data. The dotted lines emanating from the box are called whiskers.¹ The whiskers connect the box with the outermost data values that are within the fences of the box plot. The fences are not shown in the box plots produced by JMP but are computed as follows:

Upper Fence = Upper Quartile + 1.5*(Interquartile Range)

Lower Fence = Lower Quartile + 1.5*(Interquartile Range)

The premise of the fences is that most of the data should fall within this range; i.e., the data should fall between the two fences. Data that fall outside this range are “unusual” or outliers. We will return to this discussion of where values in a set of data should fall later in this chapter when we discuss the empirical rule. Outliers are shown in the outlier box plot as filled squares at either end of the whiskers. Figure 4.12 shows that there is an outlier at the lower end of the values for Age (the 4-year-old customer) and an outlier at the upper end of the distribution for Tenure (an employee who has been with INCU for 19 years). If there are no outliers, there will be no squares beyond the whiskers of the plot.

At this point, we will focus on three summary measures in this panel, the mean, the standard deviation (Std Dev in the panel), and the sample size (N in the panel). The sample size is simply the number of observations that have a value for that variable. In our data, there are no missing values so that the sample size will be 100 for all variables. Most of the results presented in the Moments panel are what are called summary measures in statistics. Summary measures are numerical values that describe a characteristic of the set of numbers. Summary measures are usually grouped into measures of central tendency, measures of dispersion, and measures of shape.

where the Σ is the summation operation. Figure 4.13 indicates that the average age of our customer sample is 44.16 years and the average length of tenure as a customer is 9.96 years. The mean has one very interesting property. This property relates to the concept of a deviation. A deviation is the difference between a value in the data and a central value such as the mean

Deviation = (X – Mean)

If we use the mean as the central value, then the following statement is always true:

Σ(X – Mean) = 0   (4.2)

With the mean, the positive and negative deviations cancel each other out. It is in this sense that the mean is the middle value of the data. We shall see in a moment that this property has important consequences.

Measures of central tendency are certainly useful in statistics, but they are not sufficient to completely describe a set of numbers. At a minimum, we also need to describe the dispersion or variability of the numbers.

X	10	10	10	10	10	10	10	10	10	10
Y	9	15	1	5	21	7	13	4	6	19

As you can verify, the mean of both sets of numbers is 10. Yet variables X and Y are certainly not equivalent. Variable Y is much more variable than X. It is this notion of variability or dispersion that we want to capture here. The standard deviation (labeled Std Dev in the Moments panel in Figure 4.13) is one common measure of variability, but there are others. For example, the interquartile range described earlier can be considered a measure of variability or dispersion. For variable X, since all of the values are the same, the quantiles would all be the same, and the interquartile range would be 0. For Y, the first quartile is 4.75, and the third quartile is 16; so the interquartile range is 11.25, indicating that the values of Y are more variable or dispersed than the values of X.

Recall that a deviation is the difference between a value in the data and a central value, usually the mean. It may be apparent that these deviations could be used as a measure of variability since they measure how far the values are from the middle. It may occur to you that the average of these deviations might be a useful measure of variability. However, there is a problem with this measure. Recall that one of the properties of the mean is that the sum of the deviations about the mean is always zero. (See equation 4.2.) This implies that the average deviation would also always be zero. There are two ways to get around the problem of the positive and negative deviations canceling each other. One is to look at the average of the absolute deviations. This measure, called the mean absolute deviation, is often used in a business environment in connection with forecasting systems. However, absolute values do not have very nice mathematical properties, and this measure is of limited use in statistics. The second way to ensure that the positive and negative deviations do not cancel each other is to square the deviations. The variance is the average of the squared deviations about the mean.

  (4.3)

The variance is a widely used measure of variability in statistics and will play a central role in our text. However, there is one small problem with the variance. The units of the variance are in the original units squared. For example, our variable age is measured in years so that the units of the variance will be in Years.² If we take the square root of the variance, then the measure will be back in the original units. This measure is called the standard deviation.

  (4.4)

The standard deviation is also reported in the Moments panel right below the mean. The mean age in our survey in Figure 4.13 is 44.16 years, and the standard deviation is 14.66 years. For the tenure variable, the mean is 9.96 months, and the standard deviation is 3.10 months. You may note that the variance is not produced in the moments report. We will see how to find this value in a moment.

At this point, you may be asking yourself why we need two very similar measures of variability, the variance and the standard deviation. After all, if we know one of these values, we know the other. If we know the standard deviation, we can square it to find the variance. On the other hand, if we know the variance, we can take the square root to find the standard deviation. The reason we need both measures is that sometimes it is mathematically useful to use the standard deviation as the measure of variability, and sometimes it is necessary to use the variance. Therefore, we will use both measures as we progress through this book.

  (4.5)

The Z score for John would then be (22 – 20)/5 = .40 and for Mary the Z score would be (1130-1000)/100 = 1.30. Therefore, Mary did better on her test than John did on his. Note that since we are subtracting the mean from each value, the mean of the Z scores would then be zero. Since we are dividing by the standard deviation, the standard deviation of the Z scores will be 1, the unit of measure. As we will see as we progress through our discussion of statistics, this process of conversion to a standard score by subtracting a value and dividing by the standard deviation is a common one in statistics.

Table 4.1 shows different values of k along with the relevant percentages from 1 to 3.

When k = 1, the results are not too interesting; but when k = 2, Chebychev’s Theorem says that for any set of numbers at least (1 – ½²) = .75 or 75% of the values must be within 2 standard deviations of the mean. The amazing thing about this theorem is that it applies to any set of numbers. In other words, you cannot make up a set of numbers that would violate this theorem. There are very few things in mathematics that apply universally.

Although Chebychev’s Theorem is remarkable in terms of its scope, we can make even more precise statements if we know something about the shape of the distribution of values. If the distribution of values is symmetrical and bell-shaped, the empirical rule applies.³ The empirical rule states that

Table 4.2 shows some values of K along with the relevant percentages for the empirical rule.

It is apparent that if we know the shape of the distribution, we can make much more precise statements about the distribution of values. In terms of the SAT example earlier, with a mean of 1000 and a standard deviation of 100, the empirical rule would tell us that 68% of the values would be between 900 and 1100, 95% would be between 800 and 1200, and virtually all of them would be between 700 and 1300. As we will see later, two standard deviations and 95% are very important values in the practice of statistics. In fact, they are so commonly accepted that they are even incorporated into civil law.⁴

Another way to think about Chebychev’s Theorem and the empirical rule is that they tell us the percent of values that should have Z scores in a particular range. For example, the empirical rule says that 68% of the values should have a Z score between -1 and +1, 95% should have a Z value between -2 and +2, and virtually all of them should have Z scores between -3 and +3.

44.16 – 2(14.66) = 14.84 years   and   44.16 + 2(14.66) = 73.48 years

Similarly, since the mean tenure of the customers in the survey was 9.96 months with a standard deviation of 3.1 months, 95% of the customers should have tenures of between

9.96 – 2(3.1) = 3.76 months   and   9.96 + 2(3.1) = 16.16 months

Skewness is important in statistics because it can impact which measures of central tendency we use. For example, both the mean and the median are measures of central tendency for quantitative variables. However, the mean is influenced by extreme values or outliers. Therefore, in a highly skewed distribution, the median may well be a better measure of a “typical” value than the mean. Most income or wealth figures are highly positively skewed because of a few high values. That is why the median is a better measure of the typical value for such variables. For example, the average salary for the New York Yankees Major League Baseball team in 2009 was over $7.7 million, but the median was only $5.2 million.⁵ This phenomenon is not restricted to the highest paying teams. The Oakland Athletics had the lowest median salary in baseball for 2009 at $410,000, but the mean salary for The A’s was over $2.2 million.

There are different measures of skewness in statistics. The skewness measure reported by JMP in Figure 4.14 is what is called moment skewness. The equation for this measure is

  (4.6)

This measure of skewness is the form most often used by statistical software but can be difficult to interpret. Zero is an absence of skewness, but rarely will the actual skewness for a set of data be exactly zero. So what value of skewness would represent “significant” or nonzero skewness? With the moment skewness measure, it is difficult to say because the size of this measure depends on the magnitude of the numbers. An approximate value for the standard error of the skewness is where n is the number of data points. If the ratio of the skewness measure to the standard error is greater than 2, then we can say that the data exhibit significant skewness. The skewness for age (-0.149) and tenure (0.271) are not significant by this criteria.⁶ Contrast this with the example of the Oakland Athletics salaries. The salary data are contained in the file Oakland Salaries.jmp. The relevant statistics for this data are shown in Figure 4.16.⁷

For this data set, the standard error of the skewness measure is Dividing the skewness of 2.36 by this value gives a ratio of about 5.1, which is much larger than 2. You can also see the large degree of skewness from the histogram and the existence of outliers from the box plot.

A second measure of skewness is conceptually simpler than the moment measure. This measure is sometimes called Pearson skewness.⁸

  (4.7)

In this equation, the key relationship is that between the mean and the median. The basic idea is that in positively skewed distributions, the mean will be pulled in an upward or positive direction and be greater than the median; and in a negatively skewed distribution, it will be pulled in a negative direction and will be less than the median. These relationships usually hold but not always. There are several counterexamples with discrete distributions that have only a few possible values.

The other measure of shape produced by the JMP output is called kurtosis. Kurtosis is a measure of how “peaked” or “flat” the distribution of values is in relation to a normal distribution. Since kurtosis is somewhat less important in statistics than skewness, we will not discuss it any further here.

The coefficient of variation is useful for comparing the variability of different sets of numbers. Since the value of the standard deviation (or variance) depends on the magnitude of the numbers, if we have two sets of data with radically different values, a simple comparison of the standard deviations would be deceptive in terms of which set of numbers was more variable. For example, in Figure 4.14, the standard deviation of the Age variable is 14.656, while the standard deviation of the Tenure variable is 3.097. This would seem to imply that age is much more variable than tenure. However, the coefficients of variation for the two variables are 33.189 and 31.099 respectively, which indicates a much smaller difference in their variability. When you want to compare the variability of two very dissimilar variables, you should use the coefficients of variation.

These options, along with the drop-down menu, are shown in Figure 4.18. One of the additional options in the drop-down menu is to display a mosaic plot. A mosaic plot presents the same basic information as the histogram but in a slightly different way. Mosaic plots will become more useful when we discuss the relationships between two qualitative variables later in the text. Figure 4.18 shows the mosaic plot for Gender. As you can see, the mosaic plot does not really add any additional information to the histogram, and that is why it is omitted by default.

As you can see, by default JMP stacks the two outputs on top of each other with the output for No Loan on the top. You can use the scroll bar on the right side of the output to scroll down to the output for those who have loans. Often it would be better to have the results side by side for easier comparison. You can do this in two ways. One is to select Horizontal Layout from the drop-down menus for each of the outputs. The second way to achieve this is to select the Stack option from the drop-down menu with Distributions No Loan. Either way will display the results as shown in Figure 4.21.

From the histogram, we can see that the distribution of ages seems to be different for those customers without loans and those who have loans. The summary measures show that the mean age of those with loans is 46.15 years while the average age of those without loans if 43.46 years. The standard deviation of the ages in the No Loan group also seems to be higher than in the Loan group. The ages of those with loans seems to be concentrated in the 30–60 year old range while the No Loan group has a broader distribution. Of course, these conclusions are appropriate only within the context of this particular sample. We cannot generalize beyond the sample to all of the customers at INCU without inferential statistics. We will see in later chapters how we can determine if these sample results can be safely generalized to the population at large.

We can also summarize quantitative variables with a BY variable. A BY variable is a qualitative variable that divides the data into groups. After viewing the results for Age, Ann Rigney wonders if there could also be differences in terms of where the customers live. In that case, a campaign targeted to different zip codes may also make sense. Figure 4.22 shows the resulting output where again, the output has been stacked for easier viewing.

From the output, it appears that those who currently have loans tend to be concentrated in the 32209 zip code (about 54%) and that a campaign focusing on the other two zip codes might be worthwhile.

For quantitative variables, important summary measures include the mean (average) value, the variance and standard deviation, and measures of shape, particularly skewness. Two different skewness measures are described: moment skewness and Pearson skewness. The standard deviation is a particularly useful measure of variability of the data. It is often used in statistics as a “unit of measure.” When used this way in standard (Z) scores, it allows us to compare values measured on different scales. It also serves as the foundation for Chebychev’s Theorem and empirical rule, which allow us to make statements about where values are likely to fall in a data distribution for quantitative variables. The coefficient of variation can be used to compare the variability of two different variables that have very different magnitudes. The concept of outliers was also introduced in this chapter. Outliers can be identified in Outlier Box Plots or by using standard scores and the empirical rule.

Qualitative variables are primarily summarized by a frequency distribution along with the associated relative frequencies or proportions. Sometimes qualitative variables are also used to split the data into groups for analysis. We can then look at how the data vary depending on the value of the qualitative variable.

1.  Analyze the data for Betty. What conclusions can you draw from the data?

2.  Betty notices that the accounts variable has been designated as a continuous variable, but it is really a discrete variable. She wonders if you should analyze this variable as an ordinal variable rather than continuous. Change the variable type and redo the analysis. Does this change your perceptions of the data? Is this a better way to analyze this variable?

3.  Liz Barton in the Marketing Department at INCU believes that there are marked differences between male and female customers of the credit union. She would like to see an analysis by gender of the variables Age, Tenure, number of accounts, and whether or not they use Bill Pay and the ATMs. Conduct the analysis for Liz. Do there appear to be any differences between males and females on these variables? What are the implications of these differences?

Andy Jordon, the director of Customer Service at INCU, is most interested in the survey results for customer satisfaction.

4.  Analyze the three customer satisfaction variables. What can you tell Andy about customer satisfaction?

5.  Andy would like to see if there are differences in customer satisfaction based on gender or location (measured by zip code). Conduct the appropriate analyses and draw up a brief report for Andy.

Betty Anderson is curious about the application of the empirical rule to their sample data.

6.  Does the empirical rule seem to hold for the age and tenure data? [Hint: One way to approach this is to create a new column based on a formula to calculate Z scores for the age and tenure variables and then sort the data on these Z scores.]