Advanced High School Statistics: Third Edition

Sometimes researchers wish to see the relationship between two variables. When we talk of a relationship or an association between variables, we are interested in how one variable behaves as the other variable increases or decreases.

A scatterplot provides a case-by-case view of data that illustrates the relationship between two numerical variables. A scatterplot is shown in Figure 2.1.1, illustrating the relationship between the number of line breaks (line_breaks) and number of characters (num_char) in emails for the email50 data set. In any scatterplot, each point represents a single case. Since there are 50 cases in email50, there are 50 points in Figure 2.1.1.

The variable that is suspected to be the response variable is plotted on the vertical (y) axis and the variable that is suspected to be the explanatory variable is plotted on the horizontal (x) axis. In this example, the variables could be switched since either variable could reasonably serve as the explanatory variable or the response variable.

The association between two variables can be positive or negative, or there can be no association. Positive association means that larger values of the first variable are associated with larger values of the second variable. Additionally, the association can follow a linear trend or a curved (nonlinear) trend.

Sometimes two variables is one too many: only one variable may be of interest. In these cases we want to focus not on the association between two variables, but on the distribution of a single, or univariate, variable. The term distribution refers to the values that a variable takes and the frequency of these values. Here we introduce a new data set, the email50 data set. This data set contains the number of characters in 50 emails. To simplify the data, we will round the numbers and record the values in thousands. Thus, 22105 is recorded as 22.

Rather than look at the data as a list of numbers, which makes the distribution difficult to discern, we will organize it into a table called a stem-and-leaf plot shown in Figure 2.1.9. In a stem-and-leaf plot, each number is broken into two parts. The first part is called the stem and consists of the beginning digit(s). The second part is called the leaf and consists of the final digit(s). The stems are written in a column in ascending order, and the leaves that match up with those stems are written on the corresponding row. Figure 2.1.9 shows a stem-and-leaf plot of the number of characters in 50 emails. The stem represents the ten thousands place and the leaf represents the thousands place. For example, 1 | 2 corresponds to 12 thousand. When making a stem-and-leaf plot, remember to include a legend that describes what the stem and what the leaf represent. Without this, there is no way of knowing if 1 | 2 represents 1.2, 12, 120, 1200, etc.

0 | 00000011111223334455566777999
1 | 0001124467
2 | 25556679
3 |
4 | 23
5 |
6 | 4
Legend: 1 | 2 = 12,000

When there are too many numbers on one row or there are only a few stems, we split each row into two halves, with the leaves from 0-4 on the first half and the leaves from 5-9 on the second half. The resulting graph is called a split stem-and-leaf plot. Figure 2.1.11 shows the previous stem-and-leaf redone as a split stem-and-leaf.

0 | 000000111112233344
0 | 55566777999
1 | 00011244
1 | 67
2 | 2
2 | 5556679
3 |
3 |
4 | 23
4 |
5 |
5 |
6 | 4
Legend: 1 | 2 = 12,000

Another simple graph for univariate numerical data is a dot plot. A dot plot uses dots to show the frequency, or number of occurrences, of the values in a data set. The higher the stack of dots, the greater the number occurrences there are of the corresponding value. An example using the same data set, number of characters from 50 emails, is shown in Figure 2.1.13.

These graphs make it easy to observe important features of the data, such as the location of clusters and presence of gaps.

Additionally, we can easily identify any observations that appear to be unusually distant from the rest of the data. Unusually distant observations are called outliers. Later in this chapter we will provide numerical rules of thumb for identifying outliers. For now, it is sufficient to identify them by observing gaps in the graph. In this case, it would be reasonable to classify the emails with character counts of 42 thousand, 43 thousand, and 64 thousand as outliers since they are numerically distant from most of the data.

Stem-and-leaf plots and dot plots are ideal for displaying data from small samples because they show the exact values of the observations and how frequently they occur. However, they are impractical for larger samples. For larger samples, rather than showing the frequency of every value, we prefer to think of the value as belonging to a bin. For example, in the email50 data set, we create a table of counts for the number of cases with character counts between 0 and 5,000, then the number of cases between 5,000 and 10,000, and so on. Such a table, shown in Table 2.1.18, is called a frequency table. Bins usually include the observations that fall on their left (lower) boundary and exclude observations that fall on their right (upper) boundary. This is called left inclusive. For example, 5 (i.e. 5000) would be counted in the 5-10 bin, not in the 0-5 bin. These binned counts are plotted as bars in Figure 2.1.19 into what is called a histogram or frequency histogram, which resembles the stacked dot plot shown in Figure 2.1.13.

Histograms provide a view of the data density. Higher bars represent where the data are relatively more common. For instance, there are many more emails between 0 and 10,000 characters than emails between 10,000 and 20,000 in the data set. The bars make it easy to see how the density of the data changes relative to the number of characters.

Sometimes questions such as the previous ones can be answered more easily with a cumulative frequency histogram. This type of histogram shows cumulative, or total, frequency achieved by each bin, rather than the frequency in that particular bin.

Knowing that 25 of the emails had fewer than a value between 5 and 10 thousand characters is useful information, but it is even more useful if we know what percent of the total 25 represents. Knowing that there were 50 total emails tells us that \(25 / 50 = 0.5 = 50\%\) of the emails had fewer than a value between 5 and 10 thousand characters. When we want to know what fraction or percent of the data meet a certain criteria, we use relative frequency instead of frequency. Relative frequency is a fancy term for percent or proportion. It tells us how large a number is relative to the total.

Just as we constructed a frequency table, frequency histogram, and cumulative frequency histogram, we can construct a relative frequency table, relative frequency histogram, and cumulative relative frequency histogram.

Frequency and relative frequency histograms are especially convenient for describing the shape of the data distribution . Figure 2.1.19 shows that most emails have a relatively small number of characters, while fewer emails have a very large number of characters. When data trail off to the right in this way and have a longer right tail , the shape is said to be right skewed.

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Other ways to describe data that are right skewed: skewed to the right, skewed to the high end, or skewed to the positive end.

Data sets with the reverse characteristic — a long, thin tail to the left — are said to be left skewed. We also say that such a distribution has a long left tail. Data sets that show roughly equal trailing off in both directions are called symmetric.

In addition to looking at whether a distribution is skewed or symmetric, histograms, stem-and-leaf plots, and dot plots can be used to identify modes. A mode is represented by a prominent peak in the distribution.

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Another definition of mode, which is not typically used in statistics, is the value with the most occurrences. It is common to have no observations with the same value in a data set, which makes this other definition useless for many real data sets.

There is only one prominent peak in the histogram of num_char.

Figure 2.1.32 shows histograms that have one, two, or three prominent peaks. Such distributions are called unimodal, bimodal, and multimodal, respectively. Any distribution with more than 2 prominent peaks is called multimodal. Notice that in Figure 2.1.19 there was one prominent peak in the unimodal distribution with a second less prominent peak that was not counted since it only differs from its neighboring bins by a few observations.

Finally, we note that the graphical summaries of this section and the numerical summaries of the next section fall into the realm of descriptive statistics. Descriptive statistics is about describing or summarizing data; it does not attribute properties of the data to a larger population. Inferential statistics, on the other hand, uses samples to generalize or to infer something about a larger population. We will have to wait until Chapter 5 to enter the exciting world of inferential statistics.