Graphing Quantitative Data – Distribution and Shape

In topic D1, you were introduced to a dataset from the U.S. Census Bureau that contained the percentages of adults completing high school and college for all states and the District of Columbia. Here are all of the data for the high school completion rates.

This data is stored in the StatCrunch data file hs_completion_rates.txt.

You may recall from Topic D1 that these completion rates were collected in order to answer questions similar to the following:

What is a typical high school completion rate for the states?
Are there sizable differences in the completion rates for states? Can states with high and low rates be identified?
Are there particular states that stand out (either in the high end or in the low end) in terms of getting their high school students to graduate?

The first step in exploring these completion rates and answering the questions is to construct a graph. For quantitative data, a graph can help illuminate patterns—such as frequently occurring values, clusters, and gaps—that are not easily visible from a table of values. In a sense, a graph gives you a way to see how the variable actually varies.

A simple graph that is easy to construct by hand is a dotplot (often called a line plot in the school curriculum). You draw a number line covering the smallest and largest data values and then you place a dot on the number line for each data value. Here is a dotplot of the high school completion rates. This dotplot was created with Fathom, a software package designed to collect, explore, and analyze data. Note that the dots are placed on a higher level when they start to overlap. For example, there is a dot placed on a second level around 86 percent and the three dots corresponding to states with graduation rates of 90.3 percent are placed on three levels.

Display D2.9: Dotplot of high school completion rates.

From the graph alone, you may begin to see answers to the questions at hand. For example, there are fairly sizeable differences—ranging between 79.6 and 91.7 percent—in the high school completion rates. And because so many values are clustered between 87 and 92 percent, a typical completion rate might be somewhere in this interval.

What you see from the dotplot are the values that the variable takes and how often each occurs; this is called the distribution of high school completion rates. What do you look for in this distribution?

SHAPE

For one thing, you look at the general shape of the data. To help understand the shape of the data, you can draw a smooth curve over the dots in the dotplot. The curve helps you focus on the general pattern of the data rather than small gaps and clusters amongst the individual values.

Display D2.10: Smooth curve approximating the shape of the distribution of high school graduation rates.

There are several common shapes that you may see in the smooth curve that you draw over the dotplot.

Uniform

In a uniform distribution, each data value occurs at roughly the same frequency. The overall shape of the distribution is flat.

Symmetric
or
Symmetric describes a distribution in which the data values drop off (or increase) at the same rate at the left and right ends. You can imagine dividing the data into two halves, where the left half is approximately a mirror image of the right.
The symmetric distribution illustrated on the left—when the majority of the data values are in the middle—is frequently described as bell-shaped in school textbooks.

Skewed

A distribution is called skewed right when the data values pile up at the low end and the frequency of data values decrease slowly as you move right toward larger values.

Conversely, skewed left is when the data values pile up for large values and the frequencies decrease slowly as you move left toward smaller values.

To help understand different shapes, consider these graphs of four batches of test scores corresponding to some hypothetical classes (Display D2.11). The shapes of the datasets are indicated by smooth curves drawn over the graphs. The scores of Class 1 have a symmetric shape with the frequencies of students scoring less than 50 dropping off in the same way as the frequencies of students scoring more than 50. Class 2's scores have a symmetric "u-shaped" distribution. For this class it was common to score close to 100 or close to 0. The scores of Class 3 are skewed right where low scores were the most common, and Class 4 scores are skewed left with a large number of high scores. You would probably be happiest with the test scores of Class 4, although the shape of the test scores for Class 1 is customary for standardized tests.

Display D2.11: Distributions of test scores for four hypothetical classes.

For the high school completion data, you see from the dotplot in Display D2.10 that there is a tight clump of states with high completion rates between 88 and 92 percent and there is a wide interval of states with low completion rates between 78 and 84 percent. So, the shape of the high school completion rates is somewhat skewed left.

CENTER

When examining a distribution, you also look for a center, or representative data value. You will learn more precise definitions in later topics, but for now look for a representative value that is located in the center of the distribution.

By looking at the dotplot in Display D2.10, you see that the center of the entire distribution is about 87. So, it might be reasonable to say that 87 percent is a representative high school completion rate for a state.

SPREAD

Along with noticing a central value, you want to say something about the spread of the percentages. The spread helps illuminate characteristics of the data that the center alone cannot. For example, consider two hypothetical classes of three students that take the same test. One class scores 40, 70, and 100, and the other class scores 69, 70, and 71. While both classes have a center of 70—and would appear to be identical if you were told only the center—describing the spread of the values would emphasize how drastically different the two classes performed.

Like the center, there are different ways of defining spread, and you will learn precise definitions later. One quick way to describe the spread is to name the interval that contains all of the values. From inspecting the data table in Display D2.8, you see that the minimum and maximum rates are 79.6 and 91.7 percent, respectively. So, the high school completion rates for all states and the District of Columbia fall in the interval [79.6, 91.7].

The interval that contains all of the data might be wide if there are one or more unusually small or large values. As an alternative, you can give an interval that contains a particular high percentage of the data. Looking at the dotplot in Display D2.9, you see that a large number of the values fall between 85 and 91 percent. Actually, 31 states have a high school completion rate greater than or equal to 85 percent but less than 91 percent. Because there are 51 states and 31/51 = 0.61, you may further describe the spread by saying, "About 61% of the states have high school completion rates between 85 and 91 percent."

INTERESTING FEATURES

In addition to talking about the shape, center, and spread of the data, you also want to point out any interesting features of the distribution. These could include

unusually small or large data values that stand out from the majority of the data; these are commonly called outliers

gaps in the data
clusters of several data values

For the high school completion rate data, no unusually small or large rates stand out. But it is interesting to note that the two smallest rates (79.6 and 79.9) correspond respectively to Texas and Mississippi, two southern states, and the three highest rates (91.6, 91.6 and 91.7) correspond to Alaska, Minnesota, and Wyoming, three northern states. This suggests that there might be a relationship between a state's high school completion rate and its location. You'll explore this relationship further in a later topic.

PRACTICE: Graphing Quantitative Data

Here are the daily high temperatures (in degrees Fahrenheit) for the city of Atlanta, Georgia, in the month of March 2005.

This data is stored in the StatCrunch data file temps_atlanta.txt.

1. Construct a dotplot of the temperatures on the scale below.

2. Draw a smooth curve over the dotplot and describe the shape of the distribution of temperatures.

3. Give a center value for the temperatures and describe the spread by giving an interval that contains a high percentage of the data.

4. Are there any interesting features about these data, such as outliers, gaps, or clusters?

5. REFLECTION Is there any possible explanation for the large spread in temperatures in the data? (Think about how the weather in Atlanta changes in March.)

6. REFLECTION If you collected and graphed the daily high temperatures for Miami, Florida, in March 2005 and compared the distribution to the one above for Atlanta, would you expect to see any differences in the shape, center, and spread? How do you think the distribution of daily high temperatures in March 2005 for Minneapolis, Minnesota, would compare? Explain.