In the previous section, What to Know About Applications of Histograms: 3D, you practiced using histograms to describe a quantitative dataset. You described the shape, estimated the center and spread, and identified any outliers in the distribution. Now it’s time to use the skills you learned on a dataset involving information collected from student evaluations of their classes at a Texas University.

What Do Students Think?

Before you begin this activity, take a moment to think about a scenario in which only a low percentage (e.g., fewer than 10%) of students in your class completes the course evaluation at the end of the semester.

In this activity, you’ll see common statistical language used to describe a distribution based on what is observed from a graphical display, which you’ll describe by identifying its shape, center, spread, and any outliers present. You’ll also see that range (the difference between the maximum and minimum values) of a distribution that contains outliers or is skewed can be a misleading representation of spread.

We will investigate the question:

In general, what percentage of students completes course evaluations?

To do so, we will use the evals dataset^[1], which contains information collected from student evaluations for a sample of 463 courses taught by 94 professors at The University of Texas at Austin. Each row has a different course, and the columns have information about the professor and summaries from the evaluations. The first 10 observations of the selected variables within the “Teaching Evaluations” dataset are displayed in the following table.

The following variables are used in this analysis:

• cls_did_eval: Number of students who completed evaluations
• cls_perc_eval: Percentage of students who completed evaluations
• age: Age of professor in years
• cls_students: Total number of students in the course
• score: Average professor evaluation score (1 to 5, where 1 is the lowest and 5 is the highest)

If we are interested in course evaluation completion, we are naturally curious about how many students completed the evaluation.

Examine the data

Now, let’s create a graph to visualize the distribution of the variable of interest.

Use your histogram summarize the description of the distribution by answering Questions 3, 4, and 5.

So far, we’ve been able to use the histogram to answer questions about the distribution of cls_perc_eval. The answers to these questions give us some information about the data; however, they do not give us a broad view of the overall distribution of the variable. In addition to visualizing the distribution with a graphical display, we can use common statistical language to describe the distribution.

Before diving into the details, consider why we might want to use words to describe a distribution.

Describe the distribution

Now, in Questions 8 – 12, use statistical terms to describe the distributions of the variable cls_perc_eval. If necessary, refer to the Describing Distributions section at the end of this activity for details about how to describe a distribution.

Recall that the spread is a measure of how much the values in a dataset tend to differ from one another. One way we can describe the spread is by finding the minimum and maximum values in the data and calculating the difference between them. This difference is called the range.

Summarize the description of a distribution

Good work. You’ve thoroughly described the variable cls_perc_eval using statistical language: shape, center, spread (range) and outliers. Now it’s your turn to try it on your own. Use the features you described in Questions 8 – 12 to describe the distribution of each of the following variables:

• age: Age of professor in years
  • Dataset = “Teaching Evaluations – Age”
• cls_students: Total number of students in the course  
  • Dataset = “Teaching Evaluations – Students”
• score: Average professor evaluation score (1 to 5, where 1 is the lowest)
  • Dataset = “Teaching Evaluations – Scores”

Record your results in Questions 13 – 19 below.

Determine the appropriate representation of the spread of a distribution

Reference: Describing Distributions

The features used to describe the distribution of a quantitative variable are the shape, center, spread, and presence of outliers.

Shape: The overall pattern (left skewed, right skewed, symmetric) and the number of peaks (unimodal, bimodal, multimodal, uniform).

Center: A measure that describes where the middle of the distribution is. The center is a number that describes a typical value. For example, one way to think about center is that it could be the point in the distribution where about half of the observations are below it and half are above it.

Spread: A measure of how far apart the data are. In this lesson, the range is used to measure spread. The range is the difference between the maximum value and minimum value.

Outliers: Unusual observations that are outside the general pattern of the distribution.

The description of shape includes two parts: (1) the overall pattern (left skewed, right skewed, symmetric) and (2) the number of peaks (unimodal, bimodal, multimodal, uniform).

The overall pattern can be described as one of the following:

Symmetric: The left and right sides of the distribution (closely) mirror each other. If you drew a vertical line down the center of the distribution and folded the distribution in half, the left and right sides would closely match one another.

Left skewed: The distribution has a longer tail to the left.

Right skewed: The distribution has a longer tail to the right.

In addition to the overall pattern, the description of shape also includes the number of peaks. This is also known as the modality. The modality can be described as one of the following:

Unimodal: There is one prominent peak.

Bimodal: There are two prominent peaks.

Multimodal: There are three or more prominent peaks.

Uniform: There are no prominent peaks.

The next feature is the center. For now, we can use the histogram to get an approximate value of the center. (In a later activity, you will learn statistics used to describe the center more precisely.)

When describing the spread of a distribution that is left skewed, right skewed, or has outliers, it can be misleading to only rely on the range to measure spread, since it is influenced by skewness and outliers. In this case, the range may make the spread appear to be larger than it is for a vast majority of the data.

If this is the case, in addition to reporting the range, you can include additional information about the spread of most of the data as well. This will give the reader a more accurate and complete picture of the true spread of the data. For example, in addition to reporting the range for the distribution of cls_perc_eval, we can also include information that most of the data are between about 50% and 100%, or within 50%. (In later activities, you will learn additional statistics to describe the typical spread of the data.)

The last feature in the description is the presence of outliers. Outliers are observations in the data that are unusual and outside the general pattern of the rest of the observations in the distribution. When working with a univariate distribution for a quantitative variable, an outlier is an observation that has an unusually high or unusually low value. It is good practice to make note of outliers, as these observations can sometimes influence the statistical results (e.g., the range).