Wiki Page, Math 13 - Perez Sandbox : Module 4 - Dotplots (2b of 2)

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2026-01-17T21:15:48-08:00

Describing Center and Spread

To describe the pattern in a distribution of a quantitative variable, we describe more than the shape. We also describe the center and spread. Later in this module, we develop more precise ways to identify the center of a distribution and to measure the spread. For now, we discuss these concepts informally.

Introduction to the Concept of Center

When we describe a distribution of a quantitative variable, it is helpful to identify a typical value. We choose a single value of the variable to represent the entire group. This is one way to think about the center of the distribution.

Introduction to the Concept of Spread

We also want to describe how much the data varies among individuals in the group. Variability is another word for spread. We describe the spread in two ways:

We look at the smallest value and the largest value to describe an overall range in the data.

We also describe an interval of typical values to represent common variable values for the group.

EXAMPLE Describing shape, center and spread

This short movie illustrates how to describe the shape, center and spread of a distribution using a dotplot. (3:52 minutes)

We will now compare and contrast two distributions. We use a categorical variable to create the two groups. For each group we have a distribution of the quantitative variable. Again we use shape, center and spread in our description.

EXAMPLE Comparing two distributions

In this example, we compare the distribution of sugar content in adult cereals and child cereals. The categorical variable is Cereal (Adult, Child). The quantitative variable is sugar content.

Compare the shapes:

The sugar content in adult cereals is skewed to the right. Many adult cereals have less than 8 grams of sugar in a serving. A smaller number of adult cereals contain high amounts of sugar. The sugar content for children’s cereals is skewed to the left. Many children’s cereals have more than 8 grams of sugar in a serving, with a smaller number of children’s cereals with low amounts of sugar.

Comment: There is nothing special about the number 8. We chose the number 8 as a convenient reference point to describe the opposite trends in these two distributions.

Compare the centers:

A typical adult cereal has 3 grams of sugar in a serving. A typical children’s cereal has 12 grams of sugar in a serving.

Comment: Here we looked at the most common value in each distribution. We develop more precise ways to describe the center of a distribution in the next section. For now, just choose a reasonable typical value to represent the group.

Compare the spreads:

Overall interval and range: Adult cereals vary from 0 to 14 grams of sugar in a serving. Children’s cereals vary from 1 to 15 grams. So both types of cereal vary over a range of 14 grams.
(Note: Overall range = highest value – lowest value. For adult cereals: 14 – 0 = 14. For children’s cereals: 15 – 1 = 14)

Typical interval: Typical adult cereals have between 0 and 6 grams of sugar in a serving, compared to 9 to 13 grams in typical children’s cereals.

Comment: Here we looked at clumps in the data to identify an interval of typical values. We develop more precise ways to describe the spread a distribution in the last two sections of this module.

Pulling it altogether:

When comparing two distributions, we usually tie all of these ideas into one paragraph.

Here is a paragraph that does a good job comparing the two sugar distributions without using formal vocabulary:

In this sample, children’s cereals have more sugar per serving than adult cereals. A typical child cereal has 12 grams of sugar in a serving compared to a typical adult cereal with only 3 grams of sugar. Of course sugar amounts vary in both groups. Child cereals vary from 1 to 15 grams of sugar in a serving, compared to 0 to 14 grams for adult cereals. However, even in light of this variability, we can still say that children's cereals have more sugar when we look at how the data falls within these intervals. For example, many adult cereals have 0 to 6 grams of sugar in a serving, but not many child cereals have these small amounts of sugar. Typical children’s cereals to have 9 to 13 grams of sugar per serving, well above the typical amounts found in adult cereals.

Here is a paragraph that uses more formal vocabulary to summarize the comparison:

In this sample, children’s cereals have more sugar per serving than adult cereals. The distribution of sugar in children’s cereals is skewed left, which means that fewer child cereals have smaller amounts of sugar. Typical children’s cereals have 9 to 13 grams of sugar per serving with a center of about 12 grams. The distribution of sugar in adult cereals is skewed right, which means that fewer adult cereals have larger amounts of sugar. Typical adult cereals have 0 to 6 grams of sugar per serving with a center of about 3 grams. Both groups of cereals have an overall range of 14 grams, with the sugar in a serving of child cereal varying from 1 to 15 grams, compared to adult cereals that vary from 0 to 14 grams. Despite the identical range in sugar amounts for each group, child cereals clearly tend to have larger amounts of sugar when we examine the shape, center and spread in these distributions.

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