Wiki Page, Math 13 - Perez Sandbox : Module 4 - Histograms (1 of 4)

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2026-01-17T21:15:49-08:00

Histograms

Here we continue our discussion of graphs that summarize the distribution of a quantitative variable.

Recall that our goal in data analysis is to describe patterns in data and create a useful summary about a group. When a graph summarizes the distribution of a variable, we can see

the possible values of the variable.

the number of individuals with each variable value or interval of values.

A histogram is another way to display the distribution of a quantitative variable. Histograms are particularly useful for large data sets. A histogram divides the variable values into equal-sized intervals. We can see the number of individuals in each interval, but we cannot see individual data values.

How histograms relate to dotplots

Let's start our introduction to histograms by looking at how they relate to dotplots.

Below we have three graphs of the same set of hip girth measurements for 507 adults who exercise regularly. (Hip girth is the measurement around the hips.) There is a dotplot, a dotplot with bins, and a histogram.

Dotplot

From the dotplot, we can see that the distribution of hip measurements has an overall range of 79 to 128 cm. For convenience, we started the axis at 75 and ended the axis at 130.

Dotplot with Bins

To create a histogram, divide the variable values into equal-sized intervals called bins. In this graph, we chose bins with a width of 5 cm. Each bin contains a different number of individuals. For example, 48 adults have hip measurements between 85 and 90 (but not including 90) cm, and 97 adults have hip measurements between 100 and 105 (but not including 105) cm.

Histogram

Here is a histogram. Each bin is now a bar. The height of the bar indicates the number of individuals with hip measurements in the interval for that bin. As before, we can see that 48 adults have hip measurements between 85 cm and 90 cm (including 85, but not including 90). Similarly, 97 adults have hip measurements between 100 cm and 105 cm (including 100, but not including 105).

In the histogram, the count is the number of individuals in each bin. For each bin, you can assume that values at the left-hand endpoint of the bin are included in the bin, but the values at the right-hand end-point are not included.

The count is also called the frequency. From these counts, we can determine a percentage of individuals with a given interval of variable values. This percentage is called a relative frequency.

See the transition from dotplot to histogram in a short movie

This short movie walks through the transition from a dotplot to a histogram to reinforce what you just read. It also gives a preview of how to use a histogram to find relative frequencies, which is covered in the examples below. (6:09 minutes)

Analyzing a histogram using percentages (relative frequencies)

The next examples will illustrate how we use frequencies (counts) to compute relative frequencies (percentages.) Here we work some examples of calculating a relative frequency from a histogram.

EXAMPLE Hip measurements

Approximately what percentage of the sample shown in the histogram above has hip measurements at least 85 cm but less than 90 cm?

Answer: Of the 507 adults in the data set, 48 have hip measurements between 85 and 90 cm (including 85 but not including 90).

48 out of 507 is .

So approximately 9.5% of the adults in this sample have hip girths between 85 and 90 cm.

EXAMPLE Men's heights

1) A sportswear manufacturer plans to produce three sizes of sweatpants for men. Size Large will fit men who are 185 cm or taller. The histogram above shows heights for a sample of 247 men. What percentage of this sample will wear size Large sweatpants?

Answer: There are 247 men in the data set. We need to determine how many are 185 cm or taller, then convert this to a percentage.

2) How many are 185 cm or taller?

Answer: In the histogram these men are in the last three bins. Add the heights of the bins to determine the number of men: 28+12+2= 42.

What percentage of the men are 185 cm or taller? 42 out of 247. Divide to get about 0.170; multiply by 100% to convert to percentage. 17.0% percent of the men will wear size Large sweatpants.

EXAMPLE Movie budgets

The histogram shows the budget distribution for a sample of movies that are not action/adventure movies. Budgets are in millions of dollars.

What percentage of the sample has a budget less than $100 million?

Answer: We can see from the histogram that movies in the first two bins have a budget less than $100 million. This is 18 movies. Now we need to determine how many movies are in the sample altogether by adding up the number of movies in all of the bins: 9+9+2+8+2+1=31.

18 out of 31 is about 0.581. Multiply by 100% to get 58.1%. About 58.1% of the movies in this sample have budgets less than $100 million.

Describing shape, center and spread using a histogram

With a histogram, we can see the shape of a distribution and we can identify an interval of typical values. But we lose some precision when describing the center and spread. Because the data is in bins, we cannot see individual data values. Therefore, it is not possible to identify a single data value to represent the center of the distribution. We also cannot calculate the range because we cannot see the distribution's minimum and maximum values in the histogram.

Later we will provide more examples and practice describing and comparing distributions using histograms.

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