Lesson 7Using Histograms to Answer Statistical Questions

Learning Goal

Let’s draw histograms and use them to answer questions.

Learning Targets

I can draw a histogram from a table of data.
I can use a histogram to describe the distribution of data and determine a typical value for the data.

Lesson Terms

center
distribution
frequency
histogram
spread

Warm Up: Which One Doesn’t Belong: Questions

Here are four questions about the population of Alaska. Which question does not belong? Be prepared to explain your reasoning.

In general, at what age do Alaska residents retire?
At what age can Alaskans vote?
What is the age difference between the youngest and oldest Alaska residents with a full-time job?
Which age group is the largest part of the population: 18 years or younger, 19–25 years, 25–34 years, 35–44 years, 45–54 years, 55–64 years, or 65 years or older?

Activity 1: Measuring Earthworms

An earthworm farmer set up several containers of a certain species of earthworms so that he could learn about their lengths. The lengths of the earthworms provide information about their ages. The farmer measured the lengths of 25 earthworms in one of the containers. Each length was measured in millimeters.

Problem 1

Using a ruler, draw a line segment for each length:

20 millimeters
40 millimeters
60 millimeters
80 millimeters
100 millimeters

Problem 2

Here are the lengths, in millimeters, of the 25 earthworms.

Complete the table for the lengths of the 25 earthworms.

length	frequency
0 millimeters to less than 20 millimeters
20 millimeters to less than 40 millimeters
40 millimeters to less than 60 millimeters
60 millimeters to less than 80 millimeters
80 millimeters to less than 100 millimeters

Problem 3

Use the grid and the information in the table to draw a histogram for the worm length data. Be sure to label the axes of your histogram.

A grid with a scale of 10 to 100 in increments of 10 on the horizontal axis and 0 to 14 in increments of 1 on the vertical axis.

Problem 4

Based on the histogram, what is a typical length for these 25 earthworms? Explain how you know.

Problem 5

Write 1–2 sentences to describe the spread of the data. Do most of the worms have a length that is close to your estimate of a typical length, or are they very different in length?

Are you ready for more?

Here is another histogram for the earthworm measurement data. In this histogram, the measurements are in different groupings.

A histogram of worm length in millimeters.

Based on this histogram, what is your estimate of a typical length for the 25 earthworms?
Compare this histogram with the one you drew. How are the distributions of data summarized in the two histograms the same? How are they different?
Compare your estimates of a typical earthworm length for the two histograms. Did you reach different conclusions about a typical earthworm length from the two histograms?

Activity 2: Tall and Taller Players

Professional basketball players tend to be taller than professional baseball players.

Here are two histograms that show height distributions of 50 male professional baseball players and 50 male professional basketball players.

Decide which histogram shows the heights of baseball players and which shows the heights of basketball players. Be prepared to explain your reasoning.
Write 2–3 sentences that describe the distribution of the heights of the basketball players. Comment on the center and spread of the data.
Write 2–3 sentences that describe the distribution of the heights of the baseball players. Comment on the center and spread of the data.

Lesson Summary

Here are the weights, in kilograms, of 30 dogs.

Before we draw a histogram, let’s consider a couple of questions.

What are the smallest and largest values in our data set? This gives us an idea of the distance on the number line that our histogram will cover. In this case, the minimum is 10 and the maximum is 34, so our number line needs to extend from 10 to 35 at the very least.
(Remember the convention we use to mark off the number line for a histogram: we include the left boundary of a bar but exclude the right boundary. If 34 is the right boundary of the last bar, it won’t be included in that bar, so the number line needs to go a little greater than the maximum value.)
What group size or bin size seems reasonable here? We could organize the weights into bins of 2 kilograms (10, 12, 14, …), 5 kilograms, (10, 15, 20, 25, …), 10 kilograms (10, 20, 30, …), or any other size. The smaller the bins, the more bars we will have, and vice versa.

Let’s use bins of 5 kilograms for the dog weights. The boundaries of our bins will be: 10, 15, 20, 25, 30, 35. We stop at 35 because it is greater than the maximum.

Next, we find the frequency for the values in each group. It is helpful to organize the values in a table.

weights in kilograms	frequency
10 to less than 15	$5$
15 to less than 20	$7$
20 to less than 25	$10$
25 to less than 30	$3$
30 to less than 35	$5$

Now we can draw the histogram.

A histogram: The horizontal axis is labeled “dog weights in kilograms” and the numbers 10 through 35, in increments of 5, are indicated. On the vertical axis the numbers 0 through 10, in increments of 2, are indicated. The data represented by the bars are as follows: Weight from 10 up to 15, 5. Weight from 15 up to 20, 7. Weight from 20 up to 25, 10. Weight from 25 up to 30, 3. Weight from 30 up to 35, 5.

The histogram allows us to learn more about the dog weight distribution and describe its center and spread.