Lesson 3Interpreting Histograms

Learning Goal

Let’s explore how histograms represent data sets.

Learning Targets

  • I can recognize when a histogram is an appropriate graphical display of a data set.

  • I can use a histogram to get information about the distribution of data and explain what it means in a real-world situation.

Lesson Terms

  • center
  • distribution
  • frequency
  • histogram
  • spread

Warm Up: Dog Show (Part 1)

Problem 1

Here is a dot plot showing the weights, in pounds, of 40 dogs at a dog show.

A dot plot for “weight in pounds.” The numbers 60 through 180, in increments of 10, are indicated. There are tick marks halfway between each indicated number. The data are as follows: 68 pounds, 1 dot. 70 pounds, 1 dot. 72 pounds, 2 dots. 75 pounds, 1 dot. 76 pounds, 1 dot. 82 pounds, 2 dots. 85 pounds, 1 dot. 90 pounds, 4 dots. 93 pounds, 1 dot. 96 pounds, 3 dots. 101 pounds, 4 dots. 106 pounds, 1 dot. 113 pounds, 1 dot. 114 pounds, 5 dots. 119 pounds, 3 dots. 123 pounds, 1 dot. 124 pounds, 2 dots. 137 pounds, 1 dot. 139 pounds, 1 dot. 146 pounds, 1 dot. 153 pounds, 2 dots. 162 pounds, 1 dot.

  1. Write two statistical questions that can be answered using the dot plot.

  2. What would you consider a typical weight for a dog at this dog show? Explain your reasoning.

Activity 1: Dog Show (Part 2)

Problem 1

Here is a histogram that shows some dog weights in pounds.

A histogram: The horizontal axis is labeled "weight in pounds" and the numbers 60 through 180, in increments of 20, are indicated. On the vertical axis the numbers 0 through 14, in increments of 2, are indicated. The data represented by the bars are as follows: Weight from 60 up to 80, 6. Weight from 80 up to 100, 11. Weight from 100 up to 120 , 14. Weight from 120 up to 140, 5. Weight from 140 up to 160, 3. Weight from 160 up to 180, 1.

Each bar includes the left-end value but not the right-end value. For example, the first bar includes dogs that weigh 60 pounds and 68 pounds but not 80 pounds.

Use the histogram to answer the following questions.

  1. How many dogs weigh at least 100 pounds?

  2. How many dogs weigh exactly 70 pounds?

  3. How many dogs weigh at least 120 and less than 160 pounds?

  4. How much does the heaviest dog at the show weigh?

  5. What would you consider a typical weight for a dog at this dog show? Explain your reasoning.

Problem 2

Discuss with a partner:

  1. If you used the dot plot to answer the same five questions you just answered, how would your answers be different?

  2. How are the histogram and the dot plot alike? How are they different?

Activity 2: Tall and Taller Players

Problem 1

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.

  1. Decide which histogram shows the heights of baseball players and which shows the heights of basketball players. Be prepared to explain your reasoning.

    1. A histogram with data at 70-72, 74-86, and 88-90.
    2. A histogram with data ranging from 66 to 80 with a peak between 72-74.

  2. Write 2–3 sentences that describe the distribution of the heights of the basketball players. Comment on the center and spread of the data.

  3. 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

In addition to using dot plots, we can also represent distributions of numerical data using histograms.

Here is a dot plot that shows the weights, in kilograms, of 30 dogs, followed by a histogram that shows the same distribution.

A dotplot and histogram for dog weights in kilograms. For the dot plot, the numbers 10 through 35, in increments of 5, are indicated. The 30 data values are as follows: 10 kilograms, 1 dot. 11 kilograms, 1 dot. 12 kilograms, 2 dots. 13 kilograms, 1 dot. 15 kilograms, 1 dot. 16 kilograms, 2 dots. 17 kilograms, 1 dot. 18 kilograms, 2 dots. 19 kilograms, 1 dot. 20 kilograms, 3 dots. 21 kilograms, 1 dot. 22 kilograms, 3 dots. 23 kilograms, 1 dot. 24 kilograms, 2 dots. 26 kilograms, 2 dots. 28 kilograms, 1 dot. 30 kilograms, 1 dot. 32 kilograms, 2 dots. 34 kilograms, 1 dot. 35 kilograms, 1 dot. For the 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.

In a histogram, data values are placed in groups or “bins” of a certain size, and each group is represented with a bar. The height of the bar tells us the frequency for that group.

For example, the height of the tallest bar is 10, and the bar represents weights from 20 to less than 25 kilograms, so there are 10 dogs whose weights fall in that group. Similarly, there are 3 dogs that weigh anywhere from 25 to less than 30 kilograms.

Notice that the histogram and the dot plot have a similar shape. The dot plot has the advantage of showing all of the data values, but the histogram is easier to draw and to interpret when there are a lot of values or when the values are all different.