Lesson 8Describing Distributions on Histograms

Learning Goal

Let’s describe distributions displayed in histograms.

Learning Targets

  • I can describe the shape and features of a histogram and explain what they mean in the context of the data.

  • I can distinguish histograms and bar graphs.

Lesson Terms

  • center
  • distribution
  • frequency
  • histogram
  • spread

Warm Up: Which One Doesn’t Belong: Histograms

Problem 1

Which histogram does not belong? Be prepared to explain your reasoning.

  1. A histogram with data between 75 and 125
  2. A histogram with data ranging between 55 and 105
  3. A histogram with data ranging from 55 to 135 and 145 to 155
  4. A histogram with data ranging from 75 to 125

Activity 1: Sorting Histograms

Problem 1

  1. Your teacher will give your group a set of histogram cards. Sort them into two piles—one for histograms that are approximately symmetrical, and another for those that are not.

  2. Discuss your sorting decisions with another group. Do both groups agree which cards should go in each pile? If not, discuss the reasons behind the differences and see if you can reach agreement. Record your final decisions.

    • Histograms that are approximately symmetrical:

    • Histograms that are not approximately symmetrical:

  3. Histograms are also described by how many major peaks they have. Histogram A is an example of a distribution with a single peak that is not symmetrical.

    Which other histograms have this feature?

  4. Some histograms have a gap, a space between two bars where there are no data points. For example, if some students in a class have 7 or more siblings, but the rest of the students have 0, 1, or 2 siblings, the histogram for this data set would show gaps between the bars because no students have 3, 4, 5, or 6 siblings.

    Which histograms do you think show one or more gaps?

  5. Sometimes there are a few data points in a data set that are far from the center. Histogram A is an example of a distribution with this feature.

    Would you describe any of the other histograms as having this feature? If so, which ones?

Activity 2: Getting to School

Problem 1

Your teacher will provide the data that your class collected on how students travel to school and their travel times.

  1. Use the data to draw a histogram that shows your class’s travel times.

    A blank histogram: The horizontal axis is labeled “travel time in minutes” and the numbers 0 through 75, in increments of 5, are indicated. The vertical axis has 30 horizontal gridlines indicated.

  2. Describe the distribution of travel times. Comment on the center and spread of the data, as well as the shape and features.

  3. Use the data on methods of travel to draw a bar graph. Include labels for the horizontal axis.

    A blank grid with 40 horizontal units and 21 vertical units.

  4. Describe what you learned about your class’s methods of transportation to school. Comment on any patterns you noticed.

  5. Compare the histogram and the bar graph that you drew. How are they the same? How are they different?

Are you ready for more?

A stem and leaf plot is a table where each data point is indicated by writing the first digit(s) on the left (the stem) and the last digit(s) on the right (the leaves). Each stem is written only once and shared by all data points with the same first digit(s). For example, the values 31, 32, and 45 might be represented like:

Key: means

Problem 1

A class took an exam and earned the scores:

  • 86,

  • 73

  • 85

  • 86

  • 72

  • 94

  • 88

  • 98

  • 87

  • 86

  • 85

  • 93

  • 75

  • 64

  • 82

  • 95

  • 99

  • 76

  • 84

  • 68

  1. Create a stem and leaf plot for this data set.

  2. How can we see the shape of the distribution from this plot?

  3. What information can we see from a stem and leaf plot that we cannot see from a histogram?

  4. What do we have more control of in a histogram than in a stem and leaf plot?

Lesson Summary

We can describe the shape and features of the distribution shown on a histogram. Here are two distributions with very different shapes and features.

Two histograms, labeled “A” and “B” where the horizontal axis has the numbers 10 through 30, in increments of 2, indicated and on the vertical axis, the numbers 0 through 6 are indicated. The data represented by the bars on histogram “A” are: 10 up to 12, 0; 12 up to 14, 1; 14 up to 16, 2, 16 up to 18, 4; 18 up to 20, 5; 20 up to 22, 6; 22 up to 24, 4, 24 up to 26, 3; 26 up to 28, 2; 28 up to 30, 1. The data represented by the bars on histogram “B” are: 10 up to 12, 5; 12 up to 14, 3; 14 up to 16, 2; 16 up to 18, 2; 18 up to 20, 1; 20 up to 22, 0; 22 up to 24, 1; 24 up to 26, 3; 26 up to 28, 2; 28 up to 30, 1.

  • Histogram A is very symmetrical and has a peak near 21. Histogram B is not symmetrical and has two peaks, one near 11 and one near 25.

  • Histogram B has two clusters. A cluster forms when many data points are near a particular value (or a neighborhood of values) on a number line.

  • Histogram B also has a gap between 20 and 22. A gap shows a location with no data values.

Here is a bar graph showing the breeds of 30 dogs and a histogram for their weights.

A bar graph and a histogram. The bar graph has three categories on the horizontal axis labeled "pugs," "beagles," and "German shepherds." The vertical axis has the numbers 0 through 12, in increments of 3, indicated. The bar labeled “pugs” has a height of 9. The bar labeled “beagles” has a height of 9” and the bar labeled “German shepherds” has a height of 12. The histogram has a horizontal axis labeled “dog weights in kilograms.” The vertical axis has the numbers 0 through 10, in increments of 2, indicated. The data represented by the bars are as follows: 10 up to 15 kilograms, 5; 15 up to 20 kilograms, 7; 20 up to 25 kilograms, 10; 25 up to 30 kilograms, 3; 30 up to 35 kilograms, 5.

Bar graphs and histograms may seem alike, but they are very different.

  • Bar graphs represent categorical data. Histograms represent numerical data.

  • Bar graphs have spaces between the bars. Histograms show a space between bars only when no data values fall between the bars.

  • Bars in a bar graph can be in any order. Histograms must be in numerical order.

  • In a bar graph, the number of bars depends on the number of categories. In a histogram, we choose how many bars to use.