Lesson 6

Areas in Histograms

Lesson Narrative

The mathematical purpose of this lesson is to apply the concepts of mean and standard deviation to data modeled using a normal distribution and to use the area under a normal curve to estimate percentiles. The work of this lesson connects to previous work because students were introduced to the normal distribution as a way to model distributions that are approximately symmetric and bell-shaped. The work of this lesson connects to upcoming work because students will use the normal distribution to estimate the proportion of data values falling within given intervals. When students make connections between histograms, normal distributions, the mean, and the standard deviation to estimate population proportions, students are reasoning abstractly and quantitatively (MP2).

Technology isn’t required for this lesson, but there are opportunities for students to choose to use appropriate technology to solve problems. We recommend making technology available.

Learning Goals

Teacher Facing

  • Describe (orally) the relationship between areas in a histogram and proportions of data.
  • Generalize (orally and in writing) that normally distributed data always have the same proportion of values in an interval with endpoints described by the mean and standard deviation.

Student Facing

  • Let’s find proportions of data in certain intervals.

Required Materials

Required Preparation

A copy of the blackline master should be made available for each group of 2 students for the activity Story Submissions

Learning Targets

Student Facing

  • I can calculate a proportion of a set of data that matches a shaded area in a histogram.
  • I recognize the patterns of proportions that occur in distributions that are approximately normal in shape.

CCSS Standards

Addressing

Building Towards

Glossary Entries

  • normal distribution

    mean = 10. standard deviation = 1

    A bell-shaped distribution.

    mean = 10. standard deviation = 2

    A bell-shaped distribution.

    mean = 8. standard deviation = 2

    A bell-shaped distribution.

    A specific distribution in statistics whose graph is symmetric and bell-shaped, has an area of 1 between the \(x\)-axis and the graph, and has the \(x\)-axis as a horizontal asymptote. 

  • relative frequency histogram
    Histogram. 

    A histogram where the height of each bar is the fraction of the entire data set that falls into the corresponding interval (that is, it is the relative frequency with which the data values fall into that interval).