Lesson 5

Normal Distributions

The mathematical purpose of this lesson is to introduce the normal distribution as a way to model distributions that are approximately symmetric and bell-shaped. The work of this lesson connects to previous work because students described the shape of different distributions and discussed ways to collect data. The work of this lesson connects to upcoming work because students will apply the concepts of mean and standard deviation to data modeled using a normal distribution, and use the area under a normal curve to estimate percentiles. Students are introduced to relative frequency histograms, which are histograms that show the proportion of the entire data set that falls into specified intervals. Students also encounter the term normal distribution, which is a distribution that is symmetric and bell-shaped, has an area of 1 between the \(x\)-axis and the curve, and has the \(x\)-axis as a horizontal asymptote. When students articulate the things they notice and wonder about histograms and a curve, they are attending to precision (MP6) in the language they use to describe what they see.

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.

Identify normal distributions as a way to model distributions that are approximately symmetric and bell-shaped and explain (orally) the reasoning.

Let’s investigate a specific type of distribution called a normal distribution.

Rulers marked with centimeters

Be prepared to show students a normal curve fit to data collected from the class in the synthesis of the activity Playing the Piano. Figure out, ahead of time, how to do this quickly. If you plan to use technology, an applet is available in the synthesis.

I can calculate a relative frequency and create a relative frequency histogram.
I know that a normal curve is defined using the mean and standard deviation.

Addressing

Building Towards

HSS-ID.A.4

normal distribution

mean = 10. standard deviation = 1

mean = 10. standard deviation = 2

mean = 8. standard deviation = 2

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

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).