# Bell-Shaped Distributions and the Normal Model

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## Objective

SWBAT use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages.

#### Big Idea

For mound-shaped, symmetric distributions, the Normal Model is a good approximation. Applying the normal model allows us to predict population percentages and percentile values using the Empirical Rule.

## Warm Up

20 minutes

For today's warm up, I ask my students to examine the posters made in the previous class.  As they look at the posters, I ask each student to fill out Warm up Poster Scavenger Hunt, to keep them looking closely at the work of their peers.

Today's lesson will focus on distributions that can be modeled with the normal curve.  The point of this warm-up is to honor the work completed by groups in the previous day's class and focus in on distributions that can be approximated with the normal model [MP5, MP7].

40 minutes

Common Core Standard S-ID.4 requires students to "use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages." This fits well with the modeling theme of Algebra 2 and I emphasize the modeling aspect of this standard [MP4].

I begin the discussion by presenting some summary statistics for our school's recent SAT math scores. I tell my students that the scores for our school was approximately bell-shaped, with a mean score of 520 and a standard deviation of 98.  I ask students to brainstorm with their table partners what they know about the scores based on what I have shared.  I hope that they will make a sketch of a bell shaped distribution and consider things like the median, minimum and maximum scores and percentiles.

As they discuss, I distribute SAT questions, a set of 8 questions about the SAT math scores at our school.  Some of the questions can be answered with the given information (what is the median of the data set? What percent of students scored above 600?) and some cannot (what percent of students scored better on math than verbal? What was the highest score?)  I like to print these questions on cardstock to that students can sort them easily.

I ask students to sort the cards into two piles: questions that can be answered with the given information and questions that cannot.  Students then work on answering the questions that are possible to answer [MP3].

My goal in presenting the normal model through this series of questions is to help students understand the point of a statistical model [MP4].  I find that my students often do not understand that a statistical model helps us "go beyond" the given information.  Knowing just the shape, mean and standard deviation of a distribution can be enough information to predict population percentages and percentile scores, as long as the distribution is symmetric and mound-shaped.

## Structured Discussion about the Normal Distribution

20 minutes

After the groups have considered the questions, we come together as a class to discuss each of the questions.  I draw the distribution of SAT scores on the board and give students notes about the normal model and the Empirical Rule [MP4].  I explain how the Empirical_Rule can be used to answer some questions about the distribution...

• What interval?
• Centered on the mean?
• Did about 95% of our students score between ____ and ____?

...but not others

• What was the score that 70% our students scored higher than?

## Wrap up and Assignment

10 minutes

After taking notes, asking questions, and discussing the Empirical Rule, students will complete Exit Ticket Empirical Rule.  This is a set of 3 questions about the SAT Math distribution presented earlier in the lesson.  Students answer these questions using what they have learned in class today about the Empirical Rule [MP4].

The assignment for the evening is an Empirical Rule Worksheet, a set of 7 multi-part questions that can be answered by applying the Empirical Rule.  The answers to this worksheet will be available to students on Edmodo. My students will also take a quiz tomorrow on describing one-variable distributions, the normal model and the Empirical Rule.