# Exploring Standard Deviation

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## Objective

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

#### Big Idea

How can we find the area under the normal curve? Students use what they know about the empirical rule and graphing calculators to estimate areas under the normal curve.

## Opening

20 minutes

I begin today's class with a warm up activity that gives students the opportunity to calculate standard deviation by hand one more time. I'll be showing them how the calculator can help them later in today's lesson.  I post or hand out the warm up activity, Snowfall in Boston Warm Up and let students get to work.  As in the previous lesson, I may want to give them a Standard Deviation Graphic Organizer to help them keep track of their calculations.

As students finish the warm up, I have them compare their answers with a partner.  I leave time at the end of the opening for students to ask questions and clear up any confusion.

Once everyone agrees on the standard deviation and the number of data items within one standard deviation of the mean, I show students how the calculator can help them find standard deviation a lot faster.  I use TI-84 Plus calculators in my classroom and use the STAT button to list data items and then CALC 1: 1-Var Stats to show the mean and standard deviation.  I like to use wabbitemu software on my Smartboard to show students how to use the graphing calculator in real time.

## Investigation

30 minutes

In the next section of class, students will be applying what they know about standard deviation and the normal distribution.  Again, I like to have students do this work by hand first, so they have a better understanding of how normal distribution works and then show them how the calculator can be a tool for them. I find if I show students how to use the calculator too early in the process, they miss out on some of the conceptual understanding of how to find the area underneath the normal curve.  I hand out SAT Scores and let students work in pairs.  I let students know that for now, they can make an estimate on Question 4, though I will be showing them how to find a more exact value later in the class.

Issues to watch for:

• If students have trouble getting started, I ask them to sketch a normal curve and fill in the information they know.
• Students may need to look back at their notes to remember what percentages fall under the different deviations of the normal curve.
• In my experience, students often have trouble calculating the percentages of half of the area that they know.  For example, in Question 1, students will need to figure out how much of the area is between the first and second deviation.  Again, I ask them to fill in the values they already know and then ask them if they know 68% and 95%, how can they figure out half of where they overlap?  They can use a similar procedure to find the remaining 2.5%.
• Keep an eye out for students who use different strategies to find the percentages. I ask them to share out their methods in the whole group share out.

## Discussion + Closing

10 minutes

I start the discussion of today's work by asking my students to share out how they found their answers for Questions 1 though 3. I focus on SMP 3: Construct viable arguments and critique the reasoning of others here.  It is important that my students understand how to find the different areas under the curve, not just the ones they are already familiar with (68 - 95 - 99.7).  I make sure to give students the opportunity to share the different ways they found those percentages.

Next, we take a look at Question #4. I ask my students to share what percentile they think the score will be in.  I may need to review the idea of percentiles and I often use something like test scores or baby weights to remind students how percentiles work.  Next, I show students how the calculator can help them to find a more exact answer to Question 4.  Before using the calculator, I let students know they will need to know the lower bound, upper bound, mean, and standard deviation for their data.  I try to elicit the lower and upper bound from students by asking them what are they want to know about.  They will probably see that the upper bound should be 750, the score they are curious about.  I let them reason about what the lower bound should be.  Of course, they already know the mean and the standard deviation for this problem. I show students how to enter these values using DISTR > normalcdf and explain that the value the calculator gives them means that 99.18% of the data is less than a score of 750.  This means the score 750 would be in the 99th percentile.

Lastly, I give the students the opportunity to practice using the calculator again to check their work on the previous questions.

I close class with an exit ticket, giving students the opportunity to reflect on today's work. I ask them to think about their work with standard deviation and normal distribution. I ask them to respond to the following prompt:

What steps in the process are you most confident about?

I might use their responses at a later time to pair students who are confident about different parts of the process and have them work on a problem together.