# Standard Deviation

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## Objective

SWBAT calculate standard deviation.

#### Big Idea

How can we measure how spread out a data set is? Students learn how to calculate standard deviation and apply it to some data sets they have been working with.

## Opening

30 minutes

I begin today's class by reminding students that they have been trying to determine how to decide how spread out a set of data is from the mean.  They have been comparing four sets of data, all with the same mean and evaluating different methods to examine how spread out they are.  I let students know that today they will learn a statistical method called standard deviation that is used in math to show much variation from the mean there is in a set of data. (For more detail on the conceptual trajectory of recent lessons, see Standard Deviation Opening).

I lead the class using Standard Deviation and I ask my students to calculate the standard deviation of the data set along with me. As we work, here are some of the issues that I keep in mind

• When calculating the difference from the mean, I let my students know it doesn't matter if they subtract the smaller value from the larger value.  I remind them of absolute value, or let them know that when the square the difference, any potential negatives would be eliminated anyway.
• My students should recognize the first steps from mean absolute deviation.  I let them know that standard deviation takes this process a step further.
• I make sure to spend a lot of time labeling the normal distribution curve with students so they understand how to calculate the values that match up with the deviation lines. I have found my students understand the idea of standard deviation, and where one, two, and three deviations away are, but they sometimes have trouble connecting those points to the data sets they are working with.
• I try to elicit from my students the idea that if 99.7% of data fall within three standard deviations of the mean, they should be able to determine what makes a data item rare or unusual.

## Investigation

25 minutes

Next,  students take some time to work on calculating a standard deviation on their own. There are a lot of steps in this process. My students may need help keeping track of them. I find a Graphic Organizer to be helpful for my students.  Students can either work on calculating the standard deviation of each data set they worked on in the previous lesson, Spread Out, or they could work on some sample data sets like the ones in Standard Deviation Practice.

Toward the end of class, I ask students how the standard deviation statistic helps them describe which data set is the most spread out and which has the least variation. If time allows, I have my students compare this new ranking with their initial rankings from the previous class.

As my students work, I circulate and check for understanding. I usually find that some students realize they can work across the table, rather than down each column.  I might bring this strategy to the attention of the other students and let them choose the method that works best for them.

I remind students that they will need to be accurate in their calculations.  We usually decide as a class how we will deal with rounding statistical measures.

## Closing

5 minutes

As I close today's class, I let students know that knowing the standard deviation of a data set will give us useful information about how rare or common a result will be.  I let students know we will be applying this tool to different problem situations in our upcoming lessons.

As a closing activity, I ask my students to write their own steps for calculating standard deviation. They can keep these in their binders for future reference.  I write the following prompt on the board:

In order to help you calculate standard deviation in the future, write down the steps to finding standard deviation in your own words.

I ask students to imagine they are creating a standard deviation "cheat sheet" for a student who has never taken this class.  How can they clearly describe how to calculate standard deviation?