Unit 3, Lesson 12

Standard Deviation

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Integrated Math 1

12.1

Warm-up

Notice and Wonder: Measuring Variability

What do you notice? What do you wonder?

<p>Dot plot from 5 to 15 by 1's. Beginning at 5, number of dots above each increment is 0, 1, 1, 2, 3, 4, 3, 2, 1, 1, 0.</p> — mean: 10, MAD: 1.56, standard deviation: 2

<p>Dot plot from 5 to 15 by 1's. Beginning at 5, number of dots above each increment is 0, 3, 3, 3, 3, 3, 3, 3, 3, 3, 0.</p> — mean: 10, MAD: 2.22, standard deviation: 2.58

<p>Dot plot from 5 to 15 by 1's. Beginning at 5, number of dots above each increment is 0, 11, 10, 8, 5, 3, 5, 8, 10, 11, 0.</p> — mean: 10, MAD: 2.68, standard deviation: 2.92

<p>Dot plot from 5 to 15 by 1's. Beginning at 5, number of dots above each increment is 0, 1, 2, 3, 7, 15, 7, 3, 2, 1, 0.</p> — mean: 10, MAD: 1.12, standard deviation: 1.61

<p>Dot plot from 5 to 15 by 1's. Beginning at 5, number of dots above each increment is 0, 2, 4, 6, 4, 3, 4, 6, 4, 2, 0.</p> — mean: 10, MAD: 2.06, standard deviation: 2.34

<p>Dot plot from 5 to 15 by 1's. Beginning at 5, number of dots above each increment is 0, 0, 0, 0, 0, 35, 0, 0, 0, 0, 0.</p> — mean: 10, MAD: 0, standard deviation: 0

Are You Ready for More?

12.2

Activity

Investigating Standard Deviation

Use technology to find the mean and the standard deviation for the data in the dot plots.

What do you notice about the mean and standard deviation that you and your partner found for the three dot plots?
Invent some data that fit the conditions. Be prepared to share your data set and reasoning for choice of values.

Partner 1

Partner 2

Dot plots:

<p>Dot plot from 0 to 11 by 1's. Beginning at 0, number of dots above each increment is 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0.</p>

<p>Dot plot from 0 to 60 by 10's. Beginning at 0, number of dots above each increment is 0, 2, 2, 2, 2, 2, 0.</p>

<p>Dot plot from 0 to 3 by 0.2's. Beginning at 0, number of dots above each increment is 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0.<br>
</p>

Dot plots:

<p>Dot plot from negative 10 to 1 by 1's. Beginning at negative 10, number of dots above each increment is 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0.</p>

<p>Dot plot from 0 to 30 by 5's. Beginning at 0, number of dots above each increment is 0, 2, 2, 2, 2, 2, 0.</p>

<p>Dot plot from negative 4 to 8 by 1's. Beginning at negative 4, number of dots above each increment is 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0.</p>

Conditions:

10 numbers with a standard deviation equal to the standard deviation of your first dot plot with a mean of 6.
10 numbers with a standard deviation three times greater than the data in the first row.
10 different numbers with a standard deviation as close to 2 as you can get in 1 minute.

Conditions:

10 numbers with a standard deviation equal to the standard deviation of your first dot plot with a mean of 12.
10 numbers with a standard deviation four times greater than the data in the first row.
10 different numbers with a standard deviation as close to 2 as you can get in 1 minute.

Are You Ready for More?

12.3

Activity

Investigating Variability

Begin with the data:

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20

Use technology to find the mean, standard deviation, median, and interquartile range.
How do the standard deviation and mean change when you remove the greatest value from the data set? How do they change if you add a value to the data set that is twice the greatest value?
What do you predict will happen to the standard deviation and mean when you remove the least value from the data set? Check to see if your prediction was correct.
What happens to the standard deviation and mean when you add a value to the data set equal to the mean? Add a second value equal to the mean. What happens?
Add, change, and remove values from the data set to answer the question: What appears to change more easily, the standard deviation or the interquartile range? Explain your reasoning.

Are You Ready for More?

Student Lesson Summary

We can describe the variability of a distribution using the standard deviation. The standard deviation is a measure of variability that is calculated using a method that is similar to the one used to calculate the MAD, or mean absolute deviation.

A deeper understanding of the importance of standard deviation as a measure of variability will come with a deeper study of statistics. For now, know that the standard deviation is mathematically important and will be used as the appropriate measure of variability when the mean is an appropriate measure of center.

Like the MAD, the standard deviation is large when the data set is more spread out, and the standard deviation is small when the variability is small. The intuition you gained about MAD will also work for the standard deviation.

Glossary