Download 3.07: Three measures of spread

Honors Math 3
Unit 3: Inferential Statistics
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Date____________________
3.07: Three measures of spread
Here are three ways of measuring how spread out a data set is. The inputs for all three measures are the
deviations (how far each data value is from the mean), but each measure combines these numbers in a
different way. For any of the three measures, a larger value indicates that a data set is more spread out.
How to calculate the mean absolute deviation
Step 1. Given a data set, calculate the mean (notation: x )
Step 2. Find the absolute deviation for each data value x by calculating | x – x |.
Step 3. Find the mean of the Step 2 numbers. The result is called the mean absolute deviation.
How to calculate the variance (a.k.a. mean squared deviation)
Step 1. Given a data set, calculate the mean.
Step 2. Find each data value’s deviation.
Step 3. Calculate the square of each number found in Step 2.
Step 4. Now calculate the mean of the numbers from Step 3. The result is called the variance.
How to calculate the standard deviation (a.k.a. root mean squared deviation)
Steps 1–4. Same as for finding the variance.
Step 5. Now take the square root of the variance. The result is called the standard deviation.
Example
Data set: {3, 3, 4, 6, 6, 6, 6, 10, 14, 15, 15, 15, 15, 16, 16}. First step is finding the mean: x = 10.
x
3
3
4
6
6
6
6
10
14
15
15
15
15
16
16
|x– x|
7
7
6
4
4
4
4
0
4
5
5
5
5
6
6
sum of data values = 150
mean = 150/15 = 10
sum of absolute deviations = 72
mean absolute deviation = 72/15 = 4.8
( x – x )2
49
49
36
16
16
16
16
0
16
25
25
25
25
36
36
sum of squares of deviations = 386
variance = 386/15 ≈ 25.733
standard deviation ≈ 25.733 ≈ 5.073
You try it
1. For the data set {3, 4, 5, 6, 9, 10, 10, 10, 10, 10, 11, 15, 15, 16, 16}, answer the following questions using
the table below. Follow the model of the page 1 example.
a. Find the mean.
b. Find the mean absolute deviation.
c. Find the variance.
d. Find the standard deviation.
e. A good way to check your answers is to find the standard deviation on your calculator. Here’s how:
type the data set into your calculator under STAT Edit L1, then do
STAT Calc 1-Var Stats
f. Compare the answers to b, c, and d with the values for the example from page 1.
According to each of the measures of spread, which data set is more spread out?
x
3
4
5
6
9
10
10
10
10
10
11
15
15
16
16
|x– x|
( x – x )2
If you have time, try the following. We will discuss this more next class.
2. Generalize using 𝚺 notation: Now suppose the data set is {x1, x2, x3, …, xn}. Write expressions using 𝚺
notation for each of the following.
mean absolute deviation =
variance =
standard deviation =