Download Measures of Variation

Measures of Variation
Describes how data are spread out
Variance, 𝜎 2 , and Standard Deviation, 𝜎
Show how much data values deviate from the mean
Finding the Variance and Standard Deviation
11-7: Standard Deviation
Algebra 2 CP
1.
2.
3.
4.
Find the mean, 𝑥,ҧ of the data set or sample.
Find the difference, 𝑥 − 𝑥,ҧ between each value 𝑥 and the mean.
Square each difference, 𝑥 − 𝑥ҧ 2 .
Find the average of (mean) of these squares. This is the variance.
σ 𝑥 − 𝑥ҧ 2
𝜎2 =
𝑛
5. Take the square root of the variance. This is the standard deviation.
𝜎=
What are the mean, variance and standard deviation of these values?
6.5 5.8 3.9 5.7 4.2
1. Find the mean: 𝑥ҧ =
2. Make a table:
𝒙
6.5+5.8+3.9+5.7+4.2
= 5.22
5
𝟐
𝒙−ഥ
𝒙
𝒙−ഥ
𝒙
6.5
6.5 − 5.22 = 1.28
1.282 = 1.6384
5.8
5.8 − 5.22 = .58
3.9 3.9 − 5.22 = −1.32
5.7
5.7 − 5.22 = .48
4.2 4.2 − 5.22 = −1.02
.582 = .3364
−1.32
σ 𝑥−𝑥ҧ 2
𝑛
=
4.988
5
The table displays the number of sales a salesperson made each month
during the past 15 months. Use a calculator to calculate the mean and
standard deviation.
Months 1
2
3
4
5
6
7
8
9 10 11 12 13 14 15
Sales 4
3
5
4
6
8
1
3
2
5
6
𝑛
4
7
5
3
1. Use STAT EDIT to enter the sales data into the lists.
2. Press STAT, move the cursor to CALC and select the 1-VAR STATS option.
3. Choose the list you entered the data in and then press ENTER.
−1.02
2
= 1.0404
Mean
4.988
Standard
Deviation
= .9976
σ 𝑥−𝑥ҧ 2
4. Find the standard deviation: 𝜎 =
= 1.7424
2
.482 = .2304
Sum
3. Find the variance: 𝜎 2 =
2
σ 𝑥 − 𝑥ҧ
𝑛
= .9976 = .9988
Use the sales data from the previous example to find how many standard
deviations of the mean all of the values fall within.
1. Draw a dot plot of the data.
2. Mark where the mean is located on the graph.
3. Mark off intervals the size of the standard deviation on either side of the
mean.
Homework: p.722 #6-13, 16, 18, 21, 22
1.82
1.82
1.82
1.82
Mean =4.4
All the data values fall within two standard deviations of the mean.
1