1. 2.4 Measures of Variation

2. Range and Deviation
Range - The difference between the maximum and the minimum data entries in the set
Range = (Max – Min)
Deviation – the difference between the entry and the mean of the data set.
Deviation of

3. Variance and Standard Deviation
Population Variance:
Population Standard Deviation:
Sample Variance:
Sample Standard Deviation:

4. Deviation Squares Heights (in inches) 70 -.3 .09 1.7 2.89 72 .7 .49 7169 73 68
-.3
1.7 .7
-1.3
2.7
-2.3
.09
2.89
.49
1.69
7.29
5.29
SSx=
2.01
1.418
2.233
1.494
Mean = 70.3

5. Empirical Rule

6. About 68% of the data lies within 1 standard deviation of the mean

7. About 95% of the data lies within 2 standard deviation of the mean

8. About 99.7 of the data lies within 3 standard deviation of the mean

10. Examples
Heights of Women in the U.S. have a mean of 64 with a standard deviation of Use the empirical rule to estimate:
The percent of the heights that are between and 64 inches.
ANS: 34%
Between what two values does about 95% of the data lie?
ANS: (58.5, 69.5)

11. Chebychev’s Theorem
The portion of any data set lying within k standard deviations (k>1) of the mean is at least
Example: k = 2 , 75% of the data is within 2 standard deviations of the mean
k = 3; 88.9% of the data lies within 3 standard deviations of the mean.

12. Sample Standard deviation for grouped data