1. Standard Deviation
Lecture 18
Sec
Tue, Oct 4, 2005

2. Deviations from the Mean
Each unit of a sample or population deviates from the mean by a certain amount.
Define the deviation of x to be (x –x). 1 2 3 4 5 6 7 8
x = 3.5

3. Deviations from the Mean
Each unit of a sample or population deviates from the mean by a certain amount.
deviation = -3.5 1 2 3 4 5 6 7 8
x = 3.5

4. Deviations from the Mean
Each unit of a sample or population deviates from the mean by a certain amount.
dev = -1.5 1 2 3 4 5 6 7 8
x = 3.5

5. Deviations from the Mean
Each unit of a sample or population deviates from the mean by a certain amount.
dev = +1.5 1 2 3 4 5 6 7 8
x = 3.5

6. Deviations from the Mean
Each unit of a sample or population deviates from the mean by a certain amount.
deviation = +3.5 1 2 3 4 5 6 7 8
x = 3.5

7. Deviations from the Mean
How do we obtain one number that is representative of the set of individual deviations?
If we add them up to get the average, the positive deviations will cancel with the negative deviations, leaving a total of 0.
That’s no good.

8. Sum of Squared Deviations
We will square them all first. That way, there will be no canceling.
So we compute the sum of the squared deviations, called SSX.
Procedure
Find the deviations
Square them all
Add them up

9. Sum of Squared Deviations
SSX = sum of squared deviations
For example, if the sample is {0, 2, 5, 7}, then
SSX = (0 – 3.5)2 + (2 – 3.5)2 + (5 – 3.5)2 + (7 – 3.5)2
= (-3.5)2 + (-1.5)2 + (1.5)2 + (3.5)2 =
= 29.

10. The Population Variance
Variance of the population – The average squared deviation for the population.
The population variance is denoted by 2.

11. The Population Standard Deviation
The population standard deviation is the square root of the population variance.
We will interpret this as being representative of deviations in the population (hence the name “standard”).

12. The Sample Variance
Variance of a sample – The average squared deviation for the sample, except that we divide by n – 1 instead of n.
The sample variance is denoted by s2.
This formula for s2 makes a better estimator of 2 than if we had divided by n.

13. Example
In the example, SSX = 29.
Therefore,
s2 = 29/3 =

14. The Sample Standard Deviation
The sample standard deviation is the square root of the sample variance.
We will interpret this as being representative of deviations in the sample.

15. Example In our example, we found that s2 = 9.667.
Therefore, s = 9.667 =

16. Example
Use Excel to compute the mean and standard deviation of the sample {0, 2, 5, 7}.
Do it once using basic operations.
Do it again using special functions.
Then compute the mean and standard deviation for the on-time arrival data.
OnTimeArrivals.xls.

17. Alternate Formula for the Standard Deviation
An alternate way to compute SSX is to compute
Note that only the second term is divided by n.
Then, as before

18. Example Let the sample be {0, 2, 5, 7}. Then  x = 14 andSo
SSX = 78 – (14)2/4
= 78 – 49
= 29,
as before.

19. TI-83 – Standard Deviations
Follow the procedure for computing the mean.
The display shows Sx and x.
Sx is the sample standard deviation.
x is the population standard deviation.
Using the data of the previous example, we have
Sx =
x =

20. Interpreting the Standard Deviation
Both the standard deviation and the variance are measures of variation in a sample or population.
The standard deviation is measured in the same units as the measurements in the sample.
Therefore, the standard deviation is directly comparable to actual deviations.

21. Interpreting the Standard Deviation
The variance is not comparable to deviations.
The most basic interpretation of the standard deviation is that it is roughly the average deviation.

22. Interpreting the Standard Deviation
Observations that deviate fromx by much more than s are unusually far from the mean.
Observations that deviate fromx by much less than s are unusually close to the mean.

23. Interpreting the Standard Deviationx

24. Interpreting the Standard Deviationx

25. Interpreting the Standard Deviation
x – s x
x + s

26. Interpreting the Standard Deviation
A little closer than normal tox
but not unusual
x – s x
x + s

27. Interpreting the Standard Deviation
Unusually close tox
x – s x
x + s

28. Interpreting the Standard Deviation
A little farther than normal fromx
but not unusual
x – 2s
x – s x
x + s
x + 2s

29. Interpreting the Standard Deviation
Unusually far fromx
x – 2s
x – s x
x + s
x + 2s

30. Let’s Do It! Let’s Do It! 5.13, p. 329 – Increasing Spread.
Example 5.10, p. 329 – There Are Many Measures of Variability.