Standard Deviation

1. Standard Deviation Lecture 23 Sec. 5.4 Wed, Feb 25, 2004

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). 0 1 2 3 5 6 7 8 x = 4

3. 0 1 2 3 5 6 7 8 x = 4 Deviations from the Mean • Each unit of a sample or population deviates from the mean by a certain amount. deviation = –4

4. 0 1 2 3 5 6 7 8 x = 4 Deviations from the Mean • Each unit of a sample or population deviates from the mean by a certain amount. dev = 1

5. 0 1 2 3 5 6 7 8 x = 4 Deviations from the Mean • Each unit of a sample or population deviates from the mean by a certain amount. deviation = 3

6. Sum of Squared Deviations • We want to add up all the deviations, but to keep the negative ones from cancelling the positive ones, so we square them all first. • Then we compute the sum of the squared deviations. • We call this quantity SSX.

7. Sum of Squared Deviations • SSX = sum of squared deviations =  (x –x)2. • For example, if the sample is {0, 5, 7}, then SSX = (0 – 4)2 + (5 – 4)2 + (7 – 4)2 = (-4)2 + (1)2 + (3)2 = 16 + 1 + 9 = 26.

8. The Population Variance • The variance of the population is the average squared deviation. • The population variance is denoted by 2. 2 = [ (x – )2]/N.

9. The Sample Variance • The variance of a sample is the average squared deviation, except that we divide by n – 1 instead of n. • The sample variance is denoted by s2. s2 = [ (x –x)2]/(n – 1). • This formula for s2 makes a better estimator of 2 than if we had divided by n.

10. Example • In the example, SSX = 26. • Therefore, s2 = 26/2 = 13.

11. The Standard Deviation • The standard deviation of a sample or population is the square root of the variance of that sample or population. • The standard deviation of the population is denoted . • The standard deviation of a sample is denoted s.

12. Example • In our example, we found that s2 = 13. • Therefore, s = 13 = 3.606.

13. Example • See Example 5.10, p. 293. • Use Excel to compute the mean and standard deviation of {0, 5, 7}.

14. Alternate Formula for the Standard Deviation • An alternate way to compute SSX =  (x –x)2 • is to compute SSX = ( x2) – ( x)2/n. • Note that only the second term is divided by n. • Then s2 = SSX/(n – 1), as before.

15. Example • Let the sample be {0, 5, 7}. • Then  x = 12 and  x2 = 0 + 25 + 49 = 74. • So SSX = 74 – (12)2/3 = 74 – 48 = 26, as before.

16. Standard Deviation on the TI-83 • Follow the instructions 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 = 3.605551275. • x = 2.943920289.

17. 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.

18. 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.

19. 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.

20. Interpreting the Standard Deviation x

21. Interpreting the Standard Deviation s s x

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

23. Interpreting the Standard Deviation Closer than normal tox x – s x x + s

24. Interpreting the Standard Deviation Farther than normal fromx x – s x x + s

25. Let’s Do It! • Let’s do it! 5.13, p. 295 – Increasing Spread. • Let’s do it! 5.14, p. 297 – Variation in Scores.

26. Assignment • Page 298: Exercises: 10, 11, 14 – 18, 20, 21. • Page 311: Exercises: 32, 36, 37, 39, 47.