Standard Deviation

# Standard Deviation

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## Standard Deviation

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1. Standard Deviation Lecture 18 Sec. 5.3.4 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). 0 1 2 3 5 6 7 8 4 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 0 1 2 3 5 6 7 8 4 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 0 1 2 3 5 6 7 8 4 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 0 1 2 3 5 6 7 8 4 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 0 1 2 3 5 6 7 8 4 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 = 12.25 + 2.25 + 2.25 + 12.25 = 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 = 9.667.

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 = 3.109.

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 SSXis 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 and  x2 = 0 + 4 + 25 + 49 = 78. • So 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 = 3.109126351. • x = 2.692582404.

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 s s x – s x x + s

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

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

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

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

28. 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.