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Standard Deviation

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tmattingly
32 diapositives · 564 Views · Jan 03, 2020

This lecture focuses on the concept of variability in populations and how the standard deviation serves as a crucial measure. Variability affects our ability to estimate parameters accurately. By exploring deviations from the mean, we define and calculate deviations, sum of squared deviations, and both population and sample variances. Additionally, we touch on interpreting standard deviations to identify unusual observations relative to the mean. Examples and formulas are provided to illustrate these statistical concepts clearly.

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Standard Deviation
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  1. Standard Deviation Lecture 18 Sec. 5.3.4 Mon, Feb 18, 2007

  2. Variability • Our ability to estimate a parameter accurately depends on the variability of the population. • What do we mean by variability in the population? • How do we measure it?

  3. 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 6 7 8 9 10 5

  4. 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 6 7 8 9 10 5 x = 6

  5. Deviations from the Mean • Each unit of a sample or population deviates from the mean by a certain amount. deviation = –5 1 2 3 4 6 7 8 9 10 5 x = 4

  6. Deviations from the Mean • Each unit of a sample or population deviates from the mean by a certain amount. dev = –2 1 2 3 4 6 7 8 9 10 5 x = 4

  7. Deviations from the Mean • Each unit of a sample or population deviates from the mean by a certain amount. dev = +1 1 2 3 4 6 7 8 9 10 5 x = 4

  8. Deviations from the Mean • Each unit of a sample or population deviates from the mean by a certain amount. dev = +2 1 2 3 4 6 7 8 9 10 5 x = 4

  9. Deviations from the Mean • Each unit of a sample or population deviates from the mean by a certain amount. deviation = +4 1 2 3 4 6 7 8 9 10 5 x = 4

  10. Deviations from the Mean • How do we obtain one number that is representative of the whole set of individual deviations? • Normally we use an average to summarize a set of numbers. • Why will the average not work in this case?

  11. Sum of Squared Deviations • We will square them all first. That way, there will be no canceling. • So we compute the sum of thesquared deviations, called SSX.

  12. Sum of Squared Deviations • Procedure • Find the average • Find the deviations from the average • Square the deviations • Add them up

  13. Sum of Squared Deviations • SSX = sum of squared deviations • For example, if the sample is {1, 4, 7, 8, 10}, then SSX = (1 – 6)2 + (4 – 6)2 + (7 – 6)2 + (8 – 6)2 + (10 – 6)2 = (–5)2 + (–2)2 + (1)2 + (2)2 + (4)2 = 25 + 4 + 1 + 4 + 16 = 50.

  14. The Population Variance • Variance of the population • The population variance is denoted by 2.

  15. The Population Standard Deviation • The population standard deviation is the square root of the population variance.

  16. The Sample Variance • Variance of a sample • The sample variance is denoted by s2.

  17. The Sample Variance • Theory shows that if we divide by n – 1 instead of n, we get a better estimator of 2. • Therefore, we do it.

  18. Example • In the example, SSX = 50. • Therefore, s2 = 50/4 = 12.5.

  19. 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.

  20. Example • In our example, we found that s2 = 12.5. • Therefore, s = 12.5 = 3.536. • How does that compare to the individual deviations?

  21. Alternate Formula for the Standard Deviation • An alternate way to compute SSXis to compute • Then, as before

  22. Example • Let the sample be {1, 4, 7, 8, 10}. • Then  x = 30 and  x2 = 1 + 16 + 49 + 64 + 100 = 230. • So SSX = 230 – (30)2/5 = 230 – 180 = 50, as before.

  23. 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 = 3.535533906. • x = 3.16227766.

  24. Interpreting the Standard Deviation • The standard deviation is directly comparable to actual deviations. • How does 3.536 compare to -5, -2, +1, +2, and +4?

  25. 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.

  26. Interpreting the Standard Deviation x

  27. Interpreting the Standard Deviation s s x

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

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

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

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

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