1. The Standard Deviation of a Discrete Random Variable Lecture 24 Section 7.5.1 Fri, Oct 20, 2006

2. The Variance and Standard Deviation Variance of a Discrete Random Variable – The variance of the values that the random variable takes on, in the long run. Variance of a Discrete Random Variable – The variance of the values that the random variable takes on, in the long run. This is the average squared deviation of the values that the random variable takes on, in the long run. This is the average squared deviation of the values that the random variable takes on, in the long run. Standard Deviation of a Discrete Random Variable – The square root of the variance. Standard Deviation of a Discrete Random Variable – The square root of the variance.

3. Why Do We Need the Standard Deviation? Our goal later will be to assign a margin of error to our estimates of a population mean. Our goal later will be to assign a margin of error to our estimates of a population mean. To do this, we need a measure of the variability of our estimator  x. To do this, we need a measure of the variability of our estimator  x. This, in turn, requires a measure of the variability of x, namely, the standard deviation. This, in turn, requires a measure of the variability of x, namely, the standard deviation.

4. The Variance of a Discrete Random Variable The variance of X is denoted by The variance of X is denoted by  X 2 or Var(X) The standard deviation of X is denoted by  X. The standard deviation of X is denoted by  X. Usually there are no other variables, so we may write  2 and . Usually there are no other variables, so we may write  2 and .

5. The Variance The variance is the expected value of the squared deviations. The variance is the expected value of the squared deviations. That agrees with the earlier notion of the average squared deviation. That agrees with the earlier notion of the average squared deviation. Therefore, Therefore,

6. The Variance Since the variance of X is the average value of (X –  ) 2, we use the method of weighted averages to compute it. Since the variance of X is the average value of (X –  ) 2, we use the method of weighted averages to compute it.

7. Example of the Variance xP(x)P(x) 00.10 10.30 20.40 30.20 Again, let X be the number of children in a household. Again, let X be the number of children in a household.

8. Example of the Variance xP(x)P(x)x – µ 00.10-1.7 10.30-0.7 20.40+0.3 30.20+1.3 Subtract the mean (1.70) from each value of X to get the deviations. Subtract the mean (1.70) from each value of X to get the deviations.

9. Example of the Variance xP(x)P(x)x – µ(x – µ) 2 00.10-1.72.89 10.30-0.70.49 20.40+0.30.09 30.20+1.31.69 Square the deviations. Square the deviations.

10. Example of the Variance xP(x)P(x)x – µ(x – µ) 2 (x – µ) 2  P(x) 00.10-1.72.890.289 10.30-0.70.490.147 20.40+0.30.090.036 30.20+1.31.690.338 Multiply each squared deviation by its probability. Multiply each squared deviation by its probability.

11. Example of the Variance xP(x)P(x)x – µ(x – µ) 2 (x – µ) 2  P(x) 00.10-1.72.890.289 10.30-0.70.490.147 20.40+0.30.090.036 30.20+1.31.690.338 0.810 =  2 Add up the products to get the variance. Add up the products to get the variance.

12. Example of the Variance xP(x)P(x)x – µ(x – µ) 2 (x – µ) 2  P(x) 00.10-1.72.890.289 10.30-0.70.490.147 20.40+0.30.090.036 30.20+1.31.690.338 0.810 =  2 0.9 =  Take the square root to get the standard deviation. Take the square root to get the standard deviation.

13. Alternate Formula for the Variance It turns out that It turns out that That is, the variance of X is “the expected value of the square of X minus the square of the expected value of X.” That is, the variance of X is “the expected value of the square of X minus the square of the expected value of X.” Of course, we could write this as Of course, we could write this as

14. Example of the Variance xP(x)P(x) 00.10 10.30 20.40 30.20 One more time, let X be the number of children in a household. One more time, let X be the number of children in a household.

15. Example of the Variance xP(x)P(x)x2x2 00.100 10.301 20.404 30.209 Square each value of X. Square each value of X.

16. Example of the Variance xP(x)P(x)x2x2 x2P(x)x2P(x) 00.1000.00 10.3010.30 20.4041.60 30.2091.80 Multiply each squared X by its probability. Multiply each squared X by its probability.

17. Example of the Variance xP(x)P(x)x2x2 x2P(x)x2P(x) 00.1000.00 10.3010.30 20.4041.60 30.2091.80 3.70 = E(X 2 ) Add up the products to get E(X 2 ). Add up the products to get E(X 2 ).

18. Example of the Variance Then use E(X 2 ) and µ to compute the variance. Then use E(X 2 ) and µ to compute the variance. Var(X) = E(X 2 ) – µ 2 Var(X) = E(X 2 ) – µ 2 = 3.70 – (1.7) 2 = 3.70 – 2.89 = 0.81. It follows that  =  0.81 = 0.9. It follows that  =  0.81 = 0.9.

19. TI-83 – Means and Standard Deviations Store the list of values of X in L 1. Store the list of values of X in L 1. Store the list of probabilities of X in L 2. Store the list of probabilities of X in L 2. Select STAT > CALC > 1-Var Stats. Select STAT > CALC > 1-Var Stats. Press ENTER. Press ENTER. Enter L 1, L 2. Enter L 1, L 2. Press ENTER. Press ENTER. The list of statistics includes the mean and standard deviation of X. The list of statistics includes the mean and standard deviation of X. Use  x, not Sx, for the standard deviation. Use  x, not Sx, for the standard deviation.

20. TI-83 – Means and Standard Deviations Let L 1 = {0, 1, 2, 3}. Let L 1 = {0, 1, 2, 3}. Let L 2 = {0.1, 0.3, 0.4, 0.2}. Let L 2 = {0.1, 0.3, 0.4, 0.2}. Compute the parameters  and . Compute the parameters  and .

21. Example: Powerball www.powerball.com www.powerball.com www.powerball.com Find the standard deviation of the value of a Powerball ticket. Find the standard deviation of the value of a Powerball ticket. x P(x)P(x)P(x)P(x) 2000000000.000000006844 2000000.0000002806 100000.000001711 1000.00007015 1000.00008384 70.003437 70.001341 40.007881 30.01450 00.9727