Mean and Standard Deviation

1. Mean and Standard Deviation Lecture 23 Section 7.5.1 Mon, Oct 17, 2005

2. The Mean and Standard Deviation • Mean of a Discrete Random Variable – The average of the values that the random variable takes on, in the long run. • Standard Deviation of a Discrete Random Variable – The standard deviation of the values that the random variable takes on, in the long run.

3. The Mean of a Discrete Random Variable • The mean is also called the expected value. • However, that does not mean that it is literally the value that we expect to see. • “Expected value” is simply a synonym for the mean or average.

4. The Mean of a Discrete Random Variable • The mean, or expected value, of X may be denoted by either of two symbols. µ or E(X) • If another random variable is called Y, then we would write E(Y). • Or we could write them as µX and µY.

5. Computing the Mean • Given the pdf of X, the mean is computed as • This is a weighted average of X. • Each value is weighted by its likelihood.

6. Example of the Mean • Recall the example where X was the number of children in a household.

7. Example of the Mean • Multiply each x by the corresponding probability.

8. Example of the Mean • Add up the column of products to get the mean. 1.70 = µ

9. Let’s Do It! • Let’s do it! 7.23, p. 430 – Profits and Weather.

10. The Variance of a Discrete Random Variable • Variance of a Discrete Random Variable – The average squared deviation of the values that the random variable takes on, in the long run. • The variance of X is denoted by 2 or Var(X) • The standard deviation of X is denoted by .

11. The Variance and Expected Values • The variance is the expected value of the squared deviations. • That agrees with the earlier notion of the average squared deviation. • Therefore,

12. Example of the Variance • Again, let X be the number of children in a household.

13. Example of the Variance • Subtract the mean (1.70) from each value of X to get the deviations.

14. Example of the Variance • Square the deviations.

15. Example of the Variance • Multiply each squared deviation by its probability.

16. Example of the Variance • Add up the products to get the variance. 0.810 = 2

17. Example of the Variance • Add up the products to get the variance. 0.810 = 2 0.9 = 

18. Alternate Formula for the Variance • 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.” • Of course, we could write this as

19. Example of the Variance • One more time, let X be the number of children in a household.

20. Example of the Variance • Square each value of X.

21. Example of the Variance • Multiply each squared X by its probability.

22. Example of the Variance • Add up the products to get E(X2). 3.70 = E(X2)

23. Example of the Variance • Then use E(X2) and µ to compute the variance. • Var(X) = E(X2) – µ2 = 3.70 – (1.7)2 = 3.70 – 2.89 = 0.81. • It follows that  = 0.81 = 0.9.

24. TI-83 – Means and Standard Deviations • Store the list of values of X in L1. • Store the list of probabilities of X in L2. • Select STAT > CALC > 1-Var Stats. • Press ENTER. • Enter L1, L2. • Press ENTER. • The list of statistics includes the mean and standard deviation of X. • Use x, not Sx, for the standard deviation.

25. TI-83 – Means and Standard Deviations • Let L1 = {0, 1, 2, 3}. • Let L2 = {0.1, 0.3, 0.4, 0.2}. • Compute the statistics. • Compute µ and  for the Indoor and Outdoor distributions in Let’s Do It! 7.23, p. 430.

26. Let’s Do It! • Return once more to Let’s Do It! 7.23, p. 430. • The standard deviation of Profit Outdoors is 23.9. • Use the original formula to compute the standard deviation of Profit Indoors. • Use the alternate formula to compute the standard deviation of Profit Indoors. • Use the TI-83 to find the standard deviation of Profit Indoors.