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. x
P(X = x)
0.10 1
0.30 2
0.40 3
0.20

7. Example of the Mean Multiply each x by the corresponding probability.
P(X = x)
xP(X = x)
0.10
0.00 1
0.30 2
0.40
0.80 3
0.20
0.60

8. Example of the Mean Add up the column of products to get the mean. x
P(X = x)
xP(X = x)
0.10
0.00 1
0.30 2
0.40
0.80 3
0.20
0.60
1.70 = µ

9. Let’s Do It!
Let’s Do It! 7.23, p. 462 – 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. x
P(X = x)
0.10 1
0.30 2
0.40 3
0.20

13. Example of the Variance
Subtract the mean (1.70) from each value of X to get the deviations. x
P(X = x)
x – µ
0.10
-1.7 1
0.30
-0.7 2
0.40
+0.3 3
0.20
+1.3

14. Example of the Variance
Square the deviations. x
P(X = x)
x – µ
(x – µ)2
0.10
-1.7
2.89 1
0.30
-0.7
0.49 2
0.40
+0.3
0.09 3
0.20
+1.3
1.69

15. Example of the Variance
Multiply each squared deviation by its probability. x
P(X = x)
x – µ
(x – µ)2
(x – µ)2P(X = x)
0.10
-1.7
2.89
0.289 1
0.30
-0.7
0.49
0.147 2
0.40
+0.3
0.09
0.036 3
0.20
+1.3
1.69
0.338

16. Example of the Variance
Add up the products to get the variance. x
P(X = x)
x – µ
(x – µ)2
(x – µ)2P(X = x)
0.10
-1.7
2.89
0.289 1
0.30
-0.7
0.49
0.147 2
0.40
+0.3
0.09
0.036 3
0.20
+1.3
1.69
0.338
0.810 = 2

17. Example of the Variance
Add up the products to get the variance. x
P(X = x)
x – µ
(x – µ)2
(x – µ)2P(X = x)
0.10
-1.7
2.89
0.289 1
0.30
-0.7
0.49
0.147 2
0.40
+0.3
0.09
0.036 3
0.20
+1.3
1.69
0.338
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. x
P(X = x)
0.10 1
0.30 2
0.40 3
0.20

20. Example of the Variance
Square each value of X. x
P(X = x) x2
0.10 1
0.30 2
0.40 4 3
0.20 9

21. Example of the Variance
Multiply each squared X by its probability. x
P(X = x) x2
x2P(X = x)
0.10
0.00 1
0.30 2
0.40 4
1.60 3
0.20 9
1.80

22. Example of the Variance
Add up the products to get E(X2). x
P(X = x) x2
x2P(X = x)
0.10
0.00 1
0.30 2
0.40 4
1.60 3
0.20 9
1.80
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.
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 parameters  and .

26. Let’s Do It! Return once more to Let’s Do It! 7.23, p. 462.
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.