1. Variance Math 115b Mathematics for Business Decisions, part II
Ekstrom Math 115b

2. Variance Variance gives a measure of the dispersion of data
Larger variance means the data is more spread out
Several formulas for different types of variables

3. Variance: Finite R.V. Finite R.V. Easiest if a table is set up x Given
Compute

4. Variance: Finite R.V.
Ex. Find the variance for the following finite R.V., X x
fX (x) 1
0.30 3
0.15 5
0.10 7 9

5. Variance: Finite R.V. First find the mean x fX (x) 1 0.30 3 0.15 5
0.10 7 9

6. Variance: Finite R.V. Calculate x-value minus the mean x 1 0.30 3 0.15
0.10 7 9 -4 -2 2 4
Calculate x-value minus the mean

7. Variance: Finite R.V. Calculate the square of x-value minus the mean x1
0.30 -4 3
0.15 -2 5
0.10 7 2 9 4 16 4
Calculate the square of x-value minus the mean

8. Variance: Finite R.V. x 1
0.30 -4 16 3
0.15 -2 4 5
0.10 7 2 9
4.80
0.60
0.00
Multiply the squared values by their respective p.d.f. values

9. Variance: Finite R.V. Variance = 10.8x 1
0.30 -4 16
4.80 3
0.15 -2 4
0.60 5
0.10
0.00 7 2 9
Variance =
10.8
Add the final column to find the variance

10. Variance: Finite R.V.
Ex. Find the variance for the following finite R.V., X x 1
0.10 3
0.25 5
0.30 7 9

11. Variance: Finite R.V. First find the mean x fX (x) 1 0.10 3 0.25 5
0.30 7 9

12. Variance: Finite R.V. Calculate x-value minus the mean x 1 0.10 3 0.25
0.30 7 9 -4 -2 2 4
Calculate x-value minus the mean

13. Variance: Finite R.V. Calculate the square of x-value minus the mean x1
0.10 -4 3
0.25 -2 5
0.30 7 2 9 4 16 4
Calculate the square of x-value minus the mean

14. Variance: Finite R.V. x 1
0.10 -4 16 3
0.25 -2 4 5
0.30 7 2 9
1.60
1.00
0.00
Multiply the squared values by their respective p.d.f. values

15. Variance: Finite R.V. Variance = 5.2x 1
0.10 -4 16
1.60 3
0.25 -2 4
1.00 5
0.30
0.00 7 2 9
Variance =
5.2
Add the final column to find the variance

16. Variance Variance is used to find standard deviation
Standard deviation is ALWAYS the square root of variance
Standard deviation is represented by sigma or
Standard deviation is the “typical amount” of variation from the mean (approx. 2/3 of all data lies within 1 standard deviation of mean)

17. Variance: Binomial R.V.
Shortcut for binomial R.V.

18. Variance: Binomial R.V.
Ex. Suppose X represents to total number of students that pass a particular class in a given semester. If there are 34 students in the class and historically 83% of the students pass, find the standard deviation of X.
Soln:

19. Variance: Binomial R.V.
A solution could also be attempted using the BINOMDIST function
Recall binomial R.V.’s are finite R.V.’s
Complete table as done in previous examples

20. Variance: Continuous R.V.
Similar formula for continuous R.V.
Value is found using Integrating.xlsm
Recall,

21. Variance: Continuous R.V.
Ex. Suppose is a p.d.f. on the interval [0, 1]. Find the mean of X, the variance of X, and the standard deviation of X.
Soln:

22. Variance: Continuous R.V.
Soln:

23. Variance: Continuous R.V.
Ex. Let T be an exponential random variable with parameter Find and
Soln:

24. Variance: Continuous R.V.
Note that for an exponential random variable, the variance is equal to and the standard deviation is equal to .
Ex. Let W be a uniform random variable on the interval [0, 30]. Find and .

25. Variance: Continuous R.V.
Soln.:
Estrom Math 115b

26. Variance: Continuous R.V.
Note that for a uniform random variable, the variance is equal to

27. Variance: Standardization
Different types of variables can have similar parameters (mean & std. dev.)
We can transform the variables
New variable S defined as

28. Variance: Standardization
We say S is the standardization of X.
The mean of S will be 0
The standard deviation of S will be 1.

29. Variance: Standardization
Determining probabilities using standardized values
Ex. Suppose X is an exponential random variable with parameter Determine where S is the standardization of X.
Soln: Recall

30. Variance: Standardization
So,
Then,

31. Variance: Sample Formula:
A sample is a collection of data from some random variable (finite or continuous)

32. Variance: Sample
Standard deviation of a sample is found by taking the square root of variance
Formula:
Ex. Find the mean, variance, and standard deviation of the sample 14, 16, 17, 21, 22.

33. Variance: Sample
Soln:
Mean:
AVERAGE function in Excel

34. Variance: Sample
Variance:
VAR function in Excel

35. Variance: Sample
Standard Deviation:
STDEV function in Excel

36. Variance: Sample Why are sample values important?
Sometimes unreasonable/impossible to achieve all values for a random variable
We can assume and
Samples help predict values for the random variable

37. Variance: Sample Mean
Formula:
Compares a group of sample means

38. Variance: Sample Mean
Ex. Find the variance and standard deviation of the sample mean for the following data set:
14, 16, 17, 21, 22
Soln:

39. Variance: Sample Mean How do you interpret ?
If there were samples of size 5, the sample mean would be about 18 (from previous calculation) and, on average, the sample mean would be within units about 2/3 of the time.

40. Variance: Project What to do:
Calculate standard deviation of errors (use STDEV function)
My standard deviation is about 13.53
We assume mean is 0 even though we calculated a value that was different