1. 3-2 Measures of Variance

2. Symbols n: sample size N: population size ∑: sum f: frequency
: sample mean µ: population mean
s 2: sample variance σ2: population variance
s : sample standard deviation
σ: population standard deviation
Quiz tomorrow on these!

3. Remember: Section 3-1 was about measures of central tendency Mean ()
Median
mode

4. Measures of Variation:
Range
Variance (2)
Standard Deviation ()

5. 1. Range: maximum - minimum
Measures of Variation
1. Range: maximum - minimum
Advantage: quick and easy
Disadvantage: only considers 2 numbers

6. Measures of Variation 2. Variance Describes the variation in the data
Units are usually meaningless

7. Measures of Variation 3.Standard Deviation
Describes the variation or average distance the data is from the mean
Gives the “average deviation”
Units are meaningful

8. Formulas Population Standard Deviation Sample Standard Deviation
*Beware: We must now start determining whether a problem is a sample or population.

9. Example 1: Here’s how the standard deviation formula works:
We will use this example. Mrs. Sigette asked 5 people at the game how much money they had in their pocket. Those people had $70, $75, $80, $85, $90.
The mean is $80.
The range is $20.

10. How much does each data point differ (or deviate) from the mean?x
$70
70 – 80 = -10
$75
75 – 80 = -5
$80
80 – 80 = 0
$85
85 – 80 = 5
$90
90 – 80 = 10

11. Since we are looking for the average of that difference, we need to add them up. The problem is they add up to zero, so mathematicians decided to eliminate the negatives by squaring all of the differences (easier than taking the absolute value). x
$70
-10
$75 -5
$80
$85 5
$90 10
 = 0
100 25
 = 250

12. The variance (σ2) is 50 square dollars.
To get the standard deviation or average of the deviations or variance, you must add all of the deviations and divide by the number of data points you have.
The variance (σ2) is 50 square dollars.
Whaaaaat??

13. This number is actually the variance, but it is not in a useful unit (square dollars?). Therefore to put it back into useful units, we will take the square root of it.

14. Stock A and B Range = ? Range = ? Mean = 61.4 Mean = 61.5
Standard Deviation = ?
Range = ?
Mean = 61.5
Standard Deviation = ?

15. Example 2: Value of Stock A56 57 58 61 63 67
Range:
Mean ( ): n:
Sample Variance: s2 =
Standard Deviation: s =

16. Example 2: Value of Stock B56 75 48 52 57 67 33 77 82 =
Range:
Mean ( ): n:
Sample Variance: s2 =
Standard Deviation: s =

17. Stock A and B Range = Mean = 61.5 Standard Deviation = Range =

18. Starting salaries for Corporation X (1000s of dollars)
Example 3: x
(x-)2 40 23 41 50 49 32 29 52 58 =
Range:
Mean ( ): N:
Population Variance: 2 =
Standard Deviation:  =