1. Section 9.2
Variability

2. Objectives:
1. To define and distinguish
measures of variability.
2. To calculate measures of variability.

3. Variability is the amount of scatter or dispersion of data from the mean.

4. Consider the following values.
28, 34, 41, 39, 34, 36, 40, 29, 33, 34, 30, 34, 37, 40, 33, 35, 32, 33, 34, 35, 39, 37, 36, 33, 34, 36, 34, 35, 29, 33, 35, 34, 36, 34

5. xi Tally Frequency
28 I 1
29 II 2
30 I 1
31 0
32 I 1
33 IIIII 5
34 IIIII IIII 9
35 IIII 4
36 IIII 4
37 II 2
38 0
39 II 2
40 II 2
41 I 1
Total 34
xi Tally Frequency
28 I 1
29 II 2
30 I 1
31 0
32 I 1
33 IIIII 5
34 IIIII IIII 9
35 IIII 4
36 IIII 4
37 II 2
38 0
39 II 2
40 II 2
41 I 1
Total 34
xi Tally Frequency
28 I 1
29 II 2
30 I 1
31 0
32 I 1
33 IIIII 5
34 IIIII IIII 9
35 IIII 4
36 IIII 4
37 II 2
38 0
39 II 2
40 II 2
41 I 1
Total 34

6. Histogram Frequencies 10- 9- 8- 7- 6- 5- 4- 3- 2- 1-
Histogram
Frequencies

7. little variability m

8. moderate variability m

9. considerable variabilitym

10. Today we are going to discuss three measures of variability:
1. Range
2. Variance
3. Standard deviation

11. A B C
xi xi xi
7 8 6
6 6 6
5 4 6
4 2 2

12. Each list has a mean and median of 6, but the lists are not the same.
The range is the highest value minus the lowest value.

13. The deviation of a data value from the mean is the difference between the data value and the mean, xi – x.

14. A B C
xi xi-x xi xi-x xi xi-x

15. The mean deviation averages the absolute values of the deviations.
 |xi – x|

16. The sum of the squared deviations (numerator) is important and is often abbreviated to SS for sum of squares:
SS = (xi – x)2 n
i=1

17. B C
xi xi-x (xi-x)2 xi xi-x (xi-x)2
SS = SS = 32

18. Defintion (xi – )2 2 = N Variance The average of squared deviation.
For a population: N
i=1
(xi – )2
2 =

19. Defintion (xi – x)2 s2 = n-1
Variance The average of squared deviation.
For a sample:
n-1 n
i=1
(xi – x)2
s2 =

20. Defintion
Variance Population variance can be estimated based on sample variance.

21. Defintion (xi - x)2 s = n - 1
Standard Deviation The square root of the variance.
s =
n - 1 n
(xi - x)2
i =1

22. Practice: Find SS, s2, and s for the data below:
25, 26, 32, 45, 51, 67
1) Find the mean.
= 246 x 41 6
246 =

23. x x - x (x - x)2
SS = 1354
=270.8
6 - 1
1354
s2 =
s = 16.5

24. Practice: Find the indicated statistics for the set of data below (by making a table). 32, 24, 18, 36, 23, 37, 29, 16, 41, 43, 37, 48 1.  n
i=1 xi
= 384

25. Practice: Find the indicated statistics for the set of data below (by making a table). 32, 24, 18, 36, 23, 37, 29, 16, 41, 43, 37, 48
2. Mean
= 32

26. Practice: Find the indicated statistics for the set of data below (by making a table). 32, 24, 18, 36, 23, 37, 29, 16, 41, 43, 37, 48
3. Order the data and find the
deviation of the fifth data value in the above list: x5 – x.
x5 – x = -3

27. Practice: Find the indicated statistics for the set of data below (by making a table). 32, 24, 18, 36, 23, 37, 29, 16, 41, 43, 37, 48 4.
(xi – x) n
i=1 
= 0

28. Practice: Find the indicated statistics for the set of data below (by making a table). 32, 24, 18, 36, 23, 37, 29, 16, 41, 43, 37, 48 5.
|xi – x| n
i=1 
= 100

29. Practice: Find the indicated statistics for the set of data below (by making a table). 32, 24, 18, 36, 23, 37, 29, 16, 41, 43, 37, 48
6. Mean deviation
≈ 8.3

30. Practice: Find the indicated statistics for the set of data below (by making a table). 32, 24, 18, 36, 23, 37, 29, 16, 41, 43, 37, 48
7. SS
= 1130

31. Practice: Find the indicated statistics for the set of data below (by making a table). 32, 24, 18, 36, 23, 37, 29, 16, 41, 43, 37, 48
8. s²
≈ 102.7

32. Practice: Find the indicated statistics for the set of data below (by making a table). 32, 24, 18, 36, 23, 37, 29, 16, 41, 43, 37, 48
9. s
≈ 10.1

33. Practice: Find the indicated statistics for the set of data below (by making a table). 32, 24, 18, 36, 23, 37, 29, 16, 41, 43, 37, 48
10. range
= 32

34. Homework: pp

35. ■ Cumulative Review
28. Given points A(9, -3) and B(2, -5), find AB, the midpoint of AB, and the point ¼ of the way from A to B.

36. ■ Cumulative Review
29. Write the equation of the line through A and B in exercise Give your answer in standard form.

37. ■ Cumulative Review
30. Which type of function is always continuous?
a. trigonometric
b. rational
c. radical
d. piece

38. ■ Cumulative Review
31. If f(x) is continuous and f(a) = 0, what happens at x = a on the graph of g(x) = ? 1
f(x)

39. ■ Cumulative Review
32. Write the equation of a hyperbola opening vertically that is centered at the origin with b = 6 and with perpendicular asymptotes.