Section 9.2 Variability

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

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

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

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

Histogram Frequencies 10- 9- 8- 7- 6- 5- 4- 3- 2- 1- 28 29 30 31 32 33 34 35 36 37 38 39 40 41 Histogram Frequencies

little variability m

moderate variability m

considerable variability m

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

A B C xi xi xi 8 10 10 7 8 6 6 6 6 5 4 6 4 2 2

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.

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

A B C xi xi-x xi xi-x xi xi-x 8 2 10 4 10 4 7 1 8 2 6 0 6 0 6 0 6 0 5 -1 4 -2 6 0 4 -2 2 -4 2 -4

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

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

B C xi xi-x (xi-x)2 xi xi-x (xi-x)2 10 4 16 10 4 16 8 2 4 6 0 0 6 0 0 6 0 0 4 -2 4 6 0 0 2 -4 16 2 -4 16 SS = 40 SS = 32

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

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 =

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

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

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

x x - x (x - x)2 25 -16 256 26 -15 225 32 -9 81 45 4 16 51 10 100 67 26 676 SS = 1354 =270.8 6 - 1 1354 s2 = s = 16.5

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

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

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

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

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

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

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

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

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

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

Homework: pp. 456-457

■ 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.

■ Cumulative Review 29. Write the equation of the line through A and B in exercise 28. Give your answer in standard form.

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

■ 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)

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