QBA 260Business Statistics

Chapter 5

Deviating from the Average Compare two datasets (reported in

inches) Dataset 1: 48, 46, 46, 49, 49, 47, 47,

46, 49 Dataset 2: 47, 51, 49, 53, 47, 45, 37,

38, 60 What is the average of each

dataset? How are they different?

Comparing Datasets

Histogram - Dataset 1

0

2

4

6

8

10

10 20 30 40 50 60

Bin Categories

Num

ber w

ithin

each

bin

Histogram - dataset 2

0

2

4

6

8

10

10 20 30 40 50 60

Bin Categories

Num

ber w

ithin

eac

h bi

n

Mean (Average) for both datasets is 47.4 inchesVariances differ: Dataset 1 = 1.78 sq. inches, Dataset 2 = 51.03 sq. inches

Mean Mean

Variance and Deviations Variance is a measure of how much the

data points in a sample deviate from the sample mean

For data: x1, x2, x3,…, xn A deviation is the difference between a data

value and a central value such as the average

XXDeviation i

Deviations Dataset 1: 48,

46, 46, 49, 49, 47, 47, 46, 49

Average = 47.4 Deviations The average

deviation is (always) zero

Xi X-Bar Deviation

48 47.4 0.56

46 47.4 -1.44

46 47.4 -1.44

49 47.4 1.56

49 47.4 1.56

47 47.4 -0.44

47 47.4 -0.44

46 47.4 -1.44

49 47.4 1.56

47.4 0.00

(Average)   (Sum)

Deviations Dataset 2: 47,

51, 49, 53, 47, 45, 37, 38, 60

Average = 47.4 Deviations The average

deviation is (always) zero

Xi X-Bar Deviation

47 47.4 -0.44

51 47.4 3.56

49 47.4 1.56

53 47.4 5.56

47 47.4 -0.44

45 47.4 -2.44

37 47.4 -10.44

38 47.4 -9.44

60 47.4 12.56

47.4 0.00

(Average)   (Sum)

Populations and Samples Population

variance

Sample variance

N

XXN

1i

2i

2

)(

1N

XXs

N

1i

2i

2

)(

Sample Variance – Dataset 1 Sample variance

1

)(1

2

2

N

XXs

N

ii

1

22.142

N

s

78.119

22.142

s

Xi X-Bar Deviation Sq Dev

48 47.4 0.56 0.31

46 47.4 -1.44 2.09

46 47.4 -1.44 2.09

49 47.4 1.56 2.42

49 47.4 1.56 2.42

47 47.4 -0.44 0.20

47 47.4 -0.44 0.20

46 47.4 -1.44 2.09

49 47.4 1.56 2.42

47.4 0.00 14.22

(Average)   (Sum) (Sum)

Sq Inches

Sample Variance – Dataset 2 Sample variance

1

)(1

2

2

N

XXs

N

ii

Xi X-Bar Deviation Sq Dev

47 47.4 -0.44 0.20

51 47.4 3.56 12.64

49 47.4 1.56 2.42

53 47.4 5.56 30.86

47 47.4 -0.44 0.20

45 47.4 -2.44 5.98

37 47.4 -10.44 109.09

38 47.4 -9.44 89.20

60 47.4 12.56 157.64

47.4 0.00 408.22

(Average)   (Sum) (Sum)

1

22.4082

N

s

03.5119

22.4082

s Sq Inches

Variance and Standard Deviation The calculated variance is in a different

unit of measure than the original dataset Original Dataset measured in inches Variance measured in inches squared (due to

the use of squared deviations before we average them)

To accommodate this, we calculate the standard deviation which is the square root of the Variance

Standard Deviation Square root of the variance

1N

XXs

N

XX

N

1i

2i

N

1i

2i

)(

)(

Excel Functions

VARP(…) Population Variance VARPA(…) Population Variance

(includes cells that contain true/false data)

VAR(…) Sample Variance VARA(…) Sample Variance (includes

cells that contain true/false data)

Excel Functions STDEVP(…) Population Standard

Deviation STDEVPA(…) Population Standard

Deviation (includes cells that contain true/false data)

STDEV(…) Sample Standard Deviation STDEVA(…) Sample Standard Deviation

(includes cells that contain true/false data)

Average Absolute Deviation

Xi X-Bar Deviation ABS

48 47.4 0.56 0.56

46 47.4 -1.44 1.44

46 47.4 -1.44 1.44

49 47.4 1.56 1.56

49 47.4 1.56 1.56

47 47.4 -0.44 0.44

47 47.4 -0.44 0.44

46 47.4 -1.44 1.44

49 47.4 1.56 1.56

47.4 0.00 10.44

(Average)   (Sum) (Sum)

Absolute Deviation = 10.44/9 = 1.16 EXCEL: AVEDEV