2.4-MEASURES OF VARIATION

• 1) Range – Difference between max & min• 2) Deviation – Difference between entry &

mean• 3) Variance – Sum of differences between

entries and mean, divided by population or sample -1.

• 4) Standard Deviation – Square root of variance

Range

• Range = (Maximum entry) – (Minimum entry)• Find range of the starting salaries (1000 of $):41 38 39 45 47 41 44 41 37 42

Range

• Range = (Maximum entry) – (Minimum entry)• Find range of the starting salaries (1000 of $):41 38 39 45 47 41 44 41 37 4247-37 = range of 10 or $10,000

Range

• Range = (Maximum entry) – (Minimum entry)• Find range of the starting salaries (1000 of $):41 38 39 45 47 41 44 41 37 4247-37 = range of 10 or $10,000 • Find range of the starting salaries (1000 of $):40 23 41 50 49 32 41 29 52 58

Range

• Range = (Maximum entry) – (Minimum entry)• Find range of the starting salaries (1000 of $):41 38 39 45 47 41 44 41 37 4247-37 = range of 10 or $10,000 • Find range of the starting salaries (1000 of $):40 23 41 50 49 32 41 29 52 58

Range

• Range = (Maximum entry) – (Minimum entry)• Find range of the starting salaries (1000 of $):41 38 39 45 47 41 44 41 37 4247-37 = range of 10 or $10,000 • Find range of the starting salaries (1000 of $):40 23 41 50 49 32 41 29 52 5858 – 23 = 35 or $35,000

Deviation

• Deviation = How far away entries are from mean. For each entry, entry – mean of data set. x = x - µ. May be positive or negative

• Population Variance = Mean of the SQUARE of the variance. σ² = Σ(x-µ)²÷N

• Sample Variance = Variance for a SAMPLE of a population. s² = Σ(x-x)²÷(n-1)

• Standard deviation = SQUARE ROOT of variance.σ = √ Σ(x-µ)² ÷ N s=√Σ(x-x)²÷(n-1)

Find mean, deviation, sum of squares, population variance & std. deviation

Salaryx

Deviationx- µ

Squares(x-µ)²

41383945474144413742Σ = Σ= Σ = SSx=sum of

Squares =

Find mean, deviation, sum of squares, population variance & std. deviation

Salaryx

Deviationx- µ

Squares(x-µ)²

41383945474144413742Σ = 415 Σ= Σ = SSx=sum of

Squares =

Find mean, deviation, sum of squares, population variance & std. deviation

Salaryx

Deviationx- µ

Squares(x-µ)²

41383945474144413742Σ = 415µ=

Σ= Σ = SSx=sum of Squares =

Find mean, deviation, sum of squares, population variance & std. deviation

Salaryx

Deviationx- µ

Squares(x-µ)²

41383945474144413742Σ = 415µ= 415/10=41.5

Σ= Σ = SSx=sum of Squares =

Find mean, deviation, sum of squares, population variance & std. deviation

Salaryx

Deviationx- µ

Squares(x-µ)²

41 41-41.5 = -.5383945474144413742Σ = 415µ= 415/10=41.5

Σ= Σ = SSx=sum of Squares =

Find mean, deviation, sum of squares, population variance & std. deviation

Salaryx

Deviationx- µ

Squares(x-µ)²

41 -.538 -3.539 -2.545 3.547 5.541 -.544 2.541 -.537 -4.542 .5Σ = 415µ= 415/10=41.5

Σ= Σ = SSx=sum of Squares =

Find mean, deviation, sum of squares, population variance & std. deviation

Salaryx

Deviationx- µ

Squares(x-µ)²

41 -.538 -3.539 -2.545 3.547 5.541 -.544 2.541 -.537 -4.542 .5Σ = 415µ= 415/10=41.5

Σ=0 Σ = SSx=sum of Squares =

Find mean, deviation, sum of squares, population variance & std. deviation

Salaryx

Deviationx- µ

Squares(x-µ)²

41 -.5 (-.5)² = .2538 -3.539 -2.545 3.547 5.541 -.544 2.541 -.537 -4.542 .5Σ = 415µ= 415/10=41.5

Σ=0 Σ = SSx=sum of Squares =

Find mean, deviation, sum of squares, population variance & std. deviation

Salaryx

Deviationx- µ

Squares(x-µ)²

41 -.5 (-.5)² = .2538 -3.5 12.2539 -2.5 6.2545 3.5 12.2547 5.5 30.2541 -.5 .2544 2.5 6.2541 -.5 .2537 -4.5 20.2542 .5 .25Σ = 415µ= 415/10=41.5

Σ=0 Σ = SSx=sum of Squares =

Find mean, deviation, sum of squares, population variance & std. deviation

Salaryx

Deviationx- µ

Squares(x-µ)²

41 -.5 (-.5)² = .2538 -3.5 12.2539 -2.5 6.2545 3.5 12.2547 5.5 30.2541 -.5 .2544 2.5 6.2541 -.5 .2537 -4.5 20.2542 .5 .25Σ = 415µ= 415/10=41.5

Σ=0 Σ = SSx=sum of Squares = 88.5

Find mean, deviation, sum of squares, population variance & std. deviation

Salaryx

Deviationx- µ

Squares(x-µ)²

41 -.5 (-.5)² = .2538 -3.5 12.2539 -2.5 6.2545 3.5 12.2547 5.5 30.2541 -.5 .2544 2.5 6.2541 -.5 .2537 -4.5 20.2542 .5 .25Σ = 415µ= 415/10=41.5

Σ=0 Σ = SSx=sum of Squares = 88.5

N = 10

σ² = SSx/N

Find mean, deviation, sum of squares, population variance & std. deviation

Salaryx

Deviationx- µ

Squares(x-µ)²

41 -.5 (-.5)² = .2538 -3.5 12.2539 -2.5 6.2545 3.5 12.2547 5.5 30.2541 -.5 .2544 2.5 6.2541 -.5 .2537 -4.5 20.2542 .5 .25Σ = 415µ= 415/10=41.5

Σ=0 Σ = SSx=sum of Squares = 88.5

N = 10

σ² = SSx/N

σ²= 88.5/10 = 8.85

σ= √σ²

Find mean, deviation, sum of squares, population variance & std. deviation

Salaryx

Deviationx- µ

Squares(x-µ)²

41 -.5 (-.5)² = .2538 -3.5 12.2539 -2.5 6.2545 3.5 12.2547 5.5 30.2541 -.5 .2544 2.5 6.25√41 -.5 .2537 -4.5 20.2542 .5 .25Σ = 415µ= 415/10=41.5

Σ=0 Σ = SSx=sum of Squares = 88.5

N = 10

σ² = SSx/N

σ²= 88.5/10 = 8.85

σ= √σ²

σ =√8.85 = 2.97

Find mean, deviation, sum of squares, population variance & std. deviation

Salaryx

Deviationx- µ

Squares(x-µ)²

40234150493241295258Σ = Σ= Σ = SSx=sum of

Squares =

Find mean, deviation, sum of squares, population variance & std. deviation

Salaryx

Deviationx- µ

Squares(x-µ)²

40234150493241295258Σ = 415 Σ= Σ = SSx=sum of

Squares =

Find mean, deviation, sum of squares, population variance & std. deviation

Salaryx

Deviationx- µ

Squares(x-µ)²

40234150493241295258Σ = 415µ = 415/10=41.5

Σ= Σ = SSx=sum of Squares =

Find mean, deviation, sum of squares, population variance & std. deviation

Salaryx

Deviationx- µ

Squares(x-µ)²

40 40-41.5=-1.5234150493241295258Σ = 415µ = 415/10=41.5

Σ= Σ = SSx=sum of Squares =

Find mean, deviation, sum of squares, population variance & std. deviation

Salaryx

Deviationx- µ

Squares(x-µ)²

40 -1.523 -18.541 -.550 8.549 7.532 -9.541 -.529 -12.552 10.558 16.5Σ = 415µ = 415/10=41.5

Σ= Σ = SSx=sum of Squares =

Find mean, deviation, sum of squares, population variance & std. deviation

Salaryx

Deviationx- µ

Squares(x-µ)²

40 -1.523 -18.541 -.550 8.549 7.532 -9.541 -.529 -12.552 10.558 16.5Σ = 415µ = 415/10=41.5

Σ=0 Σ = SSx=sum of Squares =

Find mean, deviation, sum of squares, population variance & std. deviation

Salaryx

Deviationx- µ

Squares(x-µ)²

40 -1.5 (-1.5)²=2.2523 -18.541 -.550 8.549 7.532 -9.541 -.529 -12.552 10.558 16.5Σ = 415µ = 415/10=41.5

Σ=0 Σ = SSx=sum of Squares =

Find mean, deviation, sum of squares, population variance & std. deviation

Salaryx

Deviationx- µ

Squares(x-µ)²

40 -1.5 (-1.5)²=2.2523 -18.5 342.2541 -.5 .2550 8.5 72.2549 7.5 56.2532 -9.5 90.2541 -.5 .2529 -12.5 156.2552 10.5 110.2558 16.5 272.25Σ = 415µ = 415/10=41.5

Σ=0 Σ = SSx=sum of Squares =

Find mean, deviation, sum of squares, population variance & std. deviation

Salaryx

Deviationx- µ

Squares(x-µ)²

40 -1.5 (-1.5)²=2.2523 -18.5 342.2541 -.5 .2550 8.5 72.2549 7.5 56.2532 -9.5 90.2541 -.5 .2529 -12.5 156.2552 10.5 110.2558 16.5 272.25Σ = 415µ = 415/10=41.5

Σ=0 Σ = SSx=sum of Squares = 1102.5

Find mean, deviation, sum of squares, population variance & std. deviation

Salaryx

Deviationx- µ

Squares(x-µ)²

40 -1.5 (-1.5)²=2.2523 -18.5 342.2541 -.5 .2550 8.5 72.2549 7.5 56.2532 -9.5 90.2541 -.5 .2529 -12.5 156.2552 10.5 110.2558 16.5 272.25Σ = 415µ = 415/10=41.5

Σ=0 Σ = SSx=sum of Squares = 1102.5

N=10

Find mean, deviation, sum of squares, population variance & std. deviation

Salaryx

Deviationx- µ

Squares(x-µ)²

40 -1.5 (-1.5)²=2.2523 -18.5 342.2541 -.5 .2550 8.5 72.2549 7.5 56.2532 -9.5 90.2541 -.5 .2529 -12.5 156.2552 10.5 110.2558 16.5 272.25Σ = 415µ = 415/10=41.5

Σ=0 Σ = SSx=sum of Squares = 1102.5

N=10

σ²=SSx/10

Find mean, deviation, sum of squares, population variance & std. deviation

Salaryx

Deviationx- µ

Squares(x-µ)²

40 -1.5 (-1.5)²=2.2523 -18.5 342.2541 -.5 .2550 8.5 72.2549 7.5 56.2532 -9.5 90.2541 -.5 .2529 -12.5 156.2552 10.5 110.2558 16.5 272.25Σ = 415µ = 415/10=41.5

Σ=0 Σ = SSx=sum of Squares = 1102.5

N=10

σ²=SSx/10

σ²=1102.5/10 = 110.25

Find mean, deviation, sum of squares, population variance & std. deviation

Salaryx

Deviationx- µ

Squares(x-µ)²

40 -1.5 (-1.5)²=2.2523 -18.5 342.2541 -.5 .2550 8.5 72.2549 7.5 56.2532 -9.5 90.2541 -.5 .2529 -12.5 156.2552 10.5 110.2558 16.5 272.25Σ = 415µ = 415/10=41.5

Σ=0 Σ = SSx=sum of Squares = 1102.5

N=10

σ²=SSx/10

σ²=1102.5/10 = 110.25

σ=√σ²

Find mean, deviation, sum of squares, population variance & std. deviation

Salaryx

Deviationx- µ

Squares(x-µ)²

40 -1.5 (-1.5)²=2.2523 -18.5 342.2541 -.5 .2550 8.5 72.2549 7.5 56.2532 -9.5 90.2541 -.5 .2529 -12.5 156.2552 10.5 110.2558 16.5 272.25Σ = 415µ = 415/10=41.5

Σ=0 Σ = SSx=sum of Squares = 1102.5

N=10

σ²=SSx/10

σ²=1102.5/10 = 110.25

σ=√σ²

σ=√110.25 = 10.5

Find the Sample Variance and Sample Standard Deviation

Salaryx

Deviationx- x

Squares(x-x)²

41 -.5 (-.5)² = .2538 -3.5 12.2539 -2.5 6.2545 3.5 12.2547 5.5 30.2541 -.5 .2544 2.5 6.2541 -.5 .2537 -4.5 20.2542 .5 .25Σ = 415x= 415/10=41.5

Σ=0 Σ = SSx=sum of Squares = 88.5

n = 10

Find the Sample Variance and Sample Standard Deviation

Salaryx

Deviationx- x

Squares(x-x)²

41 -.5 (-.5)² = .2538 -3.5 12.2539 -2.5 6.2545 3.5 12.2547 5.5 30.2541 -.5 .2544 2.5 6.2541 -.5 .2537 -4.5 20.2542 .5 .25Σ = 415x= 415/10=41.5

Σ=0 Σ = SSx=sum of Squares = 88.5

n = 10

s²=SSx/(n-1)

Find the Sample Variance and Sample Standard Deviation

Salaryx

Deviationx- x

Squares(x-x)²

41 -.5 (-.5)² = .2538 -3.5 12.2539 -2.5 6.2545 3.5 12.2547 5.5 30.2541 -.5 .2544 2.5 6.2541 -.5 .2537 -4.5 20.2542 .5 .25Σ = 415x= 415/10=41.5

Σ=0 Σ = SSx=sum of Squares = 88.5

n = 10

s²=SSx/(n-1)

s²=88.5/(10-1) = 88.5/9 =9.83

Find the Sample Variance and Sample Standard Deviation

Salaryx

Deviationx- x

Squares(x-x)²

41 -.5 (-.5)² = .2538 -3.5 12.2539 -2.5 6.2545 3.5 12.2547 5.5 30.2541 -.5 .2544 2.5 6.2541 -.5 .2537 -4.5 20.2542 .5 .25Σ = 415x= 415/10=41.5

Σ=0 Σ = SSx=sum of Squares = 88.5

n = 10

s²=SSx/(n-1)

s²=88.5/(10-1) = 88.5/9 =9.83

s=3.14

Find the Sample Variance and Sample Standard Deviation

Salaryx

Deviationx- x

Squares(x-x)²

40 -1.5 (-1.5)²=2.2523 -18.5 342.2541 -.5 .2550 8.5 72.2549 7.5 56.2532 -9.5 90.2541 -.5 .2529 -12.5 156.2552 10.5 110.2558 16.5 272.25Σ = 415x = 415/10=41.5

Σ=0 Σ = SSx=sum of Squares = 1102.5

n=10

Find the Sample Variance and Sample Standard Deviation

Salaryx

Deviationx- x

Squares(x-x)²

40 -1.5 (-1.5)²=2.2523 -18.5 342.2541 -.5 .2550 8.5 72.2549 7.5 56.2532 -9.5 90.2541 -.5 .2529 -12.5 156.2552 10.5 110.2558 16.5 272.25Σ = 415x = 415/10=41.5

Σ=0 Σ = SSx=sum of Squares = 1102.5

n=10

s²=SSx/(n-1)

Find the Sample Variance and Sample Standard Deviation

Salaryx

Deviationx- x

Squares(x-x)²

40 -1.5 (-1.5)²=2.2523 -18.5 342.2541 -.5 .2550 8.5 72.2549 7.5 56.2532 -9.5 90.2541 -.5 .2529 -12.5 156.2552 10.5 110.2558 16.5 272.25Σ = 415x = 415/10=41.5

Σ=0 Σ = SSx=sum of Squares = 1102.5

n=10

s²=SSx/(n-1)

s²=1102.5/(10-1) = 1102.5/9 = 122.5

Find the Sample Variance and Sample Standard Deviation

Salaryx

Deviationx- x

Squares(x-x)²

40 -1.5 (-1.5)²=2.2523 -18.5 342.2541 -.5 .2550 8.5 72.2549 7.5 56.2532 -9.5 90.2541 -.5 .2529 -12.5 156.2552 10.5 110.2558 16.5 272.25Σ = 415x = 415/10=41.5

Σ=0 Σ = SSx=sum of Squares = 1102.5

n=10

s²=SSx/(n-1)

s²=1102.5/(10-1) = 1102.5/9 = 122.5

s=√s²

Find the Sample Variance and Sample Standard Deviation

Salaryx

Deviationx- x

Squares(x-x)²

40 -1.5 (-1.5)²=2.2523 -18.5 342.2541 -.5 .2550 8.5 72.2549 7.5 56.2532 -9.5 90.2541 -.5 .2529 -12.5 156.2552 10.5 110.2558 16.5 272.25Σ = 415x = 415/10=41.5

Σ=0 Σ = SSx=sum of Squares = 1102.5

n=10

s²=SSx/(n-1)

s²=1102.5/(10-1) = 1102.5/9 = 122.5

s=√s²

s=√122.5 = 11.07

Interpreting Standard Deviation

50

5

10

5

Data012345

34567

x=5s=1.2

x=5s=0

2 3 7 801234

Series 1Series 2 x=5

s=3.0

Estimate the Standard Deviation

405

10 Series 1

N=8µ=4σ=

3 5012345

Series 1

Series 1 N=8µ=4σ=

1 3 5 70

0.5

1

1.5

2

2.5Series 1

Series 1N=8µ=4σ=

Estimate the Standard Deviation

405

10 Series 1

N=8µ=4σ=0

3 5012345

Series 1

Series 1 N=8µ=4σ=

1 3 5 70

0.5

1

1.5

2

2.5Series 1

Series 1N=8µ=4σ=

Estimate the Standard Deviation

405

10 Series 1

N=8µ=4σ=0

3 5012345

Series 1

Series 1 N=8µ=4σ=1

1 3 5 70

0.5

1

1.5

2

2.5Series 1

Series 1N=8µ=4σ=+ 1 & 3

Estimate the Standard Deviation

405

10 Series 1

N=8µ=4σ=0

3 5012345

Series 1

Series 1 N=8µ=4σ=1

1 3 5 70

0.5

1

1.5

2

2.5Series 1

Series 1N=8µ=4σ=2σ²=