Standard Deviation

& Variance

OBJECTIVES

The learners are expected to:

a. Calculate the Standard Deviation of a given set of data.

b. Calculate the Variance of a given set of data.

STANDARD DEVIATION

is a special form of average deviation from the mean.

is the positive square root of the arithmetic mean of the squared deviations from the mean of the distribution.

STANDARD DEVIATION

is considered as the most reliable measure of variability.

is affected by the individual values or items in the distribution.

Standard Deviation for Ungrouped Data

How to Calculate the Standard Deviation for

Ungrouped Data

1. Find the Mean.2. Calculate the difference

between each score and the mean.

3. Square the difference between each score and the mean.

How to Calculate the Standard Deviation for

Ungrouped Data

4. Add up all the squares of the difference between each score and the mean.

5. Divide the obtained sum by n – 1.

6. Extract the positive square root of the obtained quotient.

Find the Standard Deviation

353535353535

210Mean= 35

731149351527

210Mean= 35

Find the Standard Deviation

x x-ẋ (x-ẋ)2

35 0 0

35 0 0

35 0 0

35 0 0

35 0 0

35 0 0

∑(x-ẋ)2 0

x x-ẋ (x-ẋ)2

73 38 1444

11 -24 576

49 14 196

35 0 0

15 -20 400

27 -8 64

∑(x-ẋ)2 2680

x x-ẋ (x-ẋ)2

35 0 0

35 0 0

35 0 0

35 0 0

35 0 0

35 0 0

∑(x-ẋ)2 0

Find the Standard Deviation

How to Calculate the Standard Deviation for

Grouped Data

1. Calculate the mean.2. Get the deviations by finding

the difference of each midpoint from the mean.

3. Square the deviations and find its summation.

4. Substitute in the formula.

Find the Standard Deviation

Class Limits

(1)

F(2)

Midpoint(3)

FMp(4)

_X

_Mp - X

_(Mp-X)2

_f( Mp-X)2

28-29 4 28.5 114.0 20.14 8.36 69.89 279.56

26-27 9 26.5 238.5 20.14 6.36 40.45 364.05

24-25 12 24.5 294.0 20.14 4.36 19.01 228.12

22-23 10 22.5 225.0 20.14 2.36 5.57 55.70

20-21 17 20.5 348.5 20.14 0.36 0.13 2.21

18-19 20 18.5 370.0 20.14 -1.64 2.69 53.80

16-17 14 16.5 231.0 20.14 -3.64 13.25 185.50

14-15 9 14.5 130.5 20.14 -5.64 31.81 286.29

12-13 5 12.5 62.5 20.14 -7.64 58.37 291.85

N= 100

∑fMp= 2,014.0

∑(Mp-X)2= 1,747.08

Find the Standard Deviation

Characteristics of the Standard Deviation

1. The standard deviation is affected by the value of every observation.

2. The process of squaring the deviations before adding avoids the algebraic fallacy of disregarding the signs.

Characteristics of the Standard Deviation

3. It has a definite mathematical meaning and is perfectly adapted to algebraic treatment.

4. It is, in general, less affected by fluctuations of sampling than the other measures of dispersion.

Characteristics of the Standard Deviation

5. The standard deviation is the unit customarily used in defining areas under the normal curve of error. It has, thus, great practical utility in sampling and statistical inference.

VARIANCE

is the square of the standard deviation.

In short, having obtained the value of the standard deviation, you can already determine the value of the variance.

VARIANCE

It follows then that similar process will be observed in calculating both standard deviation and variance. It is only the square root symbol that makes standard deviation different from variance.

Variance for Ungrouped Data

How to Calculate the Variance for Ungrouped

Data

1. Find the Mean.2. Calculate the difference

between each score and the mean.

3. Square the difference between each score and the mean.

How to Calculate the Variance for Ungrouped

Data

4. Add up all the squares of the difference between each score and the mean.

5. Divide the obtained sum by n – 1.

Find the Variance

353535353535

210Mean= 35

731149351527

210Mean= 35

Find the Variance

x x-ẋ (x-ẋ)2

35 0 0

35 0 0

35 0 0

35 0 0

35 0 0

35 0 0

∑(x-ẋ)2 0

x x-ẋ (x-ẋ)2

73 38 1444

11 -24 576

49 14 196

35 0 0

15 -20 400

27 -8 64

∑(x-ẋ)2 2680

x x-ẋ (x-ẋ)2

35 0 0

35 0 0

35 0 0

35 0 0

35 0 0

35 0 0

∑(x-ẋ)2 0

Find the Variance

Variance for Grouped Data

How to Calculate the Variance for Grouped

Data

1. Calculate the mean.2. Get the deviations by finding

the difference of each midpoint from the mean.

3. Square the deviations and find its summation.

4. Substitute in the formula.

Find the VarianceClass Limits

(1)

F(2)

Midpoint(3)

FMp(4)

_X

_Mp - X

_(Mp-X)2

_f( Mp-X)2

28-29 4 28.5 114.0 20.14 8.36 69.89 279.56

26-27 9 26.5 238.5 20.14 6.36 40.45 364.05

24-25 12 24.5 294.0 20.14 4.36 19.01 228.12

22-23 10 22.5 225.0 20.14 2.36 5.57 55.70

20-21 17 20.5 348.5 20.14 0.36 0.13 2.21

18-19 20 18.5 370.0 20.14 -1.64 2.69 53.80

16-17 14 16.5 231.0 20.14 -3.64 13.25 185.50

14-15 9 14.5 130.5 20.14 -5.64 31.81 286.29

12-13 5 12.5 62.5 20.14 -7.64 58.37 291.85

N= 100

∑fMp= 2,014.0

∑(Mp-X)2= 1,747.08

Find the Variance

FOR LISTENING!!!

Thank you