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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
