The document discusses how to calculate standard deviation and variance for both ungrouped and grouped data. It provides step-by-step instructions for finding the mean, deviations from the mean, summing the squared deviations, and using these values to calculate standard deviation and variance through standard formulas. Standard deviation measures how spread out numbers are from the mean, while variance is the square of the standard deviation.

2. 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.

3. 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.

4. STANDARD DEVIATION
 is considered as the most
reliable measure of variability.
 is affected by the individual
values or items in the
distribution.

5. Standard Deviation for
Ungrouped Data

6. 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.

7. 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.

8. Find the Standard
Deviation
35 73
35 11
35 49
35 35
35 15
35 27
210 210
Mean= 35 Mean= 35

9. Find the Standard
Deviation
x x-ẋ (x-ẋ)2 x x-ẋ (x-ẋ)2
35 0 0 73 38 1444
35 0 0 11 -24 576
35 0 0 49 14 196
35 0 0 35 0 0
35 0 0 15 -20 400
35 0 0 27 -8 64
∑(x-ẋ)2 0 ∑(x-ẋ)2 2680

10. Find the Standard
Deviation

11. 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.

12. Find the Standard
Deviation
Class F Midpoint FMp _ _ _ _
Limits (2) (3) (4) X Mp - X (Mp-X)2 f( Mp-X)2
(1)
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= ∑fMp= ∑(Mp-X)2=
100 2,014.0 1,747.08

13. Find the Standard
Deviation

14. 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.

15. 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.

16. 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.

17. 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.

18. 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.

19. Variance for Ungrouped
Data

20. 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.

21. 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.

22. Find the Variance
35 73
35 11
35 49
35 35
35 15
35 27
210 210
Mean= 35 Mean= 35

23. Find the Variance
x x-ẋ (x-ẋ)2 x x-ẋ (x-ẋ)2
35 0 0 73 38 1444
35 0 0 11 -24 576
35 0 0 49 14 196
35 0 0 35 0 0
35 0 0 15 -20 400
35 0 0 27 -8 64
∑(x-ẋ)2 0 ∑(x-ẋ)2 2680

24. Find the Variance

25. Variance for Grouped Data

26. 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.

27. Find the Variance
Class F Midpoint FMp _ _ _ _
Limits (2) (3) (4) X Mp - X (Mp-X)2 f( Mp-X)2
(1)
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= ∑fMp= ∑(Mp-X)2=
100 2,014.0 1,747.08

28. Find the Variance