Measures of Spread

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RangeMean Deviation

VarianceStandard Deviation

Range• For a set of data is found by subtracting

the smallest value in the given set of data.• The difference between the highest score

to lowest score in a distribution.• The simplest and the crudest measure of

variation

Importance of Range

Tells you the

distance from the

smallest to largest.

Ungrouped Data R= HS-LS

Whereas: R- RangeHS- Highest scoreLS- Lowest score

Find the range of the two groups of score distribution.

Group A Group B10 (LS) 15 (LS)

12 1615 1617 1725 1726 2328 2530 26

35 (HS) 30 (HS)

Group ARA= HS-LSRA =35-10RA =25

Group BRB= HS-LSRB= 30-15RB= 15

Grouped DataR= HSUB-LSLB

Whereas:R- Range HSUB – Upper boundary of highest

score LSLB – Lower boundary of lowest

score

Find the value of range of the scores of 50 students in Mathematics achievement

testx f

25-32 333-40 741-48 549-56 457-64 1265-72 673-80 881-88 389-97 2

n=50

LS= 25LSLB= 24.5

HS= 97HSUB= 97.5

R= HSUB-LSLB

R=97.5-24.5R= 73

Deviation ean

Mean DeviationMeasures the average deviation of the values from the arithmetic mean. It gives equal weight to the deviation of every score in the distribution.

A. Mean Deviation for Ungrouped Data

MD =

Where,

MD = mean deviation valueX = individual score

= sample meanN = number of cases

Steps in Solving Mean Deviation for Ungrouped Data

1. Solve the mean value.2. Subtract the mean value from each

score.3. Take the absolute value of the

difference in step 2.4. Solve the mean deviation using the

formula MD =

Example 1: Find the mean deviation of the scores of 10 students in a Mathematics test. Given the scores: 35, 30, 26, 24, 20, 18, 18, 16, 15, 10

Analysis:The mean deviation of the 10 scores of students is 6.04. This means that on the average, the value deviated from the mean of 212 is 6.04.

B. Mean Deviation for Grouped Data

where ,MD = mean deviation valuef = class frequency

x = class mark or midpoint of each category

x = mean value n = number of cases

Steps in Solving Mean Deviation for Grouped Data

1. Solve for the value of the mean.2. Subtract the mean value from each midpoint or class mark.3. Take the absolute value of each difference.4. Multiply the absolute value and the corresponding class

frequency.5. Find the sum of the result in step 4.6. Solve for the mean deviation using the formula for grouped

data.

Example 2: Find the mean deviation of the given below.

Analysis:The mean deviation of the 40 scores of students is 0.63. This means that on the average, the value deviated from the mean of 33.63 is 10.63.

ariance

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.

VARIANCEOne of the most

important measures of variation.It shows variation

at the mean.

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 Variance353535353535

210Mean= 35

731149351527210

Mean= 35

Find the Variancex 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 0Mean=35

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 69.89 279.5626-27 9 26.5 238.5 20.14 6.36 40.45 364.0524-25 12 24.5 294.0 20.14 4.36 19.01 228.1222-23 10 22.5 225.0 20.14 2.36 5.57 55.7020-21 17 20.5 348.5 20.14 0.36 0.13 2.2118-19 20 18.5 370.0 20.14 -1.64 2.69 53.8016-17 14 16.5 231.0 20.14 -3.64 13.25 185.5014-15 9 14.5 130.5 20.14 -5.64 31.81 286.2912-13 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

TANDARD DEVIATION

Standard Deviation• Is the most important measures of variation.

• It is also known as the square root of the variance.

• It is the average distance of all the scores that deviates from the mean value.

Formula of

Standard Deviation of Ungrouped Data

POPULATION STANDARD DEVIATION

SAMPLE STANDARD DEVIATION

Steps in solving STANDARD DEVIATION of UNGROUPED DATA

1.Solve the mean value.2.Subtract the mean value from each

score.3.Square the difference between the

mean and each score.4.Find the sum of step 3.5.Solve for the population standard

deviation or sample standard deviation using the formula for ungrouped data.

EXAMPLEX X - X (X – X)²

19 4.4 19.3617 2.4 5.7616 1.4 1.9616 1.4 1.9615 0.4 0.1614 -0.6 0.3614 -0.6 0.3613 -1.6 2.5612 -2.6 6.7610 -4.6 21.16

Ʃx= 146 Ʃ(x-x)²= 60.40x= 14.6

Formula

of Standar

d Deviatio

n of Grouped Data

Steps in solving the STANDARD DEVIATION of GROUP DATA

1. Solve the mean value.2. Subtract the mean value from each midpoint

or class mark.3. Square the difference between the mean

value and midpoint or class mark.4. Multiply the squared difference and the

corresponding class frequency.5. Find the sum of the results in step 4.6. Solve the population standard deviation or

sample standard deviation using the formula for grouped data.

EXAMPLEx f x

15-20 3 17.5 52.5 33.7 -16.2 262.44 787.32

21-26 6 23.5 141 33.7 -10.2 104.04 624.24

27-32 5 29.5 147.5 33.7 -4.2 17.64 88.2

33-38 15 35.5 532.5 33.7 1.8 3.24 48.6

39-44 8 41.5 332 33.7 7.8 60.84 486.72

45-50 3 47.5 142.5 33.7 13.8 190.44 571.32

n=40 = 1348

Ʃf (Xm-X)²= 2606.4

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Measures of Spread

Education

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