Measures of

VariationPrepared by:

Rona C. AdduCPTE - IA

is a single value that is used to described the spread of the scores in a distribution.

Variation is also known as variability or dispersion.

Measures of Variation

Range Inter-quartile Range and quartile Deviation Mean Deviation Variance and Standard Deviation

Ways of Describing Variation

Range(R)

-is the difference between the highest score and the lowest score

in the distribution.- the simplest and the crudest

measure of variation.

Range for Ungrouped Data

Formula:

R=HS-LS

where,

R = range valueHS = highest scoreLS = lowest score

Example: Find the range of the two groups of scoredistribution.

Section A Section B10 1512 1615 1617 1725 1726 2328 2530 2635 30

Formula:

R = HSυв - LSLв

Where,R = range valueHSUB = Upper boundary of

the highest scoreLSLB = Lower boundary of

the lowest score

X f25-32 333-40 741-48 549-56 457-64 1265-72 673-80 881-88 389-97 2

n=50

Range for Grouped Data

Formula:

IQR = Q3 – Q1

Inter-quartile Range (IQR)And

Quartile Deviation (QD)

Inter-quartile RangeIs the difference between the third quartile and the first quartile.

Quartile Deviation

indicates the distance we need to go above and below the median to include the middle 50% of the scores.

Formula: Q3-Q1 QD= 2

Example:Using the given data 6,8,10,12,12,14,15,16,20, find

the quartile deviation.

Quartile Deviation of Ungrouped Data

X (Score)

68

10121214151620

Solve for Q1Solve for Q3Solve for IQRSolve for QD

n=9# of cases

Quartile Deviation for Grouped Data

Example:

The data given below are the scores of fifty (50) students in Filipino class. Solve for the value of quartile deviation (QD).

Quartile Deviation of Grouped Data

X f Cf<

25-32 3 333-40 7 1041-48 5 1549-56 4 1957-64 12 3165-72 6 3773-80 8 4581-88 3 4889-97 2 50

n=50

Mean Deviation (MD)Measures the average deviation of the values from the mean.

It gives equal weight to the deviation of every score in the distribution.

Formula:

Ʃ/x-X/MD= n

Where,

MD =mean deviation valueX =individual scoreX =sample meann =number of cases

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

Mean Deviation (MD) ungrouped

X X - X /x – x /35 18.8 18.830 8.8 8.826 4.8 4.824 2.8 2.820 -1.2 1.218 -3.2 3.218 -3.2 3.216 -5.2 5.215 -6.2 6.210 -11.2 11.2

Ʃx=212 /21.2 Ʃ/X-x/=604

Mean Deviation for Grouped Data

X f Xm fXm Xm-x /Xm-x/ f/Xm-x/10-14 5 12 60 -21.63 21.63 108.1515-19 2 17 34 -16.63 16.63 33.2620-24 3 22 66 -11.63 11.63 34.8925-29 5 27 135 -6.63 6.63 33.1530-34 2 32 64 -1.63 1.63 3.2635-39 9 37 333 3.37 3.37 30.3340-44 6 42 252 8.37 8.37 50.2245-49 3 47 141 13.37 13.37 40.1150-54 5 52 260 18.37 18.37 91.85

n=40

ƩfXm=1,345 Ʃf/Xm-x/=425.22

Variance and Standard Deviation

Variance Deviationis one of the important measures of variation. It shows variation about the mean.Standard Deviationis the most important measures of variation. It shows variation.

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

Formula:

Ʃ(X-x)²ơ²=

N

Variance of Ungrouped DataX X-X (X-X)²19 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=60.40

X=14.6

Variance of Grouped Data

Example:

Score distribution of the test results of 40 students consisting of 50 items. Solve the variance and standard deviation.

X f Xm fXm X Xm-X (Xm-X)²

f(Xm-X)²

15-20 3 18.5 55.5 33.7 -1.2 262.44

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 88.233-38 15 35.5 532.5 33.7 1.8 3.24 48.639-44 8 41.5 332 33.7 7.8 60.84 486.7245-50 3 47.5 142.5 33.7 13.8 190.4

4571.32

n=40

fXm=1348

Ʃf(Xm-X)²= 2 606.4

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