measures of variability

Lesson 6Measures of Variability

Measures of Variability

Consider two sets of distributions. Compute for their means.

a. 70, 76, 78, 81, 85, 88, 89 x = 81

b. 77, 78, 79, 80, 83, 85, 85 x = 81


IMPORTANT!!!The measure of averages alone is not

sufficient to describe the distribution.

Averages locate the center of the distribution but they do not tell us how are the observations are positioned in relation to the center.


Absolute Measures of Variability The range, the quartile deviation, the average

or mean deviation, the standard deviation, and the variance give us the distances of the scores from the measures of central tendency. They are categorized as the measures of absolute variability since their units are the same as those of the original units

Relative Measures of VariabilityIf two or more distributions of different units are

to be compared, it is more appropriate to use the measures of relative variability. These measures are the coefficient of variation and the standard scores.

Measures of VariabilityThe RangeThe range is the simplest and the easiest to

compute among the measures of variability because it depends only on the pair of extreme values

But it is also the most unstable since its value easily fluctuates with the change in either of the highest or lowest observations.

It is also the most unreliable because it does not give the dispersion or spread of the observations between the two extreme values.


NOTE:A more reliable measure should include all the values in the distribution to give us an adequate spread of all the scores from the average.


Exclusive Range It is the difference between the

highest and the lowest score in a distribution.

The exclusive range is used for ungrouped data.


Example: If the lowest score in a distribution is 24 and the highest is 39, what is the exclusive range?

Measures of VariabilityInclusive range. o It is the difference between the

lower class boundary of the lowest observation and the upper class boundary of the highest observation.

oGenerally, the inclusive range is used for grouped data.


Example:

5 – 9 lower scores 10 - 14 15 - 19

20 – 24 higher scores

Find the inclusive range.


Solution:

R = highest UCB - lowest LCB = 24.5 - 4.5R = 20 (inclusive range)

Measures of VariabilityQuartile DeviationReduces the effect of the extremely low

and high observations on the measure of dispersion.

The quartile deviation is used when the median is the preferred measure of the central tendency, that is, if there are scattered or extreme observations in the distribution.

It gives the spread of the observations around the median or the middle 50% of the cases in a distribution.


Interquartile Range - the spread of distribution used the third and first quartiles corresponding to the 75th and 25th percentiles, respectively.

Interquartile Range = Q3 Q1

Measures of VariabilitySemi-interquartile range (also known

as quartile deviation) of a set of data is defined

Semi-interquartile Range =

where Q1 and Q3 are the first and third

quartiles of the data respectively.

213 QQ

QD

Measures of VariabilityExample:

Using the given set of observation, determine the

Interquartile range ( IR )Quartile deviation ( QD )

Sets of Observations:A: 10, 8, 6, 4, 14, 11 , 16, 7

B: 9, 4, 8, 7, 9, 8, 10, 17

Measures of VariabilitySolution:

16 14 11 10 8 7 6 4Solving for Q3 and Q1

Interquartile Range = Q3 – Q1 = 13.25 – 6.25 = 7

Quartile Deviation =

75.6100

1875100

1

nP

25.2100

1825100

1 nP

25.133 Q

25.6Q

5.327

213

QQ

Measures of VariabilityFor grouped data, consider the example below.

Example :

The following data are frequency distributions of two sections in Statistics in a 100 item examination. Set A Set B

f f70-74 4 71-75 375-79 12 76-80 880-84 15 81-85 1385-89 13 86-90 790-94 6 91-95 595-99 5 96-100 4

Solving for Q1 or P25 and Q3 or P75


The formula for Q a quartile point is

Quartile Deviation

cfi

cfin

lPbi

ii

100

213 QQ

QD


Set A

LL – UL f <cf 70-74 4 4 75-79 12 16 80-84 15 31 85-89 13 44 90-94 6 50 95-99 5 55 c = 5 n = 55

Measures of Variability For Set A

For Set B

512

475.135.7425

p 513

3125.415.8475

p

93975.42

5625.78442.88

QD

58

3105.7525

p 57

24305.8575

p

956.42

875.79786.89

QD

5625.7825 p 442.8875 p

875.7925 p 786.8975 p

Measures of VariabilityThe Average or Mean Deviation

The average or mean deviation takes into account all the values in a given distribution.

The formula for mean deviation is

where is read as “the absolute value of x and x = score of value = mean n = number of cases

n

XXMD

XXX

X


Example 4. Given the two sets of distribution,

let us solve for the mean deviations.

Set A: 28, 29, 32, 37 and 39 Set B: 25, 32, 33, 40 and 45

Measures of VariabilitySet A

X 28 29

32 37 39

This means that on the average, the scores deviated from the mean by 4.

xx 3328 3329

3332

3337 3339

165x 20

n

XXMD

4

520 or

Measures of VariabilitySet B

X 25 32

33 40 45

This means that on the average, the scores deviated from the mean by 6.

xx

n

XXMD

3525 3532 3533

3540

3545

175 30

65

30 or


Sets Mean Mean Deviation

A 33 4

B 35 6

If the mean deviations for sets A, and B, are 4, and 6 respectively, it can be concluded that B has greater variability than A.

Measures of VariabilityGrouped Data

For the grouped data the mean deviation can be calculated using the following formula:

nxxf

MD i 1

where xi = class marksx = grouped data meann = total frequencyfi = frequency per class intervals


Example 5. Compute the mean deviation for the data in Example 3. Set A Set B

LL – UL f LL – UL f70-74 4 71-75 375-79 12 76-80 880-84 15 81-85 1385-89 13 86-90 790-94 6 91-95 595-99 5 96-100 4c = 5 n = 55 c = 5 n = 40

Measures of Variability(1)CI

LL - UL

(2)fi

(3)xi

(4)fixi

(5) (6)

70 – 74 4 72 288 11.82 47.28

75 – 79 12 77 924 6.82 81.84

80 – 84 15 82 1230 1.82 27.3

85 – 89 13 87 1131 3.18 41.34

90 – 94 6 92 552 8.18 49.08

95 – 99 5 97 485 13.18 65.9

c = 5 n = 55 4610 312.74

82.8355

4610

nxf

x ii 69.555

74.312

nxxf

MD ii

Measures of VariabilityVariance and Standard Deviation

Using the absolute value symbol, the influence of the negative signs is eliminated in the computation of the mean deviation or average deviation.

Another way would be to square the deviation from the mean. Adding these squared deviations and dividing the sum by the number of items, the variance, which is a basic measure of dispersion, is obtained.

Extracting the square root of the variance yields the standard deviation, the most commonly used measure of dispersion or variability.


1

2

n

xxi

1

1

2

1

n

XXS

n

i

Variance and Standard Deviation of Ungrouped Data

The formula for variance ungrouped data is

s2 =

The standard deviation for ungrouped data is the square root of the variance


If the standard deviation of the population is needed, S is changed

to σ. The formula for σ2 is given by

where N = size of the population

Measures of VariabilityExample 3. To illustrate the computation for standard deviation of the sample of ungrouped data, consider the distribution below.

Score Deviation Squared Deviations (X) from the Mean from the Mean

7 7 – 10 = -3 (-3)2 = 9

8 8 – 10 = -2 (-2)2 = 4 10 10 – 10 = 0 (0)2 = 0 12 12 – 10 = 2 (2)2 = 4 13 13 – 10 = 3 (3)2

= 9

XX 2XX

50 X 0 XX 262

XX

105

50 orX


S2 =

1

2

n

xxi

= 6.5

1

1

2

1

n

XXS

n

i 55.2S

An equivalent formula does away with computing for the mean and the deviations of the observations.

variance = S2

1

2

2

nnx

x

standard deviation =

1

2

2

nnX

XS

=


Example: Solve for the variance and standard deviation

X (X2)

10 102 = 100 9 92 = 81 7 72 = 49 5 52 = 25 4 42 = 16 2 22 = 4

Σx2 = 275

37 X


Solution:

1

2

2

nnx

x

166

372752

2

s

s2 =

5834.46

s2 =

06.3s

= 9.366


An Efficient Way of Computing the Sample Standard Deviation of Grouped Data

The short formula for standard deviation for grouped data is

22

nfd

ndf

cs

where c = class size f = class frequencyd = class deviationn = number of cases or total frequency


Example 4.To illustrate the preceding formula, take for example the grouped data which shows the wages per day

of the laborers in a certain construction.

Wages No. of

Laborers

(f)

d fd (d)2 f(d)2

80-84 8 2 16 4 32

75-79 12 1 12 1 12

70-74 16 0 0 0 0

65-69 13 -1 -13 1 13

60-64 9 -2 -18 4 3658 f 3' fd 93)( 2df


22)(

nfd

ndf

cs

2

583

58935

s

325.6s


Another formula that can be used to solve the variance and standard deviation of grouped data is the long method.

variance

1

2

2

n

xxfs

standard deviation = S =

1

2

nxxf


58 f 4161 fX 1208.2321)( 2XXf

Using the preceding data, let us compute the variance using this second formula.


nfX

X 74.7158

4161 orX

1

2

nXXf

s158

1208.2321

s

326.6s


Coefficient of Variation

The coefficient of variation is used to express the standard deviation as a percentage of the mean.

To compare two distributions with different means and standard deviation we have to compute for the coefficient of variation to express them in the same unit, that is, in terms of percentage.

To illustrate we use the formula

%)100(xsCV

whereCV stands for the coefficient of

variation; S, the standard deviation;and X is the mean


Example 8Department store A has a mean weekly

sales of 340 bags with a standard deviation of 12. Department store B has a mean weekly sales of 550 bags with a standard deviation of 15. In relative terms, which store has the greater variability in their weekly sales?

Measures of Variability C

C

Example 9:The mean score of a statistics test

of class A is 75 with a standard deviation of 13 while class B hasa mean score of 86 with a standard

deviation of 16.

Which class has a larger variation from the mean?

Measures of Variability C

C

%33.17)100(7513

%60.18)100(8616

Solution:

Class A:

Class B:

Interpretation:

Thus, Class B has a larger variation from the mean.This means that the students are more

homogenous in Class A than in Class B. This further implies that the teacher in Class B

has to exert more effort to meet the individual needs of the students.

END OF THE SHOW

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