measure of variability (windri). In research include example

Measures of Variability
Windri Dewi Ayu

• The terms variability, spread, and dispersion are
synonyms, and refer to how spread out a distribution
is.
Mean = 7 in both the cases
Scores are more densely packed Scores are more spread out

• How far the scores have shown spread out from
the mean?
• Dispersion within a dataset can be measured or
described in several ways by using Measures of
Variability.
• It will make the distribution and interpretation
more meaningful.
• It shows the specific nature of distribution of data.

There are four major “Measures of
Variability”:
1) The Range
2) The Quartile Deviation
3) The Mean or Average Deviation
4) The Standard Deviation

1. Range: The range is a measure of the distance between
highest and lowest.
The simplest and most straightforward measurement of variation is the
range which measures variation in interval-ratio variables.
Range = highest score – lowest score
R= H – L
Example:
Range of temperature:
Papua : 34° – 15° 19°

Range: Examples
If the oldest person included in a study was 79 and the youngest was
18, then the range would be ......... years.
Or, if the most frequent incidences of disturbing the peace among 6
communities under study is 18 and the least frequent incidences
was 4, then the range is .......

Limitations
• It is very sensitive to the smallest and
largest data values.
• It is not a stable statistics as its value can
differ from sample to sample drawn from
the same population.
• In order to reduce the problems caused
by outliers in a dataset, the inter-quartile
range is often calculated instead of the
range.

Quartiles
The extensions of the Median concept because
they are values which divide a set of data into
equal parts.
• Median : Divides the distribution into two equal
parts.
• Quartile : Divides the distribution into four
equal parts.
• Decile : Divides the distribution into ten equal
parts.
• Percentile : Divides the distribution into one
hundred equal parts.

(2) : The Quartile Deviation : Q
Q₁ Q₂ Q₃
Inter-quartile Range
Median
25th Percentile 75th Percentile
Since IQR includes middle 50 % of scores, the value of
Q gives clear picture of spread / dispersion.
Q₁ : 1st Quartile
The point below
Which 25th
per cent of
the scores lie
Q₃ : 3rd Quartile
The point below
Which 75th
per cent of the
scores lie

The Quartile Deviation : Q
• When the extreme scores in the given
distribution are very high and very low, the
range will be very high.
• The inter-quartile range provides a clearer
picture of the overall dataset by
removing/ignoring the outlying values.
• The Quartile deviation is one-half the scale
distance between the 75th and 25th
percentiles in a frequency distribution.
(i.e. Semi-interquartile Range)

• If the middle 50% of scores in the distribution
are densely packed, quartiles will be nearer
to each other & value of Q will be less.
• If the middle 50 % of scores in the
distribution are more spread out, quartiles
will be far from each other & value of Q will
be high.

e.g. (i) 10,10,65,100,120, 180,200, 270,300,500 (n =
10)
• Upper half
180,200, 270,300,500 Q₃ = 270
• Lower half
10,10,65,100,120 Q₁ = 65
• IQR = Q₃ − Q₁ = (270 − 65 ) = 205
• Q = (Q₃ − Q₁ ) / 2 = (270 − 65 ) / 2 = 205 / 2 =
102.5
Mathematically, Q = (Q₃ − Q₁ ) / 2

(For ungrouped data)
e.g. (ii)
22,25,34,35,41,41,46,46,46,47,49,54,54,59,60 (n = 15)
• Upper half (including Median)
46,46,47,49,54,54,59,60 Q₃ = 49 + 54 / 2 = 51.5
• Lower half (including Median)
22,25,34,35,41,41,46,46 Q₁ = 35 + 41 / 2 = 38
• IQR = Q₃ − Q₁ = 51.5 − 38 = 13.5
• Q = (Q₃ − Q₁ ) / 2 = (51.5 − 38 ) / 2 = 13.5 / 2 = 6.75

(For Grouped Data)
Scores Exact Units of
Class Interval
f F
52 – 55 51.5 – 55.5 1 65
48 – 51 47.5 – 51.5 0 64
44 - 47 43.5 – 47.5 5 64
40 - 43 39.5 – 43.5 10 59
36 – 39 35.5 – 39.5 20 49 @
32 - 35 31.5 – 35.5 12 29
28 - 31 27.5 – 31.5 8 17 #
24 – 27 23.5 – 27.5 2 9
20 – 23 19.5 – 23.5 3 7
16 - 19 15.5 – 19.5 4 4
N = 65
# : Which contains the Q₁ @ : Which contains the Q₃

Selection and Application of the Q
The Quartile Deviation is used when;
 only the median is given as the measure of
central tendency;
 there are scattered or extreme scores which
would influence the S.D. excessively;
 the concentration around the Median, the
middle 50 % scores , is of primary interest.

A Deviation score
• A score expressed as its distance from the
Mean is called a deviation score.
x = ( X − )
e.g. 6, 5, 4, 3, 2, 1 Mean ( ) = 21/6 = 3.50
[ e.g. 6 – 3.50 = 2.5 is a deviation score of
6 ]
 Sum of deviations of each value from the
mean :
2.5 + 1.5 + 0.5 + (- 0.5) + (- 1.5 ) + (- 2.5 ) = 0
) = 0 ∑ x = 0

(3) : The Average Deviation : AD or
Mean Deviation (MD)
 AD is the mean of the deviations of all observations
taken from their mean.
 In averaging deviations, to find AD, the signs
( + and − ) are not taken into consideration
i.e. all the deviations are treated as positive.

The Average Deviation : AD
X :
Marks
obtained
x
Deviatio
n
│ x │
18 − 5 5
19 − 4 4
21 − 2 2
19 − 4 4
27 + 4 4
31 + 8 8
22 − 1 1
25 + 2 2
28 + 5 5
20 − 3 3
∑ X = ∑ x = 0 ∑ │x│ = 23
Mean = ∑ X / N
= 230 / 10
= 23
Average Deviation = ∑ │x│ / N
= 23 / 10
= 2.3

The Average Deviation : AD
(For grouped data) : (Under Assumed Mean Method)
Scores
Class
Interval
Exact units
of Class
Interval
Mid -
Point
x
f x‘
Devi.
fx'
60-69 59.5 – 69.5 64.5 1 3 3
50-59 49.5 – 59.5 54.5 4 2 8
40-49 39.5 – 49.5 44.5 10 1 10
30-39 29.5 – 39.5 34.5 15 0 0
20-29 19.5 – 29.5 24.5 8 – 1 – 8
10-19 9.5 – 19.5 14.5 2 – 2 – 4
N = 40 ∑│fx ’│ =
33
Average Deviation = ∑│fx’│ / N = 33 / 40 = 0.825

Selection and Application of the AD
AD is used when:
• It is desired to consider all deviations
from the mean according to their size;
• Extreme deviations would effect
standard deviation excessively.

Limitations : A.D.
• It is based on all deviations, therefore it may
be increased because of one or more
extreme deviation/s.
• All the deviations are treated as positive.
• Needs long mathematical calculations.
Hence, it is rarely used.

The Variance
The sum of the squared deviations from the mean,
divided by N, is known as the Variance.
:
OR
 This value describes characteristics of distribution.
 It will be employed in a number of very important
statistical tests.
 This value is too large to represent the spread of
scores because of squaring the deviations.

(4) : The Standard Deviation : σ
• The S.D. is the most general and stable measure of
variability.
• The S.D. is the positive square root of the variance.
• The Standard Deviation is a measure of how spread
out numbers are.
• The symbol for Standard Deviation is σ (the Greek
letter sigma).

The Standard Deviation : Formulas
• The Population Standard Deviation:
•
• The Sample Standard Deviation:
• The important change is "N-1" instead of
"N" (which is called "Bessel's correction”-
Friedrich Bessel ).
• [ The factor n/(n − 1) is itself called Bessel's correction.]

Calculation of SD
• Example: Ram has 20 Rose plants. The
number of flowers on each plant is
9, 2, 5, 4, 12, 7, 8, 11, 9, 3, 7, 4, 12, 5, 4,
10, 9, 6, 9, 4
Step 1. Work out the mean
In the formula above μ (the Greek letter
"mu") is
the mean

Calculation of SD
• Mean (µ) = ∑ X / N = 140 / 20 = 7
Step 2. Then for each number: subtract
the
Mean and square the result
This is the part of the formula that says:
Example (continued):
• (9 - 7)2 = (2)2 = 4
• (2 - 7)2 = (-5)2 = 25
• (5 - 7)2 = (-2)2 = 4 ……… etc….

Calculation of SD
Step 3. Then work out the mean of those
squared differences.
=
4+25+4+9+25+0+1+16+4+16+0+9+25+4+9+9+4
+1
+4+9 = 178
Mean of squared differences = (1/20) × 178
= 8.9
• (Note: This value is called the "Variance")

Calculation of SD
Step 4. Take the square root of the
Variance:
• Example (concluded):
σ = √(8.9) = 2.983...
• But,
... sometimes our data is only a sample of
the whole population.

Calculation of SD (For the Sample)
• Example: Ram has 20 rose plants, but
what if Ram only counted the flowers on 6
of them?
• The "population" is all 20 rose plants, and
the "sample" is the 6 he counted.
Let us say they are: 9, 2, 5, 4, 12, 7
= 6.5
s = √(13.1) = 3.619...

Comparison
Comparison
of…
N Mean Standard Deviation
Population 20 7 2.983
Sample 06 6.5 3.619
 Sample Mean is wrong by 7%
 Sample Standard Deviation is wrong by 21%
 When we take a sample, we lose some
accuracy.

Calculation of SD
Score (X) x or X − x²
15 1 1
10 − 4 16
15 1 1
20 6 36
8 − 6 36
10 − 4 16
25 11 121
9 − 5 25
∑ x² = 252
Mean ( ) = ∑ X / N
= 112 / 8
= 14
= 252 / 8
= √ 31.8 = 5.64

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