Here are the steps to solve this problem: (i) Given data: 41, 47, 48, 50, 51, 53, 60 Mean (x̅) = ∑x/n = (41 + 47 + 48 + 50 + 51 + 53 + 60)/7 = 350/7 = 50 Q1 = 48, Q3 = 53 IQR = Q3 - Q1 = 53 - 48 = 5 Q = (Q3 - Q1)/2 = (53 - 48)/2 = 2.5 AD = ∑|x - x̅|/n = (9 + 3 + 2 + 0 + 1 + 3 + 10)/7 = 28/7 =

1. Measures of Variability:
(The range, Quartile Deviation, Average
Deviation and standard deviation )
By: -
Dr. Satish P. Pathak
Department of Education (CASE)
Faculty of Education and Psychology,
The Maharaja Sayajirao University of Baroda,
Vadodara (Gujarat)
………………………………………………………………………

2. Measures of Variability
• 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

3. Measures of Variability
• 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.

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

5. (1) : The Range
• The simplest measure of variability
• Range = The difference between the highest
score and lowest score
• The range is useful for showing the spread
within a dataset and for comparing the
spread between similar datasets.

6. Selection and Application of Range
The Range is used when;
 the data are too scant (little) or too
scattered
 only an idea of extreme scores or of total
spread is wanted

7. 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.
[IQR : It is the range for the middle 50% of the data. It
overcomes the sensitivity to extreme data values. ]

8. Quantiles
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.

9. (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

10. 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)

11. The Quartile Deviation : Q
• 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.

12. The Quartile Deviation : Q
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

13. The Quartile Deviation : Q
(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

14. The Quartile Deviation : Q
(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

15. The Quartile Deviation : Q
(For Grouped Data)
Q₁ = L + N /4 − F x i
f
Q₃ = L + 3N /4 − F x i
f
N / 4 = 65 / 4 = 16.25
3N / 4 = 3x65 / 4 = 48.75
Where,
L = The exact lower limit of the
interval in which the Quartile
falls
i = The length of the interval
F = Cumulative frequency below
the interval which contains
the Quartile
f = The frequency of the interval
containing the Quartile
N = Total number of
observations

16. The Quartile Deviation : Q
(For Grouped Data)
Q₁ = L + N /4 − F x i
f
= 27.5 + 16.25 − 9 x 4
8
= 27.5 + 3.625
= 31.125
Q₃ = L + 3N /4 − F x i
f
= 35.5 + 48.75 − 29 x 4
20
= 35.5 + 3.95
= 39.45
Q = (Q₃ − Q₁ ) / 2 = 39.45 − 31.125 = 4.16
2

17. 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.

18. 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
i.e. ∑ ( X − ) = 0 ∑ x = 0
Definition of the Mean : The Mean is that value in a distribution
around which the sum of the deviation score equals zero.

19. (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.

20. The Average Deviation : AD
(For ungrouped data)
X : Marks
obtained
x
Deviation
│ 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 = 230 ∑ x = 0 ∑ │x│ = 23
Mean = ∑ X / N
= 230 / 10
= 23
Average Deviation = ∑ │x│ / N
= 23 / 10
= 2.3

21. 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

22. 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.

23. 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.

24. 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.

25. (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).

26. 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.]

27. 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
Work out the Standard Deviation.
************
Step 1. Work out the mean
In the formula above μ (the Greek letter "mu") is
the mean

28. 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….

29. 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")

30. 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.

31. 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...

32. 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.

33. Calculation of SD
(For ungrouped data)
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

34. Exercise
(i) Calculate the Mean, Quartile deviation,
Average deviation and Standard deviation for
the given ungrouped data.
41, 47, 48, 50, 51, 53, 60
Reveal your answer.
(ii) Compute S.D. for the given data:
18, 25, 21, 19, 27, 31, 22, 25, 28, 20

35. Calculation of SD
( Direct method without using deviation)
Raw Scores : x x²
15 225
10 100
15 225
20 400
8 64
10 100
25 625
9 81
∑ x = 112 ∑ x² = 1820
σ = √N ∑ x ² − ( ∑ x )²
N
= √ 8 x 1820 − (112)²
8
= 5.612

36. Calculation of Mean and SD
(For grouped data : Based on Frequency Distribution)
C.I. Midd.
Pt. : X
f x:
Devi.
fx fx²
80-84 82 5
75-79 77 6
70-74 72 8
65-69 67 10
60-64 62 16
55-59 57 20
50-54 52 12
45-49 47 9
40-44 42 8
35-39 37 6
100
σ = √N ∑ f x ²
N
x : Deviation of each Middle
point from Mean
Mean = ∑ f . X / N = 58.55
σ = 11.78

37. Calculation of Mean and SD
(For grouped data : Assumed Mean Method)
C.I. Mid. Pt. : X f x´ fx´ fx´²
52-55 53.5 1 4 4 16
48-51 49.5 0 3 0 0
44-47 45.5 5 2 10 20
40-43 41.5 10 1 10 10
36-39 37.5 A.M. 20 0 0 0
32-35 33.5 12 −1 −12 12
28-31 29.5 8 −2 −16 32
24-27 25.5 2 −3 −6 18
20-23 21.5 3 −4 −12 48
16-19 17.5 4 −5 −20 100
N = 65
σ = i √N ∑ f x´ ² − (∑ f x´ ) ²
N
σ = 7.51
i = length of class interval
…………. COMPLETE IT

38. Selection and Application of S.D.
S.D. is used when:
1) the statistics having greatest stability is
required;
2) extreme deviations exercise a proportionally
greater effect upon the variability;
3) co-efficient of correlation and other
statistics are subsequently computed.

39. Thank You