Central tendency and Variation or Dispersion

1. Central Tendency
and Variation
By;
Mr. Johny Kutty Joseph
Asstt. Professor

2. Measures of Central Tendency
• An average value within the range of the
entire data that is used to represent all of
the values of the series. Such indexes are
called measures of central tendency. It is of
three types;
• A. Arithmetic Mean or Mean
• B. Median
• C. Mode

3. Mean
• It is the most common method of central
tendency.
• Mean is computed by dividing the sum of all
the values by the total number of values. It
is represented by X
• The Mean can be of (Discrete data, discrete
frequency data, continuous frequency data.
• It is calculated by the formula (Discrete data)
X = sum of the values (Σx)
number of the values (n)

4. Mean (Discrete)
• Example: The hemoglobin levels of 10
women are given. (12.5, 13,10,11.5,
11,14,9,7.5,10,12) Calculate the Mean (X)
• X = 110.5/10 = 11.05.
• Calculate Mean from Discrete frequency
table.
• X = Σfx/Σf (where “x” = corresponding
value variable, “f” = frequency of the
variable.)

5. Mean (Frequency)
• Example: Given data gives age of 100 girls.
Find the mean age.
• X = Σfx/Σf (where “x” = corresponding
value variable, “f” = frequency of the
variable.)
• X = 1713/100 = 17.13 years.
Age in years (x) No. of students (f)
16 35
17 31
18 20
19 14
Σf=100 (n)
xf
560
527
360
266
Σxf=1713

6. Mean (Grouped Data)
• Calculate Mean from Continuous frequency Table.
• X = Σxf /Σf (where “x” = mid point of class interval.
“f” is the frequency of class interval. Mid point of
the class interval is calculated by “lower limit +
upper limit/2”)
• Calculate the Mean age of following people.
• Mean = 3762.5/135=27.87 years
Age in years (x) No. of people (f)
15-20 15
20-25 20
25-30 40
30-35 60
Σf=135 (n)
Mid point (x)
17.5
22.5
27.5
32.5
xf
262.5
450
1100
1950
Σxf=3762.5

7. Merits & Demerits of Mean
Merits
• It is simple average and easy to compute.
• It is affected by values of each item in the series.
• It is fixed mathematical formula and always gives
the same answer.
Demerits
• Very small and very large items usually affect the
value of average.
• Mean cannot be computed in case of open ended
classes (with no upper/lower limit).
• Not a good measure of central tendency.

8. Median
• It is the middle most value when the data is
arranged in ascending order of magnitude.
• It divides the observation into two equal
parts.
• One half will have items larger than the
Median value.
• One half will have items smaller than the
Median value
• It is denoted by “M”.

9. Median for Discrete Data
• Formula: M = n + 1/ 2 th observation .
• The following data gives 7 person weight in
pounds: 158,167,143,169,172,146,151.
Calculate the Median.
• Arrange the data: 143,146,151,158,167,169
and 172.
• M=n + 1/2 = 7 + 1/2 = 4th observation = 158
• If “n” is an even number: Add the values of
the observation before and after the fraction
result and divide with 2 (ie. 3.5; 3 + 4 /2).

10. Median for Discrete Frequency Data
• Formula: M = n + 1/2 th observation .
• First calculate cumulative frequency.
• Consider the final cumulative frequency as “n”.
Calculate “M”
• The value of the observation will be in reference
to the cumulative frequency column.
• In case the calculated “M” does not exactly
match with the “cf” in the table, the next
higher cumulative frequency to the calculated
“M” may be considered as observation for
median.

11. Median for Discrete Frequency Data
• Example: Calculate median for the following
frequency table data.
• M= n+1/2 = 53+1/2 = 54/2 = 27th
observation = M= 200
Income/day (x) 100 150 200 250 300 350
Number of People (f) 5 19 03 11 06 09
Cumulative
Frequency (cf)
5 24 27 38 44 53

12. Median for Grouped Data
• In case of continuous frequency table, Median
can be calculated using the formula:
• Where; “n” = Σf, “cf” (some time denoted as
“m”)= the cumulative frequency of the class
just before the median class, f = frequency of
the median class, w = the width of the class
and Lm = lower limit of the median class.

13. Median for Grouped Data
• Calculate the median for the following
grouped/ frequency data.
• First calculate the cumulative frequency.
Income/da
y (x)
100- 150 150 - 200 200 -
250
250 -
300
300 -
350
350 -
400
Number of
People (f)
5 19 03 11 06 09
Income/da
y (x)
100- 150 150 - 200 200 -
250
250 -
300
300 -
350
350 -
400
Number of
People (f)
5 19 03 11 06 09
Cum.
frequency
(cf)
5 24 27 38 44 53

14. Median for Grouped Data
• Find the median class using the formula n + 1/2th
observation where n = cf.
• In the given example; n m+ 1 /2th = 53 + 1 / 2 =
27th class of observation ie 200 – 250.
• Apply the formula
• M = (53/2 – 24) / 3 x 50 + 200.
• M= 2.5/3 x 50 + 200 = 2.5/3x50+200 = 241.7
• Therefore M = 241.7. The value of median always
falls in the median class.
• Now, n = 54 = median class will be 54 + 1 /2 th
class; i.e. 27.5th class; In this case the class that
includes 27.5 is cumulative frequency of 38.

15. Merits & Demerits of Median
Merits
• It is useful in case of open ended and unequal
classes.
• Extreme values do not affect the median.
• Most appropriate in case of qualitative data.
• Can be represented graphically.
Demerits
• Arranging data in frequency is required.
• Median is not generally calculated for
quantitative data since it is not useful in further
algebraic treatments.

16. Mode
• It is the value which is has the highest
frequency.
• It is noted by “ Z”.
• Calculate the mode of the following discrete
data. (1,7,4,1,4,5,3,4,7,9,6,5,4,3,2,4,5,4,4,3)
• In the given series Z = 4

17. Mode Discrete Frequency Data
• Calculate “Z” for the given data.
• Highest frequency is 19 and hence the
corresponding figure “ 150” is the mode.
Income/da
y (x)
100 150 200 250 300 350
Number of
People (f)
5 19 03 11 06 09

18. Mode for Grouped Data
• In case of continuous frequency data table, Mode
can be calculated using the formula:
• Z = l + (f – f1)w
2f – f1 – f2
Where;
l = lowest limit of the modal class
f = frequency of the modal class
w = width of the modal class
f1= frequency of the class just before the modal class.
f2 = frequency of the class just after the modal class.
• Modal class is the one which has the highest
frequency.

19. Mode for Grouped Data
• Example: Calculate the mode for the following;
• Apply the formula;
Z = l + (f – f1)w
2f – f1 – f2
• l = 150, f = 19, w = 50, f1=5, f2=3
• 150 + (19-5)x50/2x19-5-3
• 150 + 14x50/38-8 ; 150+ 700/30 = 150 + 23.33
• Z = 173.33
Income/da
y (x)
100- 150 150 - 200 200 -
250
250 -
300
300 -
350
350 -
400
Number of
People (f)
5 19 03 11 06 09

20. Mode for Grouped Data
• There may be no mode if no value appears
more than any other.
• There may also be two modes (bimodal), three
modes (trimodal), or four or more modes
(multimodal).
• If there are more classes with same
frequency then the mode will be more.

21. Mode by Grouping Method
• A common approach to this is grouping
method or finding the relation of mean,
median and mode.
• The relation can be explained as Mode = 3 *
Median – 2 * Mean
• This method can be used calculate accurate
mode for bimodal, tri-modal type of data.

22. Mode by grouping method.
X f 1
15 5
16 2
17 7
18 6
19 7
20 2
21 3
22 4
f 2
7
13
9
7
f 3
9
13
5
f 6
20
9
f 5
15
12
f 4
14
15

23. Mode by Grouping method
• Arrange the data in a table for analysis.
• Mode is = 18
X f1 f2 f3 f4 f5 f6
15
16 
17    
18     
19    
20 
21
22

24. Measures of Dispersion/Variation
• The observation deviating
from the central value
• It is also called as variability
of the data.
• The Means are same.
• The blue and green lines are
more heterogeneous and red
is homogenous.
• Variability or dispersion is
the measure to express the
extent to which the scores
are different from each
other.

25. Measures of Dispersion
• The different measures of
variability or dispersion
are the following
• Range
• Mean Deviation
• Standard Deviation
• Quartile Deviation

26. Range
• Range is the difference between the highest and
lowest value in the data.
• R = H – L (H = highest value, L = Lowest Value).
• Example: 3,5,6,8,9,3,4,5,6,9,12,13: R = 13-3=10.
• Example: 0-10, 10-20, 20-30, 30-40; R = 40-0=40
• Merits of Range: simple to understand and easy
to calculate.
• Demerits of Range: not suitable for deep
analysis and in case of extreme values.

27. Standard Deviation (SD)
• It is the positive square root of mean of the
squared deviations of values from the arithmetic
mean.
• Most commonly used and denoted by “SD” or σ
(sigma)
• Formula for Discrete Data
• Formula for Grouped/continuous Data

28. Standard Deviation (SD)
• In most of the cases n-1 is used instead of n
as denominator in SD formula.
• It is called as Bessel's correction as it gives
an unbiased estimator of the population
variance.
• N is appropriate for sample but n-1 creates
more accuracy in terms or generalization to
population.

29. Standard Deviation (SD)
• Given are the data of blood cholesterol levels
of 10 persons. Calculate SD.
• ( 240, 260, 290, 245, 255, 288, 272, 263,
277, 257)
• Prepare a table
• Calculate the Mean of the Data.
• In this case Mean = 264.7.
• Compute Σ (X – x )2

30. Standard Deviation (SD)
x
240
245
255
257
260
263
272
277
288
290
Σx =
2647
(x - x)
240 – 264.7 = - 24.7
245 – 264.7 = - 19.7
255 – 264.7 = - 9.7
257 – 264.7 = - 7.7
260 – 264.7 = - 4.7
263 – 264.7 = - 1.7
272 – 264.7 = 7.3
277 – 264.7 = 12.3
288 – 264.7 = - 23.3
290 – 264.7 = - 25.3
X = 264.7
(x – x )2
610.09
388.09
94.09
59.29
22.09
2.89
53.29
151.29
542.89
640.09
Σ(x – x )2
= 2564.1
Compute SD by
using the
formula
SD = √ 2564.1
10
= √256.41
SD= 16.01

31. Standard Deviation (SD)
• Given are the data of weight of adolescent
girls. Calculate SD for the continuous
frequency data/grouped data.
• Prepare the table and compute Σ (X – x )2
• Calculate the Mean using the formula
Σxf/Σf. (can be performed in the table)
Weight 60-64 64 - 68 68 - 71 71 - 74
No. of
subjects
10 09 07 02

32. Standard Deviation (SD)
x f
60 - 64 10
64 - 68 09
68 – 72 07
72 – 76 02
Σf = 28
(x - x)
- 4.14
00
3.86
7.86
(x – x )2
17.14
00
14.9
61.78
Midpoint
x
62
66
70
74
xf
620
594
490
148
Σxf = 1852
f(x – x )2
171.40
00
104.3
123.56
Σf(x – x )2 = 399.26
Compute SD by using
the formula
SD = √ 399.26
28
= √14.26
SD= 3.78

33. Characteristics of Good variation
• It should be rigidly defined.
• Easy to understand and calculate.
• It should be based on all observations.
• It should be amenable for further algebraic
treatment.
• It should be affected by sampling
fluctuations.
• Considering all the above factors SD is
considered as the good measure of variation.