The document discusses different measures of central tendency including the mean, median, and mode. It provides formulas and examples for calculating each measure. The mean is the average value and is calculated by summing all values and dividing by the total number. The median is the middle value when values are arranged in order. The mode is the value that appears most frequently in a data set.

1. Measures of Central
Tendency
DR. NAVNATH DNYANDEO INDALKAR
ASSISTANT PROFESSOR,
SMT. PUTALABEN SHAH COLLEGE OF EDUCATION, SANGLI

3. Central Tendency
 Central tendency is defined as “the statistical measure that identifies a single value as
representative of an entire distribution.”
 Central tendency is a descriptive summary of a dataset through a single value that reflects
the center of the data distribution.
 It aims to provide an accurate description of the entire data.
 It is the single value that is most typical/representative of the collected data.

4. Normal Possible Curve

6. Mean (Average)
 Mean is the sum of the separate scores divided by their numbers.
 Mean locate the Centre of distribution.
 Also known as arithmetic mean.
Formula
𝑀 =
Ʃ𝑋
𝑁
( arithmetic mean calculated from ungrouped data)

7. Mean ( Ungrouped Data)
Formula
𝑀 =
Ʃ𝑋
𝑁
Example : Find the mean of given data.
46,35,22,47,36,26,33

8. Computation of Mean from
Classified Scores
Formula
Mean - 𝑀 =
Ʃ𝑓𝑋
𝑁

9. Scores 10 9 8 7 6 5 4 3 2 1
Frequency 1 0 4 4 9 12 7 9 2 2
Scores (X) Frequency(f) fx
10 1 10
9 0 00
8 4 32
7 4 28
6 9 54
5 12 60
4 7 28
3 9 27
2 2 04
1 2 02
Total N = 50 Ʃfx = 245
Mean - 𝑀 =
Ʃ𝑓𝑋
𝑁
𝑀 =
245
50
= 4.90

10. Mean of Class distribution Data
𝑀 = 𝐴𝑀 + (
Ʃ𝑓𝑑
𝑁
) × i
M = Mean
AM = Assumed Mean
Ʃ = Sign of Simmation
d = Deviation
F = Frequency
N = No. of cases
i = Length of Class Interval

11. Example
CI Xm f d fd
90-99 1 5 5
80-89 3 4 12
70-79 3 3 9
60-69 4 2 8
50-59 6 1 6
40-49 44.5 10 0 0
30-39 5 -1 -5
20-29 2 -2 -4
10-19 3 -3 -9
0-9 3 -4 -12
N = 40 Ʃfd = 10

12. 𝑀 = 𝐴𝑀 + (
Ʃ𝑓𝑑
𝑁
) × i
AM = 44.5
Ʃ𝑓𝑑 = 10
N = 40
I = 10
𝑀 = 44.5 + (
10
40
) × 10
= 44.5 + 2.5 = 47 Mean = 47

13. CI f
135-149 1
120-134 2
105-119 4
90-104 7
75-89 6
60-74 7
45-59 7
30-44 5
15-29 3
0-14 2
Calculate the Mean of Following frequency distribution.

14. Combined Mean ( Mcomb)
Formula Mcomb=
Ʃ𝑀𝑁
Ʃ𝑁
ƩMN = Addition of Multiplication of Mean and number of each group
ƩN = Total number of all groups

15. Group M N Multiplication
Group 1 58.40 40 2336
Group 2 45.62 50 2281
Ʃ𝑁 = 90 Ʃ𝑀𝑁 = 4617
Mcomb=
Ʃ𝑀𝑁
Ʃ𝑁
Mcomb=
4617
90
= 51.30

16. Merits of Mean
 It is easy to understand and easy to calculate.
 It is based upon all the observations
 It is familiar to common man and rigidly defined.
 It is capable of further mathematical treatment. Correlation,
Standard deviation etc.
 Stable and reliable.

17. Demerits of the Mean
 It cannot be determined by inspection.
 Arithmetic mean cannot be obtained if a single observation os missing or lost.
 Arithmetic mean is very much affected by extreme values.

19. Median
 If the scores in a series are arranged in ascending or descending
order, then the point, which comes exactly at the center of the
series, is known as the Median.
 When ungrouped scores or other measures are arranged in order of
size, the median is the midpoint in the series.
 When the scores in any series are arranged in the ascending or
descending order, the point, which divides the series into two equal
parts, is called the Median.

20. Median
Mdn. =
𝑁+1
2
Series –
5,10,16,25,30,
12,15,16,17,18,19, 26, 28, 35, 45,48
Solve the below example.
10, 8, 12, 15, 6, 25, 5

21. Compute Median for the series given below.
8, 12, 4, 16, 6, 10, 16, 8
Ascending order
4, 6, 8, 8, 10, 12, 16, 16
Mdn. =
𝑁+1
2
=
8+1
2
=
9
2
= 4.5
This means the Median of the series falls at 4.5th order. Therefore the 4.5th score will
be –
8+10
2
=
18
2
= 9
The Median is 9

22. Computation of Median form frequency
Distribution
Formula Mdn.
Mdn.= L +
𝑁
2
−𝑓𝑏
𝑓𝑚
× 𝑖
L = lower limit of that CI in which Mdn. Falls
N/2 = Half of N
Fb= Cumulative frequency below that CI in which mdn. falls.
Fm = Frequency of that CI in which mdn. falls.
i = length of CI

23. CI f cf
37-39 1 54
34-36 3 53
31-33 4 50
28-30 3 46
25-27 8 43
22-24 5 35
19-21 9 (fm) 30
16-18 7 21 (fb)
13-15 5 14
10-12 9 9
Mdn.= L +
𝑁
2
−𝑓𝑏
𝑓𝑚
× 𝑖
Mdn.= 18.5 +
54
2
−21
9
× 3
18.5 + 2 = 20.50
Mdn. = 20.50

24. CI 124
-
131
116
-
123
108
-
115
100
-
107
92-
99
84-
91
76-
83
68-
75
60-
67
52-
59
f 3 6 9 12 14 9 7 5 3 2
Compute Median for the given data .

25. Merits of Median
 It is easy to understand and easy to calculate.
 It is not at all affected by extreme values.
 It can be calculated for distributions with open-end classes.
 Median is the only average to be used while dealing with qualitative data .
 Can be determined graphically.

26. Demerits of the Median
In case of even number of observations median cannot be determined exactly.
It is not based on all the observations.
It is not capable of further mathematical treatment.

27. Mode
A score, which appears maximum times in a series, is known as Mode.
A score, which repeats itself maximum times in series is called Mode.
Mode is the perhaps the easiest measure of the Central Tendency.

28. Find out the Mode of the following series
.
5, 10, 10, 4, 10, 5, 3, 10
In this series the score, which appears most of the times is ‘10’. Its frequency is 4, which is
highest in the series, and therefore, ’10’ is the Mode of the series.

29. Computation of Mode for Frequency
Distribution
Formula
Mode = 3Mdn. – 2Mean

30. CI f d fd
80-86 2 3 6
73-79 8 2 16
66-72 11 1 11/ 33
59-65 15 0 0
52-58 12 -1 -12
45-51 7 -2 -14
38-44 5 -3 -15/-41
N=60 Ʃ𝑓𝑑 = -8
𝑀 = 𝐴𝑀 + (
Ʃ𝑓𝑑
𝑁
) × i
𝑀 = 62 + (
−8
60
) × 7
62+(-0.933)
M= 61.067