This document discusses measures of central tendency including the mean, median, and mode. It defines each measure and provides examples of calculating them using individual, discrete, and continuous data series. The mean is the average value obtained by dividing the sum of all values by the total number of values. The median is the middle value when data is arranged in order. The mode is the value that occurs most frequently. Formulas and examples are given for calculating each measure using different types of data sets. Merits and demerits of each measure are also outlined.

1. Dr. Mitali Gupta
Dr. Ambedkar Institute of Management Studies &
Research
Measures of Central
Tendency

2.  A measure of central tendency is a single value that
attempts to describe a set of data by identifying the
central position within that set of data. As such,
measures of central tendency are sometimes called
measures of central location.
Definition

3. Measures of Central Tendency
 Mean-mean is the total of all the values, divided
by the number of values.
 Median-median is the middle number in a list of
numbers ordered from smallest to largest.
 Mode-mode is the value that appears most often
in a set of data.

4. Frequency
 A frequency is the number of times that a particular
data value occurs in a data set.
A series of data that is formed along with the
frequencies of their occurrences is called a frequency
series. A frequency series is again, of three types viz.
1. Individual series
2. Discrete series, and
3. Continuous series.
Frequency Series

5. Mean
 The mean (or average) is the most popular and well
known measure of central tendency. It can be used
with both discrete and continuous data, although its
use is most often with continuous. The mean is
equal to the sum of all the values in the data set
divided by the number of values in the data set. So,
if we have values in a data set and they have values
x1,x2,x3…xn, the sample mean, usually denoted by
(pronounced "x bar"), is:

6. Merits of Arithmetic Mean:
 1. It is a mathematical mean.
 2. It is easy to calculate.
 3. It is always definite.
 4. It establishes a simple relation among the values
of data series.

7. Demerits of Arithmetic Mean:
 1. It cannot be determined by simple visual
observation of data.
 2. It cannot be presented by chart or graph.
 3. It is not suitable for qualitative studies.
 4. Sometimes it does not show representation of
series.

8. Arithmetic Mean for Individual Series
(Marks)
X
65
55
42
58
94
86
∑X=400

9. Arithmetic Mean for Discrete Series
(Marks) Freq fx
X F X.F
20 8 160
30 12 360
40 20 800
50 10 500
60 6 360
70 4
N=60
280
∑fx=2460
Discrete Series

10. Arithmetic Mean (Continuous Series)
Marks (X) F MV F.MV
0-10 5 5 25
10-20 10 15 150
20-30 25 25 625
30-40 30 35 1050
40-50 20 45 900
50-60 10
N=100
55 550
∑fmv=3300

11. Median
 The median is that value of the variable which
divides the group into two equal parts, one part
comprising all values greater and the other values
less than the median.”
 Median is the middle value of the series when items
are arranged either in ascending or descending
order.

12. Merits of Median
 It is easy to calculate and simple to understand.
 In many situations median can be located simply by
inspection.
 It is not affected by the extreme values i.e. the
largest and smallest values. Because it is a
positional average and not dependent on magnitude.
 Median is the best measure of central tendency
when we deal with qualitative data, where ranking is
preferred instead of measurement or counting.

13. Demerits
 It is not based on all the observations of the series.
 It is not capable of further algebraic treatment like
mean, geometric mean and harmonic mean.

14. Calculation of Median (Individual Series)
 Wages (in ascending order): 108, 110, 112, 115, 116,
120, 140
From the following data of the wages of 7 workers compute
the median wage

15. Calculation of Median (Discrete
Series)
From the following data find the value ofmedian:
Income (Rs) No. person (f) Cumulative freq (cf)
80 16 16
100 24 40
150 26 66
180 30 96
200 20 116
250 6 122
N=122

16. Calculation of Median (Continuous
Series)
Calculate the median for the following frequency di
Marks No. students(f) Cumulative freq (cf)
5-10 7 7
10-15 15 22
15-20 24 46
20-25 31 77
25-30 42 119
30-35 30 149
35-40 26 175
40-45 15 190
45-50 10 200
N=200

17. Mode
 The mode is the number that appears most
frequently in a set. A set of numbers may have one
mode, more than one mode, or no mode at all.

18. Merits of Mode
 It is easy to understand and simple to calculate.
 It is not affected by extremely large or small values.
 It can be located just by inspection in ungrouped
data
 It can be located graphically.

19. Demerits of Mode
 It is not well defined.
 It is not based on all the values.
 It is stable for large values so it will not be well
defined if the data consists of a small number of
values.
 It is not capable of further mathematical treatment.

20. Calculation of Mode (Individual
Series)
From the following data calculate the value ofmode:
3 X No. of timesoccurred
5 3 2
8 4 2
5 5 3 Max
4 8 1
5 9 1
9
3 Mo=5
4

21. Calculation of Mode (Discrete
Series)
Calculate the mode from the following data:
size of garments No. people
28 10
29 20
30 40
31 65 Max Mo=31
32 50
33 15

22. Calculation of Mode for Continues Series

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