This document discusses different measures of central tendency including the mode, median, and mean. It provides examples of how to calculate each measure using both raw and grouped data. The mode is the most common value, and is appropriate for qualitative or nominal level data. The median is the middle value when data is ordered from lowest to highest, and is used for ordinal or interval level data. The mean is the average and is calculated by summing the product of each value and its frequency, divided by the total number of values. It requires interval level data. The appropriate measure depends on the level of measurement and research objective.

1. Measures of central tendency
Mean
Mode
Median

2. Mode
 Distribution of application for admission to
M.B.A. by discipline,
Findings,
 The commerce category is the most
predominant.
Discipline No. of students
Science 55
Arts 60
Commerce 101
Engineering 45
Medical 5
Total 266

3. Mode – Grouped data.
Procedure,
1. Find the category or the class interval which
has the greatest frequency.
2. The midpoint on this category is the mode.
Age Group Frequency
1--20 15
21--40 32
41--60 54
61--80 30
81--100 19
Total 150

4.  Mode frequency is 54 which is associated with
the modal class interval of 41-60.The
midpoint of this class interval is 50.5 i.e.
(41+60)/2 = 50.5
 Mode of this distribution is 50.5
Age Group Frequency
1--20 15
21--40 32
41--60 54
61--80 30
81--100 19
Total 150

5. Use:
 Useful measure for qualitative data.
 Appropriate measure for nominal
(qualitative) level of data.
 E.g. include gender, nationality, ethnicity,
language, genre, style, biological species, and
form.
 Application of data can be in distribution like
ethnic classification of occupational
classification.
 For finding out the typical category.

6. Median
Grouped data
 Procedure,
= l + ((n/2 – cf) / f )*i
 Find out mid point (n/2)
 Find the class interval of mid point from the
cumulative frequency column (cf)
 l is the lower limit of the median category.
 cf is the cumulative frequency up to but not
median category.
 i is the size(range) of the median class
interval.

7. AgeGroup Frequency CumulativeFrequencies
1--20 15 15
21--40 32 47
41--60 54 101
61--80 30 131
81--100 19 150
Total 150

8. n/2= 150/2=75, Class interval= 41-60, f=54
l= 41, Cf=47, i=20
= l + ((n/2 – cf) / f )*i
=41+((75-47)/54)*20
Mean =51.4
Use:
 Used for ordinal (where the order matters but
not the difference between values)or interval
(where the difference between two values is
meaningful)level data but not for nominal level
data.

9. Arithmetic mean
= ∑ (fx) / n
 X=any value
 f= the frequency of a value
 ∑=sum
 n=the number of value
Daily wages (f) Frequency (x) Fx
6 4 24
7 8 56
8 6 48
9 12 108
10 7 70
12 4 48
15 2 30
Total 43 384

10.  Arithmetic mean= = ∑ (fx) / n
 =384/43=8.9

11. Grouped data
= ∑ (fm) / n
 m=midpoint of each class interval
 f=frequency of a value
 fm=midpoint multiplied by its frequency
 n=number of cases

12. =460/20=23.
Score Frequency(f) Midpoint(m) fm
11--15 3 13 39
16--20 4 18 72
21--25 7 23 161
26--30 3 28 84
31--35 2 33 66
36--40 1 38 38
Total 20 460

13. Choice of an appropriate average
The choice depends upon the consideration of
several factors:
 The level of measurement
 The research objective

14. Level of measurement:
 Mode requires only a frequency count, it can
be applied to any set of data at the nominal,
ordinal or interval level of data.
 Median requires an ordering of items form
the highest to the lowest or vice versa.
Hence it can be obtained from an ordinal or
interval level of data and not from nominal
data like party affiliation, caste or religion.
 Mean is exclusively restricted to interval
data such as income, age, wage rate & test
score.

15. Research objective:
 Mode is useful to find out most common
category. E.g. test score, caste, age
 Mean is useful for further mathematical
manipulations.
 Median is useful to find out mid values.