“No human mind is capable of grasping in its entirety the meaning of any considerable quantity of numerical data. We want to be able to express all the relevant information contained in the mass by means of comparatively few numerical values. This is a purely practical need which the science of statistics is able to some extent to meet” (Fisher, 1950 p 7).

1. Measures of Central Tendency
Dr Rajesh Verma
Assistant Professor in Psychology
Govt. College Adampur, Hisar (Haryana)

2. Short Cut to
Mastering
Statistics is .........

3. Short Cut to
Mastering
Statistics is Doing

4. “No human mind is capable of
grasping in its entirety the meaning of
any considerable quantity of numerical
data. We want to be able to express all
the relevant information contained in the
mass by means of comparatively few
numerical values. This is a purely
practical need which the science of
statistics is able to some extent to
meet” (Fisher, 1950 p 7).

5. Definition
The tendency of data to move towards the center.
The single
numerical value that
indicates the orientation
of data towards the
calculated central value
of distribution. This
value is sometimes
called as nuclear
value of the data.

6. Introduction
In statistics, especially social research, central
tendency is a kind of average. Averages are generally
three types
 mean,
 median and
 mode.
In other words central tendency is a single value
generally located
towards the middle or center
of a distribution where
most of the data to be
concentrated (Levin &
Fox, 2006).

7. Purpose of Calculating Averages
1. It is shorthand description of a mass of
quantitative data.
2. Average makes data
meaningful and represents
it.
3. Describes the
population from which
the sample is drawn.
4. To compare two or
more groups in terms of
typical performance.

8. Table 1.1: Hypothetical Data

9. Measures of Central Tendency
The central tendency is measured by: -
1. Mean or arithmetic mean
2. Median
3. Mode

10. 1. Arithmetic Mean – Most commonly used measure of central
tendency and denoted by 𝑿commonly known as X-bar. It is the
sum of the separate scores or measures divided by their number.
Mean = 𝑿 =
∑𝑿
𝑵
(For ungrouped data)
Where 𝑿= mean
∑ = the sum of
x = score or measurement
N = number of scores or measurements
For example we have some hypothetical data such as 21, 33, 29,
31, 20, 18, 32, 23, 27, 36.
∑X = 21+33+29+31+20+18+32+23+27+36=270
N = 10
therefore 𝑿 =
𝟐𝟕𝟎
𝟏𝟎
= 27

11. Mean = 𝑿 =
∑𝒇𝑿
𝑵
(For grouped data)
Where 𝑿= mean
∑ = the sum of
f = Frequency
x = midpoint of class interval
fx = Product of frequency and midpoint
N = number of scores or measurements
As per table 1.1
∑fx = 1480
N = 50
therefore 𝑿 =
𝟏𝟒𝟖𝟎
𝟓𝟎
= 29.60
Note: Mean is a point on the distribution not a
score

12. Other Types of Mean
1. Weighted mean.
2. Harmonic mean.
3. Geometric mean.
4. Arithmetic-Geometric mean.
5. Root-Mean Square mean.
6. Heronian mean.

13. 2. Median – The median is defined as that point on
the scale of measurement above which are exactly half
the cases and below which are other half (Guilford &
Fruchter, 1978). In other words median is middle most
point in a distribution. (Levin & Fox, 2006). It divides
the distribution in two parts as a white strip (median
strip) divides the highway into two parts.
(i) Calculation of Median of ungrouped data: -
Step I – Arrange the scores in ascending order
Step II – Then count the total scores
(a) If scores are odd then the middle score is
median
For example: 18, 20, 21, 23, 27, 29, 31, 32, 33

14. (b) If scores are even then follow the following rule.
𝑴𝒆𝒅𝒊𝒂𝒏 =
(𝑵+𝟏)
𝟐
𝐭𝐡 score
For example: 18, 20, 21, 23, 27, 29, 31, 32, 33, 36
𝑴𝒆𝒅𝒊𝒂𝒏 =
𝟏𝟎+𝟏
𝟐
th score
= 11/2 = 5.5th score counting from the either end of
series.
So, Median is above 27
(5th score) and below 29
(6th score) i.e. 28

15. (ii) Calculation of Median of grouped data: -
Step I – Find the N/2 (50/2 = 25)
Step II – Start at the lowest score end of class interval
column and count off the
scores in order up to the
exact lower limit (l) of the
interval which contains the
median. The sum of these
scores is F. Or just check
the cumulative frequency
of class interval previous to
the class interval in which
median falls.
therefore F or CF = 16
Class
Interval
Frequency
(f)
cf
55-59 1 50
50-54 1 49
45-49 3 48
40-44 4 45
35-39 6 41
30-34 7 35
25-29 12 28
20-24 6 16
15-19 8 10
10-14 2 2

16. Step III – Compute the number of scores necessary to fill out
N/2, i.e. compute N/2-F. Divide the quantity by the frequency
(fm) on the interval which contains the median; and multiply
the result by the size of the
class interval (i)
N/2-F = 25-16 = 9
=
𝟗
𝟏𝟐
𝑿𝟓 = 3.75
Step IV – Add the amount
obtained by the calculation in
(3) to the exact lower limit (l) of
the interval which contains the
median. 25.5+3.75 = 29.25
Note: Median is a point on the
distribution not a score
Class
Interval
Frequency
(f)
cf
55-59 1 50
50-54 1 49
45-49 3 48
40-44 4 45
35-39 6 41
30-34 7 35
25-29 12 28
20-24 6 16
15-19 8 10
10-14 2 2

17. Formula to Calculate Median of grouped data
Median = 𝒍 +
𝑵
𝟐
−𝑪𝑭
𝒇𝒎
𝒊
Where, l = Exact lower limit of the class in which median lies
N = Number of scores
CF = Cumulative frequency of the previous class interval
to the class interval in which median falls.
fm = frequency within the interval upon which the median
falls
i = size of class interval
l = 25.5, N/2 = 50/2 = 25, F or CF = 16,fm= 12, i = 5
Substituting the values = 25.5+
𝟐𝟓−𝟏𝟔
𝟏𝟐
5
= 25.5 + (9/12) 5 = 25.5 +(.75) 5
= 25.5 + 3.75 = 29.25

18. 3. Mode – The measure of central tendency which is
most frequent occurring value of a distribution. The
mode is the only measure of central tendency available
for nominal level variables (Levin & Fox, 2006). In an
ungrouped data the single score which occurs most
frequently is the mode.

19. (a) For ungrouped data
For example in this distribution 11, 23, 25, 25, 30, 32, 36,
36, 36, 45, 48, 51 the most frequently occurring score is 36
which is mode [Crude].
(b) For grouped data
The mode is usually
taken to be the midpoint of
that interval which contains
the largest frequency. In this
table the midpoint of class
interval of 25-29 is 27 which is
mode [crude].
Formula for Mode [True] = 3 Median – 2 Mean
Class Interval Mid point of
Class interval
Frequency
(f)
55-59 57 1
50-54 52 1
45-49 47 3
40-44 42 4
35-39 37 6
30-34 32 7
25-29 27 12
20-24 22 6
15-19 17 8
10-14 12 2

20. References:
1. Guilford, J. P. & Fruchter, B. (1978).
Fundamental Statistics in Psychology and Education.
Tokyo: McGraw Hill.
2. Garrett, H. E. (2014). Statistics in Psychology and
Education. New Delhi: Pragon International.
3. Levin, J. & Fox, J. A. (2006). Elementary
Statistics. New Delhi: Pearson.
4. Upton, G. & Cook,
I. (2008). Oxford
Dictionary of Statistics,
OUP ISBN
978-0-19-954145-4.

21. vermasujit@yahoo.com
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