Measures of Central Tendency-Mean, Median , Mode- Dr. Vikramjit Singh

M E A S U R E S O F
C E N T R A L
T E N D E N C Y Dr. Vikramjit Singh

• Measures of central tendency are also usually
called as the averages.
• They give us an idea about the concentration
of the values in the central part of the
distribution.
The following are the five measures of
central tendency that are in common use:
(i) Arithmetic mean, (ii) Median, (iii) Mode,
(iv) Geometric mean, and (v) Harmonic mean
(vi) Weighted mean
Measures of Central Tendency

Mean (Average)
Mean locate the centre of distribution.
Also known as arithmetic mean
Most Common Measure
The mean is simply the sum of the values
divided by the total number of items in the set.
Measures of Central Tendency: Mean

Merits:
• 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.
It is affected by sampling fluctuations. Hence
it is more stable.

Demerits
• It cannot be determined by inspection.
Arithmetic mean cannot be used if we are
dealing with qualitative characteristics,
which cannot be measured quantitatively
like caste, religion, sex.
Arithmetic mean cannot be obtained if a
single observation is missing or lost
• Arithmetic mean is very much affected by
extreme values.

Age of children : 13, 12.5,13, 14, 15, 16,12,16.5
Mean Age of Children=
(13+12.5+13+14+15+ 16+12+16.5)/8 = 14

Q. A Survey of 100 families each having five children,
revealed the following distribution
No. of male children=
No. of Families=
Find the Mean of male children.
0 1 2 3 4 5
9 24 35 24 6 2
Mean x = 200/100 =2

If Odd n, Middle Value of Sequence
If Even n, Average of 2 Middle Value
The median is determined by sorting the data set
from lowest to highest values and taking the data
point in the middle of the sequence.
Middle Value In Ordered Sequence
Not Affected by Extreme Values
Measures of Central Tendency: Median

It is rigidly defined.
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.
Merits:
• Can be determined graphically.

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
Demerits:

If total no. of observations 'n' is even then used the
following formula for median = arithmetic mean of two
middle observations.
For ungrouped data:-
Step-1
Arranged data in ascending or descending order.
Step:-2
If total no. of observations 'n' is odd then used the
following formula for median (n+1) /2 th observation.
Step:-3

If X1, X2, X3......Xn are n
Observation arranged in ascending
or descending order of magnitude.

So median is : 7+1/2= 4th value= 5
Calculate the median for the
following data- 5, 2, 3, 4,5,1,7
Arrange in ascending order:
1,2,3,5,5,7

So median is : 7+1/2= 4th value= 4
Calculate the median for the follw
Arrange in ascending order:
1,2,3,4,5,5,7

74+75 Median = = 74.5
The data on pulse rate per minute of 10 heal
individuals are 82, 79, 60, 76, 63,81, 68, 74, 60, 75. n=
10
60, 60, 63, 68, 74,75, 76, 79, 81, 82
Xn/2 + X(n/2)+1/2

For Grouped data
Median = L+ [(N/2 - C) * h]/ f
where-
L = Lower limit of the median class
N= Total Observation
C= Cumulative frequency of the class preceeding the
frequency class
h= Class height
f= frequency of the median class

Find the median weight of the 590 infants born in a
particular year in a hospital.

So putting in the above formula
Median = L+ [(N/2 - C) * h]/ f
Median = 3 + [(295-154)*0.5]/207
=3+0.34 = 3.34 Kg

Measures of Central Tendency: Mode
Mode is the most frequent
occurring data value in a set of
observation.
There may be no mode or several
mode.

Merits:
Mode is readily comprehensible and easy to
calculate.
Mode is not at all affected by extreme values.
Mode can be conveniently located even if
the frequency distribution has class intervals
of unequal magnitude
Open-end classes also do not pose any
problem in the location of mode.
Mode is the average to be used to find the
ideal size.
1.

Mode is ill defined.
It is not based on all the
observation.
It is not capable of any further
mathematical treatment.
As compared with mean, mode is
affected by fluctuations of
sampling.
Demerits:

For Grouped Data

Age Group 20-30 30-40 40-50 50-60 60-70
No. of
Person
3 20 27 15 9
CI Frequency
20-30 3
30-40 27
40-50 27
50-60 15
60-70 9
Find Mode
F1
Fm
F2

Mode=43.68

R E L A T I O N S H I P B E T W E E N M E A N ,
M E D I A N A N D M O D E
Mode= 3 Median-2 Mean
Summary

G E O M E T R I C M E A N
Geometric mean is defined as the positive root of the
product of observations. Symbolically,
G = (X1,X2,X3 ......Xn) 1/n
It is also often used for a set of numbers whose values are
meant to be multiplied together or are exponential in nature
such as data on the growth of the human population or
interest rates of a financial investment.

G E O M E T R I C
M E A N
What is the geometric mean
of 2,3,and 6?
Step 1. First, multiply the numbers
together 2*3*6=36
Step 2. and then take the cubed root √36= 3.30

H A R M O N I C M E A N
Harmonic mean (formerly sometimes called the
subcontrary mean) is one of several kinds of
average.
The harmonic mean is a very specific type of
average.
It's generally used when dealing with averages of
units, like speed or other rates and ratios.

The Harmonic mean H of positive real number
X1,X2,X3,...Xn is defined as

What is the harmonic mean of 1,5,8,10?
Here,N = 4
H = 4 / (1/1) + (1/5) + (1/8) + (1/10)
H = 4/1.425
H = 2.8

W E I G H T E D M E A N
A weighted mean is a kind of average. Instead of
each data point contributing equally to the final
mean, some data points contribute more "weight"
than others.
If all the weights are equal, then the weighted mean
equals the arithmetic mean (the regular "average"
you're used to).
Weighted means are very common in statistics,
especially when studying populations.

Steps:
1.Multiply the numbers in your data set by the
weights.
2.Add the numbers in Step 1 up. Set this number
aside for a moment.
3.Add up all of the weights.
4. Divide the numbers you found in Step 2 by the
number you found in Step 3.

You take three 100-point exams in your statistics
class and score 80, 80 and 95. The last exam is
much easier than the first two, so your professor
has given it less weight. The weights for the three
exams are:
•Exam 1: 40% of your grade. (Note: 40% as a
decimal is .4.)
•Exam 2: 40% of your grade.
•Exam 3: 20% of your grade

1. Multiply the numbers in your data setby the
weights:
.4(80) = 32
.4(80) = 32
.2(95) = 19
2. Add the numbers up. 32 + 32 + 19 = 83
3. (0.4 + 0.4 + 0.2) = 1
4. 83/1 = 83

The arithmetic mean is best used when the sum of
the values is significant. For example, your grade in
your statistics class. If you were to get 85 on the
first test, 95 on the second test, and 90 on the third
test, your average grade would be 90.
Why don't we use the geometric mean here?
What about the harmonic mean?

What if you got a 0 on your first test and 100 on the
other two?
The arithmetic mean would give you a grade of
66.6.
The geometric mean would give you a grade of O!!!
The harmonic mean can't even be applied at all
because 1/0 is undefined.

Measures of Central Tendency-Mean, Median , Mode- Dr. Vikramjit Singh

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