Measures of Central Tendency
1
Mean
Mean: It is the central value around which the
observations are scattered.
Mean = Total of all observations
Total no. of observations
Mean =
x

x
i
n
2
Median
Median: It is that value which divides the
observations into two equal parts when all
observations are arranged in ascending or
descending order.
3
Mode
Mode: It is the observation which occurs most
frequently in the data.
4
Geometric Mean
Sometimes when we are dealing with quantities
that change over a period of time, we need to
know an average rate of change, such as average
growth rate over a period of several years. In
such cases, the simple arithmetic mean is
inappropriate, because it gives a wrong answer.
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The geometric mean of a data set
a1, a2, ..., an is given by


  ai 
 i 1 
n
1
n
 n a1.a2 .....an
6
E.G. A person has invested `5000 in the stock
market. At the end of the first year the amount has
grown to `6250; he has had a 25% profit. If at the
end of the second year his principle has grown to
`8750, the rate of increase is 40% for the year. What
is the average rate of increase of his investment
during the two years?
*If we calculate the arithmetic mean, it would be
(25+40)/2 = 32.5
*Applying this rate he gets interest
= 5000(1+32.5/100)^2= `8778
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* But the actual rate of interest can be calculated
as `8750 = 5000(1 + r/100)^2
On simplification gives r = 32.2876
* Now this can be explained with the concept of
geometric mean as √(1.25 x 1.40) = 1.322876
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We use the geometric mean to show
multiplicative effects over time in
compound
interest
and
inflation
calculations.
The geometric mean must be used when
working with percentages (which are
derived from values), whereas the standard
arithmetic mean will work with the values
themselves.
9
Harmonic Mean
The harmonic mean is a better "average" when
the numbers are defined in relation to some unit.
The common example is averaging speed, i.e. the
harmonic mean is used when averaging rates or
ratios.
The harmonic mean of the positive real numbers
x1, x2, ..., xn is defined to be
n
1
1
1

 .... 
x1
x2
xn

n
n

i 1
1
xi
10
E.G. An investor buys `1200 worth of shares in a
company each month. During the first five
months he bought the shares at a price of `10,
12, 15, 20 and 24 per share. After 5 months what
is the average price paid for the shares by him?
*If we calculate the arithmetic mean, it would be
= (10+12+15+20+24)/5 = 16.2
*The actual average price per share
= Total amount spent / Total number of shares
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Amount Spent
Price per Share
Number of Shares
1200
10
120
1200
12
100
1200
15
80
1200
20
60
1200
24
50
Total = 6000
Total = 410
The actual average price per share
= 6000/410 = 14.63
* Now this can be explained with the concept of
harmonic mean as
5
 14.63
1
1
1
1
1




10 12 15 20 24
12
Percentile
Percentile: It is that value which divides the
observations into 100 equal parts when all
observations are arranged in ascending or
descending order.
The kth percentile is the value which divides the
data such that k% of the data lies below the value
and (100-k)% of the data lies above the value.
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