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MEAN
DEFINITION
In general sense, mean is one of the many forms of averages.
In mathematics and statistics, the arithmetic mean is the sum of a set of values divided by the number of values.
The geometric mean is the product of n values raised to the power of 1/n.
The harmonic mean is the reciprocal of the arithmetic mean of the reciprocals of values.
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Arithmetic mean:

FORMULA AND DERIVATION
Arithmetic mean:-

Consider a set of observations x1,x2,x3………xn. The arithmetic mean, x ̅, is the sum of these observations divided by the number of observations, n.



In case each xi is repeated fi number of times, i.e. if each xi has a frequency fi as shown in the following frequency distribution:-



then, each xi must be added fi number of times, which is the same as xi× fi. So, the total sum of values is nothing but the sum all xi fi. Therefore, the formula for the arithmetic mean in this case is:-



Geometric mean:-

For a given set of observations x1,x2,x3………xn, the geometric mean considers the product of these values(as against the arithmetic mean, which considers the sum) raised to the power 1/n. In simpler words, the geometric mean of n observations is the nth root of the product of values.



∏ is the notation used to denote the product of values.

Harmonic mean:-

Harmonic mean first considers the arithmetic mean of the reciprocals of all values under consideration. Further calculating the reciprocal of this value gives us the harmonic mean.



EXAMPLES
Example 1

Let us suppose that a batsman scores 50, 60, 70, 80 and 90 runs respectively in his first five cricket matches.

We could calculate the batsman’s average runs using arithmetic mean, as follows:-



We could state that the batsman scored 70 runs on average in his first five matches.

Consider the following marks of 20 students in a class:-



The arithmetic mean can be found as follows:-





Example 3

The rate of increase in the price of a company’s share was 10%, 20% and 30% in the last three years. The average growth needs to be determined.

The growth factors would 1.10, 1.20 and 1.30 respectively for the three years. The geometric mean would:-



Since the geometric mean is 1.1972, it indicates an average growth of 19.72% per year.
It means that, a growth rate of 19.72% per year for three years is the same as growth rates of 10%, 20% and 30% respectively for three years.

If arithmetic mean was used for this example, we would get a different average growth rate, which would be If the average growth rate is considered to be 20%, then the total growth rate over three years would have to be 1.2 ×1.2 ×1.2=1.728, or 72.8%, which is not actually the case. The actual growth over three years is 1.1 ×1.2 ×1.3=1.716, or 71.6%.
Using arithmetic mean to find averages for growth rates, tends overstate the average value.

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