FORMULA AND DERIVATION
Consider a set of observations x1
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
. So, the total sum of values is nothing but the sum all xi
. Therefore, the formula for the arithmetic mean in this case is:-
For a given set of observations x1
, 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 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.
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:-
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.