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.

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.

Arithmetic mean:

Consider a set of observations x

In case each x

then, each x

For a given set of observations x

∏ 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.

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.