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 $$x_1,x_2,x_3\dots \dots x_n$$. The arithmetic mean, $$\overline{x}$$, is the sum of these observations divided by the number of observations, $$n$$.
$\overline{x}=\frac{\sum{x}}{n}$
In case each $$x_i$$ is repeated $$f_i$$ number of times, i.e. if each $$x_i$$ has a frequency $$f_i$$ as shown in the following frequency distribution:-

$$x_i$$ $$f_i$$
$$x_1$$ $$f_1$$
$$x_2$$ $$f_2$$
... ...
... ...
$$x_n$$ $$f_n$$
$\sum^n_{i=1}{f_i=n}$

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

$\overline{x}=\frac{\sum{x_if_i}}{n}$
Geometric mean:-

For a given set of observations $$x_1,x_2,x_3\dots \dots x_n$$, 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 $$n^{th}$$ root of the product of values.
$GM={\left(\prod^n_{i=1}{x_i}\right)}^{{1}/{n}}$ $=\sqrt[n]{x_1x_2\dots \dots \dots x_n}$
$$\Pi$$ 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.
$HM=\frac{n}{\frac{1}{x_1}+\frac{1}{x_2}\dots \dots \dots \frac{1}{x_n}}=\frac{n}{\sum{\frac{1}{x}}}$
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:-
$\overline{x}=\frac{50+60+70+80+90}{5}$ $\ \ \ =\frac{350}{5}$ $\ \ \ =70\ runs$
We could state that the batsman scored 70 runs on average in his first five matches.

Example 2

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

Marks $$(x_i)$$ Number of students$$(f_i)$$
$$20$$ $$3$$
$$40$$ $$5$$
$$60$$ $$6$$
$$80$$ $$4$$
$$100$$ $$2$$
$$\sum{f=20}$$

The arithmetic mean can be found as follows:-

Marks $$(x_i)$$ Number of students$$(f_i)$$ $$x_if_i$$
$$20$$ $$3$$ $$60$$
$$40$$ $$5$$ $$200$$
$$60$$ $$6$$ $$360$$
$$80$$ $$4$$ $$320$$
$$100$$ $$2$$ $$200$$
$$\sum{f=20}$$ $$\sum{xf=1140}$$
$\overline{x}=\frac{\sum{x_if_i}}{n}$ $\ \ \ =\frac{1140}{20}$ $\ \ \ =57\ marks$
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:-
$GM=\sqrt{1.1\times 1.2\times 1.3}$ $\ \ \ \ \ \ \ =\sqrt{1.716}$ $\ \ \ \ \ \ \ =1.1972$
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 $$\frac{10+20+30}{3}=20\%$$ If the average growth rate is considered to be $$20\%\$$ , then the total growth rate over three years would have to be $$1.2\times 1.2\times 1.2=1.728$$, or $$72.8\%$$ , which is not actually the case. The actual growth over three years is $$1.1\times 1.2\times 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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