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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 \(x_1,x_2,x_3\dots \dots x_n\). 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:-
\[GM=\sqrt[3]{1.1\times 1.2\times 1.3}\] \[\ \ \ \ \ \ \ =\sqrt[3]{1.716}\] \[\ \ \ \ \ \ \ =1.1972\]
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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