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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