Question

For grouped data, Arithmetic mean by direct method: -
(a) SfX / Sf
(b) Sd / N
(c) SX / N
(d) None of these

Answer 1

Hint: Consider \[{{X}_{i}}\] as the mid – point of the interval of observations and \[{{f}_{i}}\] as the frequencies of the observations in that particular interval. Here, i = 1, 2, 3, ……n. To find the mean using direct method, for grouped data, multiply the mid – point of intervals (\[{{X}_{i}}\]) with the frequency (\[{{f}_{i}}\]) of that interval and take their sum denoted by \[\sum{{{X}_{i}}{{f}_{i}}}\]. Now, divide this expression with the sum of frequencies, denoted by \[\sum{{{f}_{i}}}\].

Complete step by step answer:
Here, we have to find the arithmetic mean by direct method, for grouped data. First, let us understand what is a grouped data.
Group data are data formed by aggregating individual observations of a variable into groups, so that a frequency distribution of these groups serves as a convenient means of summarizing or analyzing the data.
Let us consider an example to find the formula for mean by direct method. Let us consider the following observations in a grouped form: -

Marks obtained	\[{{X}_{i}}\]	\[{{f}_{i}}\]
0 – 10	\[{{X}_{1}}=5\]	\[{{f}_{1}}=1\]
10 – 20	\[{{X}_{2}}=15\]	\[{{f}_{2}}=3\]
20 – 30	\[{{X}_{3}}=25\]	\[{{f}_{3}}=2\]
30 – 40	\[{{X}_{4}}=35\]	\[{{f}_{4}}=5\]
40 – 50	\[{{X}_{5}}=45\]	\[{{f}_{5}}=2\]

The following table indicates the marks obtained by 18 students. Here, marks are classified in intervals. This is an example of grouped data.
Now, \[{{X}_{i}}\] is the mid – point of marks intervals and \[{{f}_{i}}\] is the frequency or number of students in that interval of marks. So, here the mean is calculated by taking the ratio of \[\sum{{{X}_{i}}{{f}_{i}}}\] and \[\sum{{{f}_{i}}}\].
\[\Rightarrow \] Mean = \[\dfrac{\sum{{{X}_{i}}{{f}_{i}}}}{\sum{{{f}_{i}}}}\]
\[\Rightarrow \] Mean = \[\dfrac{{{f}_{1}}{{X}_{1}}+{{f}_{2}}{{X}_{2}}+{{f}_{3}}{{X}_{3}}+{{f}_{4}}{{X}_{4}}+{{f}_{5}}{{X}_{5}}}{{{f}_{1}}+{{f}_{2}}+{{f}_{3}}+{{f}_{4}}+{{f}_{5}}}\]
\[\Rightarrow \] Mean = \[\dfrac{\left( 5\times 1 \right)+\left( 15\times 3 \right)+\left( 25\times 2 \right)+\left( 35\times 5 \right)+\left( 45\times 2 \right)}{1+3+2+5+2}\]
\[\Rightarrow \] Mean = \[\dfrac{5+45+50+175+90}{13}\]
\[\Rightarrow \] Mean = \[\dfrac{365}{13}\approx 28.08\]
So, the formula for mean using direct method, for grouped data is \[\dfrac{\sum{{{X}_{i}}{{f}_{i}}}}{\sum{{{f}_{i}}}}=\dfrac{\sum{fX}}{\sum{f}}\] (written without indices)

So, the correct answer is “Option A”.

Note: One may note that, in the option we have been provided with \[\dfrac{SfX}{Sf}\]. Actually, here ‘S’ denotes the summation, that is ‘\[\sum{{}}\]’. So, we must not think that ‘S’ is any particular term or variable. Also, note that \[\sum{f}\] can be also written as N, which denotes the total number of observations (here it means students). So, the required formula can also be written as: - Mean = \[\dfrac{\sum{fX}}{N}\].