Question

Choose the formula used for arithmetic mean of grouped data by shortcut method is .
A $\mathop x\limits^\_ = A - \dfrac{{\sum\limits_{i = 1}^n {fd} }}{{\sum\limits_{i = 1}^n f }}$
B $\mathop x\limits^\_ = A + \dfrac{{\sum\limits_{i = 1}^n {fd} }}{{\sum\limits_{i = 1}^n f }}$
C $\mathop x\limits^\_ = A \times \dfrac{{\sum\limits_{i = 1}^n {fd} }}{{\sum\limits_{i = 1}^n f }}$
D $\mathop x\limits^\_ = A \div \dfrac{{\sum\limits_{i = 1}^n {fd} }}{{\sum\limits_{i = 1}^n f }}$

Answer 1

Hint: For the short-cut method of mean we have to take deviation $A$ take at any point ${d_i} = {x_i} - A$ , where, $i = 1,2,3......n$ hence the mean formula is equal to deviation $A$ plus summation of ${f_i}{d_i}$ divided by summation of frequency.

Complete step-by-step answer:
In the short cut method to finding the mean of the given data following methods involve
In this method we take deviations from an arbitrary point.
${x_1},{x_2},..........{x_n}$ are observations with respective the frequencies of grouped data is ${f_1},{f_2},............{f_n}$ .
Let deviation $A$ take at any point, we have
${d_i} = {x_i} - A$ , where, $i = 1,2,3......n$
So mean by this method is given by
These are the following steps involved to find the mean of grouped data .
1) Prepare a frequency table.
2) Choose $A$ and take deviations ${d_i} = {x_i} - A$ .
3) Multiply ${f_i}{d_i}$ and find the sum of all the given data .
And at last use the formula that is , mean $\mathop x\limits^\_ = A + \dfrac{{\sum\limits_{i = 1}^n {fd} }}{{\sum\limits_{i = 1}^n f }}$
A= Assumed mean of the given data
∑f= Summation of the frequencies given in the grouped data
∑fd= Summation of the frequencies and deviation of a given mean data
d= deviation of a mean data
$\mathop x\limits^\_ $ = arithmetic mean
Hence option B is the correct answer .

Note: As for the finding of the mean of grouped data through direct method , Mean = $\dfrac{{\sum {f \times } X}}{{\sum f }}$ where X is the midpoint of group and f is frequency of that and Midpoint = $\dfrac{{{\text{Lower limit + Upper Limit}}}}{2}$
As for finding the mode of the grouped data we use formula $L + \dfrac{{{f_m} - {f_{m - 1}}}}{{({f_m} - {f_{m - 1}}) + ({f_m} - {f_{m + 1}})}} \times w$ where L is the lower class boundary of the modal group ,${f_{m - 1}}$ is the frequency of the group before the modal group ,${f_m}$ is the frequency of the modal group , ${f_{m + 1}}$ is the frequency of the group after the modal group , w is the group width.