Question

What is the value of mean in frequency distribution if value \[1,\dfrac{1}{2},\dfrac{1}{3},\dfrac{1}{4},\dfrac{1}{5},....,\dfrac{1}{n}\] occur at frequencies \[1,2,3,4,5,....n\] respectively.
A. \[1\]
B. \[n\]
C. \[\dfrac{1}{n}\]
D. \[\dfrac{2}{{n + 1}}\]

Answer 1

Hint:
In this question, we use the concept of mean, median and mode in statistics. Here we use the mean concept when data in grouped is given to find out the value of mean in frequency distribution.



Formula Used:
The formula use in the question for find out the value of mean when data in grouped is given:
\[Mean = \dfrac{{\sum\limits_{i = 1}^n {{f_i}{x_i}} }}{{\sum\limits_{i = 1}^n {{f_i}} }}\]
Where \[{f_{1,}}{f_{2,}}{f_{3,....}}{f_n}\] are frequencies of observations and \[{x_1},{x_2},{x_3},....{x_n}\] are value of given observations.
Here \[\sum\limits_{i = 1}^n {{f_i}{x_i}} \] denotes the sum of product of frequencies and value of observations from 1 to n.
Also, \[\sum\limits_{i = 1}^n {{f_i}} \] denotes the sum of all frequencies from 1 to n observations.



Complete step-by-step answer:
In the given question, there are the values \[1,\dfrac{1}{2},\dfrac{1}{3},....\dfrac{1}{n}\] occur at frequencies \[1,2,3,....,n\] respectively in frequency distribution.
So,
\[Mean = \dfrac{{\sum\limits_{i = 1}^n {{f_i}{x_i}} }}{{\sum\limits_{i = 1}^n {{f_i}} }}\]
\[ = \dfrac{{1.1 + 2.\dfrac{1}{2} + 3.\dfrac{1}{3} + .... + n.\dfrac{1}{n}}}{{1 + 2 + 3 + .... + n}}\]

\[ = \dfrac{{1 + 1 + 1 + .... + 1(n times)}}{{1 + 2 + 3 + .... + n}}\]
\[ = \dfrac{n}{{\dfrac{{n(n + 1)}}{2}}}\]
  [Since, Sum of first n natural numbers\[(1 + 2 + 3 + .... + n) = \dfrac{{n(n + 1)}}{2}\]]
\[ = \dfrac{2}{{n + 1}}\]
Hence, option D is the correct answer:



Note:
In these types of questions, students need to read the statements carefully then solve it. If the data in ungrouped (There are five numbers given \[1,2,3,4\]and \[5\].Find the mean of these observations), then the formula \[Mean = \dfrac{{sum {\rm{ }}of{\rm{ }}all{\rm{ }}observations\;}}{{Total\;numbers{\rm{ }}of{\rm{ }}observations}}\] is used.
If the data in grouped is given just like given in this question then the formula \[Mean = \dfrac{{\sum\limits_{i = 1}^n {{f_i}{x_i}} }}{{\sum\limits_{i = 1}^n {{f_i}} }}\] is used here.