Question

If the arithmetic mean of the following series is 115.86, find the missing value:

Wages	110	112	113	117	?	125	128	130
No. of workers	25	17	13	15	14	8	6	2

Answer 1

Hint: Let the wages be x and no. of workers be f and the missing value be M. So find the corresponding products of x and f and add all of them, say this as N. Now count the total no. of workers, say c. Divide N by c to get the arithmetic means and equate this arithmetic mean to 115.86 to get the value of M.

Complete step-by-step answer:
We are given that the arithmetic mean of the above given data series is 115.86. We have to find the missing value of wages.
Arithmetic mean is the average value of the wages given on a day. It can be calculated by dividing the summation of products of wages and no. of workers by the summation of total no. of workers.
$\begin{array}{*{20}{c}}
  {S.No}&{Wages\left( {{x_i}} \right)}&{No\_of\_workers\left( {{f_i}} \right)}&{{x_i}{f_i}} \\
  1&{110}&{25}&{2750} \\
  2&{112}&{17}&{1904} \\
  3&{113}&{13}&{1469} \\
  4&{117}&{15}&{1755} \\
  5&M&{14}&{14M} \\
  6&{125}&8&{1000} \\
  7&{128}&6&{768} \\
  8&{130}&2&{260} \\
  {Total}& - &{100}&{9906 + 14M}
\end{array}$
Now, we got the summation values
Arithmetic mean is equal to $\dfrac{{\mathop \sum \limits_{i = 1}^n {x_i}{f_i}}}{{\mathop \sum \limits_{i = 1}^n {f_i}}}$
Summation of ${x_i}{f_i}$ is $9906 + 14M$ and summation of ${f_i}$ is 100.
Therefore, Arithmetic mean is $\dfrac{{9906 + 14M}}{{100}}$ which is equal to 115.86.
$ \Rightarrow \dfrac{{9906 + 14M}}{{100}} = 115.86$
$ \Rightarrow 9906 + 14M = 11586$
$ \Rightarrow 14M = 11586 - 9906 = 1680$
$ \Rightarrow M = \dfrac{{1680}}{{14}} = 120$
Therefore, the missing value is 120.
So, the correct answer is “120”.

Note: Do not confuse between mean and median as mean is the average value and median is the middle value. When the data is given in class intervals, then in the place of ${x_i}$ we have to take the middle point of the class interval and then find the summation.