Statistics | Exercise 13.1

Question 2

Consider the following distribution of daily wages of 50 workers of a factory.

Find the mean daily wages of the workers of the factory by using an appropriate method.

Solution

Understand the Question

To calculate the mean for large numerical values of class intervals and frequencies, the Step Deviation Method simplifies the arithmetic: $\bar{x} = a + \left( \dfrac{\sum f_i u_i}{\sum f_i} \right) \times h$ * Class Mark ( $x_i$ ): Midpoint of each class interval, calculated as $x_i = \dfrac{\text{Upper limit} + \text{Lower limit}}{2}$ .
Assumed Mean ( $a$ ): A central class mark chosen to simplify calculations ( $a = 550$ ).
Class Size ( $h$ ): The width of each class interval ( $h = 20$ ).
Deviation ( $u_i$ ): Scaled deviation given by $u_i = \dfrac{x_i - a}{h}$ .

Step 1 · Construct the Frequency Distribution Table

Class size $h = 520 - 500 = 20$ . Let the assumed mean be $a = 550$ .

\begin{array}{|c|c|c|c|c|c|} \hline \text{Daily wages (in ₹)} & \text{Number of workers } (f_i) & \text{Class Mark } (x_i) & d_i = x_i - 550 & u_i = \dfrac{d_i}{20} & f_i u_i \\ \hline 500 - 520 & 12 & 510 & -40 & -2 & -24 \\ \hline 520 - 540 & 14 & 530 & -20 & -1 & -14 \\ \hline 540 - 560 & 8 & 550 & 0 & 0 & 0 \\ \hline 560 - 580 & 6 & 570 & 20 & 1 & 6 \\ \hline 580 - 600 & 10 & 590 & 40 & 2 & 20 \\ \hline \text{Total} & \sum f_i = 50 & & & & \sum f_i u_i = -12 \\ \hline \end{array}

Step 2 · Calculate the Mean

From the table: $\sum f_i = 50, \quad \sum f_i u_i = -12, \quad a = 550, \quad h = 20$

Using the step deviation formula:

\begin{aligned} \text{Mean } (\bar{x}) &= a + \left( \dfrac{\sum f_i u_i}{\sum f_i} \right) \times h \\[0.6em] &= 550 + \left( \dfrac{-12}{50} \right) \times 20 \\[0.6em] &= 550 + \left( \dfrac{-240}{50} \right) \\[0.6em] &= 550 - 4.8 \\[0.6em] &= 545.20 \end{aligned}

Answer

$\text{₹}545.20$

Common Mistakes

Sign Errors in $\sum f_i u_i$ : Pay close attention to negative signs when summing negative and positive values (e.g., $(-24 - 14) + (6 + 20) = -38 + 26 = -12$ ).
Omitting Class Size ( $h$ ): Forgetting to multiply $\dfrac{\sum f_i u_i}{\sum f_i}$ by $h$ before adding to the assumed mean $a$ .

More questions in Exercise 13.1