Statistics | Exercise 13.1

Question 5

In a retail market, fruit vendors were selling mangoes kept in packing boxes. These boxes contained varying number of mangoes. The following was the distribution of mangoes according to the number of boxes.

Find the mean number of mangoes kept in a packing box. Which method of finding the mean did you choose?

Solution

Understand the Question

The given class intervals are discontinuous (gap of $1$ between consecutive intervals). To find the class boundaries, subtract $0.5$ from each lower limit and add $0.5$ to each upper limit.
The class mark $x_i = \dfrac{\text{Lower limit} + \text{Upper limit}}{2}$ gives the mid-point of each interval.
Since the values of frequencies $f_i$ and class marks $x_i$ are relatively large and have a common class size ( $h = 3$ ), the step-deviation method is chosen to make calculations simpler and less prone to arithmetic errors.

Step 1 · Prepare the Frequency Distribution Table

Let Assumed Mean $a = 57$ and class size $h = 3$ .

Step deviation $u_i = \dfrac{x_i - a}{h} = \dfrac{x_i - 57}{3}$$$ \begin{array}{|c|c|c|c|c|c|} \hline \text{Class interval} & \text{Number of boxes } (f_i) & \text{Class mark } (x_i) & d_i = x_i - 57 & u_i = \dfrac{d_i}{3} & f_i u_i \ \hline 49.5 - 52.5 & 15 & 51 & -6 & -2 & -30 \ \hline 52.5 - 55.5 & 110 & 54 & -3 & -1 & -110 \ \hline 55.5 - 58.5 & 135 & 57 & 0 & 0 & 0 \ \hline 58.5 - 61.5 & 115 & 60 & 3 & 1 & 115 \ \hline 61.5 - 64.5 & 25 & 63 & 6 & 2 & 50 \ \hline \textbf{Total} & \sum f_i = 400 & & & & \sum f_i u_i = 25 \ \hline \end{array}

Step 2 · Calculate the Mean using Step-Deviation Method

Using the step-deviation formula $\bar{x} = a + \left( \dfrac{\sum f_i u_i}{\sum f_i} \right) \times h$

Substitute $a = 57$ , $\sum f_i u_i = 25$ , $\sum f_i = 400$ , and $h = 3$

\begin{aligned} \bar{x} &= 57 + \left( \dfrac{25}{400} \right) \times 3 \\[0.6em] &= 57 + \left( \dfrac{1}{16} \right) \times 3 \\[0.6em] &= 57 + 0.0625 \times 3 \\[0.6em] &= 57 + 0.1875 \\[0.6em] &= 57.1875 \approx 57.19 \end{aligned}

Answer

Mean number of mangoes $= 57.19$ (using step-deviation method)

Common Mistakes

Incorrect Class Size ( $h$ ): Calculating $h = 52 - 50 = 2$ directly from original limits instead of using continuous class limits $h = 52.5 - 49.5 = 3$ .
Forgetting the Multiplier ( $h$ ): Omitting the factor of $h$ in the step-deviation formula $\bar{x} = a + \left(\dfrac{\sum f_i u_i}{\sum f_i}\right) \times h$ .
Sign Errors in Deviation: Mishandling negative signs when summing the products $f_i u_i$ for values below the assumed mean.

Question 5

More questions in Exercise 13.1