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 non-continuous (e.g., $50-52, 53-55$ ). We make them continuous by subtracting $0.5$ from the lower limits and adding $0.5$ to the upper limits, giving class width $h = 3$ .
For each class, the class mark is $x_i = \dfrac{\text{Upper limit} + \text{Lower limit}}{2}$ .
Since the frequencies and class marks are large, we use the Step-Deviation Method with assumed mean $a = 57$ : $\bar{x} = a + \left( \dfrac{\sum f_i u_i}{\sum f_i} \right) \times h, \quad \text{where } u_i = \dfrac{x_i - a}{h}$

Step 1 · Construct the Frequency Distribution Table

Make the class intervals continuous and let the assumed mean be $a = 57$ with class size $h = 3$ .

\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 \mathbf{\text{Total}} & \sum f_i = 400 & & & & \sum f_i u_i = 25 \\ \hline \end{array}

Step 2 · Calculate Mean using Step-Deviation Method

From the table, $a = 57$ , $h = 3$ , $\sum f_i = 400$ , and $\sum f_i u_i = 25$ .

Using the step-deviation formula:

\begin{aligned} \bar{x} &= a + \left( \dfrac{\sum f_i u_i}{\sum f_i} \right) \times h \\[0.6em] &= 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

The mean number of mangoes is $57.19$ , obtained using the Step-Deviation Method.

Common Mistakes

Incorrect Class Size ( $h$ ): Taking $h = 52 - 50 = 2$ from the discontinuous limits instead of finding $h = 52.5 - 49.5 = 3$ from the continuous class boundaries.
Omitting $h$ in Step-Deviation: Forgetting to multiply by $h = 3$ in the final formula $\bar{x} = a + \left(\frac{\sum f_i u_i}{\sum f_i}\right) \times h$ .
Rounding Errors: Rounding intermediate fractions too early rather than converting $\frac{3}{16} = 0.1875$ and rounding at the final step.

Question 5

More questions in Exercise 13.1