Statistics | Exercise 13.1

Question 3

The following distribution shows the daily pocket allowance of children of a locality. The mean pocket allowance is ₹18. Find the missing frequency $f$ .

Solution

Understand the Question

We are given a frequency distribution with an unknown frequency $f$ and a known mean pocket allowance $\bar{x} = ₹18$ .
To find $f$ , we calculate the class marks $x_i = \dfrac{\text{Upper limit} + \text{Lower limit}}{2}$ for each class interval.
Using the Assumed Mean Method with assumed mean $a = 18$ , we compute deviations $d_i = x_i - a$ and the products $f_i d_i$ .
Finally, substitute $\sum f_i$ and $\sum f_i d_i$ into the mean formula $\bar{x} = a + \dfrac{\sum f_i d_i}{\sum f_i}$ and solve the linear equation for $f$ .

Step 1 · Calculate Class Marks, Deviations, and Totals

Let the assumed mean be $a = 18$ .

Class mark formula: $x_i = \dfrac{\text{Upper limit} + \text{Lower limit}}{2}$

\begin{array}{|c|c|c|c|c|} \hline \text{Daily pocket allowance (in ₹)} & \text{Number of children } (f_i) & \text{Class mark } (x_i) & d_i = x_i - 18 & f_i d_i \\ \hline 11 - 13 & 7 & 12 & -6 & -42 \\ \hline 13 - 15 & 6 & 14 & -4 & -24 \\ \hline 15 - 17 & 9 & 16 & -2 & -18 \\ \hline 17 - 19 & 13 & 18 & 0 & 0 \\ \hline 19 - 21 & f & 20 & 2 & 2f \\ \hline 21 - 23 & 5 & 22 & 4 & 20 \\ \hline 23 - 25 & 4 & 24 & 6 & 24 \\ \hline \text{Total} & \sum f_i = 44 + f & & & \sum f_i d_i = 2f - 40 \\ \hline \end{array}

Sum of frequencies:

\begin{aligned} \sum f_i &= 7 + 6 + 9 + 13 + f + 5 + 4 \\ &= 44 + f \end{aligned}

Sum of products $f_i d_i$ :

\begin{aligned} \sum f_i d_i &= -42 - 24 - 18 + 0 + 2f + 20 + 24 \\ &= -84 + 2f + 44 \\ &= 2f - 40 \end{aligned}

Step 2 · Apply Assumed Mean Formula and Solve for $f$

Given mean $\bar{x} = 18$ .

Using the assumed mean formula: $\bar{x} = a + \dfrac{\sum f_i d_i}{\sum f_i}$

$18 = 18 + \dfrac{2f - 40}{44 + f}$

Subtract $18$ from both sides: $0 = \dfrac{2f - 40}{44 + f}$

Multiply both sides by $(44 + f)$ :

\begin{aligned} 2f - 40 &= 0 \\ 2f &= 40 \\ f &= 20 \end{aligned}

Answer

20

Common Mistakes

Frequency Summation Error: Writing $\sum f_i = 44f$ instead of $44 + f$ when adding the unknown frequency $f$ to the numerical values.
Sign Confusion in Deviations: Incorrectly calculating $d_i = 18 - x_i$ instead of $d_i = x_i - 18$ , which reverses all negative and positive signs in the $f_i d_i$ column.
Algebraic Simplification Error: Incorrectly solving $0 = \dfrac{2f - 40}{44 + f}$ by subtracting terms across the fraction incorrectly instead of recognizing that the numerator must equal zero.

Question 3

More questions in Exercise 13.1