9. The following table gives the literacy rate (in percentage) of 35 cities. Find the mean literacy rate.

Understand the Question We are given the literacy rate (in percentage) categorized into class intervals along with the number of cities (frequencies ). To find the mean literacy rate , we use the Step-Deviation Method: where: is the class mark. is the assumed mean (chosen from the center of ). is the class size (). . Step 1 · Prepare the Frequency Distribution Table Let assumed mean and class size . From the table: Step 2 · Calculate the Mean Using the step-deviation formula: Substitute the values: Answer > Common Mistakes Forgetting Class Size (): A common mistake in the step-deviation formula is omitting the multiplication by . Sign Errors in : Negative terms () must be carefully added to positive terms () to get , not . Units in Final Answer: Forgetting to write the unit as percentage () in the final answer.

Statistics | Exercise 13.1

Question 9

The following table gives the literacy rate (in percentage) of 35 cities. Find the mean literacy rate.

Solution

Understand the Question

We are given the literacy rate (in percentage) categorized into class intervals along with the number of cities (frequencies $f_i$ ).
To find the mean literacy rate $\bar{x}$ $\overset{x}{ˉ}$ , we use the Step-Deviation Method: $\bar{x} = a + \left( \dfrac{\sum f_i u_i}{\sum f_i} \right) \times h$ $\overset{x}{ˉ} = a + (\frac{\sum f _{i} u _{i}}{\sum f _{i}}) \times h$ where:
- $x_i = \dfrac{\text{Lower limit} + \text{Upper limit}}{2}$ is the class mark.
- $a$ is the assumed mean (chosen from the center of $x_i$ ).
- $h$ is the class size ( $h = 10$ ).
- $u_i = \dfrac{x_i - a}{h}$ .

Step 1 · Prepare the Frequency Distribution Table

Let assumed mean $a = 70$ and class size $h = 10$ . Diagram 1

\begin{array}{|c|c|c|c|c|c|} \hline \text{Literacy rate (in \%)} & \text{Number of cities } f_i & x_i & d_i = x_i - 70 & u_i = \dfrac{d_i}{10} & f_i u_i \\ \hline 45 - 55 & 3 & 50 & -20 & -2 & -6 \\ \hline 55 - 65 & 10 & 60 & -10 & -1 & -10 \\ \hline 65 - 75 & 11 & 70 & 0 & 0 & 0 \\ \hline 75 - 85 & 8 & 80 & 10 & 1 & 8 \\ \hline 85 - 95 & 3 & 90 & 20 & 2 & 6 \\ \hline \text{Total} & \sum f_i = 35 & & & & \sum f_i u_i = -2 \\ \hline \end{array}

From the table: $\sum f_i = 35$ $\sum f_i u_i = -2$

Step 2 · Calculate the Mean

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

Substitute the values:

\begin{aligned} \bar{x} &= 70 + \left( \dfrac{-2}{35} \right) \times 10 \\[0.6em] &= 70 - \dfrac{20}{35} \\[0.6em] &= 70 - \dfrac{4}{7} \\[0.6em] &= 70 - 0.5714 \dots \\[0.6em] &\approx 69.43 \end{aligned}

Answer

$69.43\%$

Common Mistakes

Forgetting Class Size ( $h$ ): A common mistake in the step-deviation formula is omitting the multiplication by $h$ .
Sign Errors in $\sum f_i u_i$ : Negative terms ( $-6 + -10 = -16$ ) must be carefully added to positive terms ( $8 + 6 = 14$ ) to get $-2$ , not $+2$ .
Units in Final Answer: Forgetting to write the unit as percentage ( $\%$ ) in the final answer.

Question 9

More questions in Exercise 13.1