Question

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

Literacy rate (in %)	45-55	55-65	65-75	75-85	85-95
No. of cities	3	10	11	8	3

Answer 1

There are several kinds of mean in mathematics, especially in statistics. For a data set, the arithmetic mean, also called the expected value or average, is the central value of a discrete set of numbers: specifically, the sum of the values divided by the number of values.
The Arithmetic Mean is the average of the numbers: a calculated "central" value of a set of numbers.
To calculate it:
Add up all the numbers,
Then divide by how many numbers there are.

Complete step-by-step solution:
Given data about class intervals and their frequencies would lead us to the calculation of mean using the standard formula where an additional term will add known as ${X_i}$ as follows:
$\overline x = \dfrac{{\sum {fx} }}{{\sum f }}$
Where,
${X_i}$ = mean class marks of each interval
${f_i}$ = number of cities
To find the class mark (${X_i}$) for each interval, using the following relation:
(${X_i}$) = (upper class limit+ lower class limit)/2
From the given values and to apply the formula lets first draw a table for computing ${f_i}{X_i}$ term easily:

Literacy rate (in %)	No. of cities(${f_i}$)	Class mark(${X_i}$)	${f_i}{X_i}$
45-55	3	50	150
55-65	10	60	600
65-75	11	70	770
75-85	8	80	640
85-95	3	90	270
Total=35		Total=2430

From the table above we can compute the required mean literacy rate using the standard mean formula as shown:
$\overline x = \dfrac{{\sum {{f_i}{x_i}} }}{{\sum {{f_i}} }} = \dfrac{{2430}}{{35}} = 69.428\%$
$\overline x = 69.43\%$

Thus after the calculation and simplification we arrive at the value for mean value to be 69.43%.

Note: We know that finding the mean of literacy rate could not be performed directly because of the class intervals given due to which we have to take the mean of each interval. Thus it is the direct method in which class marks would serve as representative of the whole class and are represented by ${X_i}$ while calculating the mean of literacy rate.