Question

Calculate the mean of the following data, using a direct method.

Class	25-35	35-45	45-55	55-65	65-75
Frequency	6	10	8	12	4

Answer 1

Hint: In this question, we are given a continuous series where we are given different classes along with their frequency and we have to find the mean of the data using the direct method. For doing this, we have to draw frequency distribution table having column as class, frequency $\left( {{f}_{i}} \right)$ class mark $\left( {{x}_{i}} \right)$ and product of frequency with classmark $\left( {{x}_{i}}{{f}_{i}} \right)$ for making table, we will first understand meaning of class mark and then calculate it for every class. Then we will draw the table and take the total sum of all frequencies and the total sum of all ${{f}_{i}}{{x}_{i}}'s$ (product of frequency with class mark). At last, we will apply the formula of mean to find the mean of the data.
\[\text{Mean}=\dfrac{\sum{{{f}_{i}}{{x}_{i}}}}{\sum{{{f}_{i}}}}\]
Where, $\sum{{{f}_{i}}{{x}_{i}}}$ denotes summation of all marks and $\sum{{{f}_{i}}}$ denotes summation of all frequencies.

Complete step-by-step solution:
Before drawing the frequency distribution table for the given continuous series, let us calculate the class marks for given classes.
As we know, class marks are defined as mid values of classes, hence, they can be evaluated by formula:
\[\text{Class mark}=\dfrac{\text{Upper limit}+\text{Lower limit}}{2}\]
For 25-35 class, the class mark becomes $\dfrac{25+35}{2}=\dfrac{60}{2}=30$.
For 35-45 class, the class mark becomes $\dfrac{35+45}{2}=\dfrac{80}{2}=40$.
For 45-55 class, the class mark becomes $\dfrac{45+55}{2}=\dfrac{100}{2}=50$.
For 55-65 class, the class mark becomes $\dfrac{55+65}{2}=\dfrac{120}{2}=60$.
For 65-75 class, the class mark becomes $\dfrac{65+75}{2}=\dfrac{140}{2}=70$.
Now, let us draw frequency distribution table where ${{f}_{i}}$ denotes frequency for ${{i}^{th}}$ interval, ${{x}_{i}}$ denotes class mark for ${{i}^{th}}$ interval.

Class	Frequency $\left( {{f}_{i}} \right)$	Class mark $\left( {{x}_{i}} \right)$	${{f}_{i}}{{x}_{i}}$
25-35	6	30	180$\left( 6\times 30 \right)$
35-45	10	40	400$\left( 10\times 40 \right)$
45-55	8	50	400$\left( 8\times 50 \right)$
55-65	12	60	720$\left( 12\times 60 \right)$
65-75	4	70	280$\left( 4\times 70 \right)$

Now, let us calculate the sum of all ${f}_{i}$ and sum of all ${{f}_{i}}{{x}_{i}}$.
Sum of all fi becomes 6+10+8+12+4 = 40
Hence, we get $\sum{{{f}_{i}}}=40$
Sum of all ${{f}_{i}}{{x}_{i}}$ becomes 180+400+400+720+280 = 1980
Hence, we get $\sum{{{f}_{i}}{{x}_{i}}}=1980$
Now, mean is the average of the data denoted by $\overline{x}$. Mean for continuous series is given by $\overline{x}=\dfrac{\sum{{{f}_{i}}{{x}_{i}}}}{\sum{{{f}_{i}}}}$
Putting values of $\sum{{{f}_{i}}{{x}_{i}}}\text{ and }\sum{{{f}_{i}}}$ we get:
$\overline{x}=\dfrac{1980}{40}=49.5$. Hence, mean of the given data is 49.5

Note: Students should carefully find the class marks for all the classes. Also, they should take care while taking the sum of all terms. Try to avoid calculations while taking the product and calculating the sum. Students should always learn formulas for calculating mean, median, and mode for different types of data given.