Question

The marks obtained out of 50, by 102 students in a physics test are given in the frequency table below.

Marks $({x_i})$	$15$	$20$	$22$	$24$	$25$	$30$	$33$	$28$	$45$
Frequency $({f_i})$	$5$	$8$	$11$	$20$	$23$	$18$	$13$	$3$	$1$

Find the average number of marks.

Answer 1

In this question, we are given the marks scored by 102 students and we have been asked to find the mean number of marks. You can either use the assumed mean method or direct method to find the answer. In the assumed mean method, assume a number out of ${x_i}$ as mean. Then, find the deviations from that mean and sum up the deviations. Now, simply put them in the formula of assumed mean.
In direct method, find ${f_i}{x_i}$ and sum them up and put them in the formula $\dfrac{{\sum {{f_i}{x_i}} }}{{\sum {{f_i}} }}$ to find the required answer.

Formula used: 1) $\bar X = A + \dfrac{{\sum {{f_i}{d_i}} }}{{\sum {{f_1}} }}$
2) ${d_i} = {x_i} - A$
3) $\bar X = \dfrac{{\sum {{f_i}{x_i}} }}{{\sum {{f_i}} }}$

Complete step-by-step solution:
We are given the marks of 102 students out of 50 and we have been asked to find the average marks. Let us use the assumed mean method for this.
First, we have to assume a mean out of ${x_i}$. Let us assume $25$. After this, we have to find the deviations,${d_i}$. It can be founded using the formula- ${d_i} = {x_i} - A$. After you have calculated ${d_i}$, find ${f_i}{d_i}$. Let us make the table below:

Marks $({x_i})$| $15$| $20$| $22$| $24$| $25$=$A$| $30$| $33$| $28$| $45$|
---|---|---|---|---|---|---|---|---|---|---
Frequency $({f_i})$| $5$| $8$| $11$| $20$| $23$| $18$| $13$| $3$| $1$| $\sum {{f_i} = 102}$
${d_i} = {x_i} - A$ $(A = 25)$| $- 10$| $- 5$| $- 3$| $- 1$| $0$| $5$| $8$| $3$| $20$|
${f_i}{d_i}$| $- 50$| $- 40$| $- 33$| $- 20$| $0$| $90$| $104$| $9$| $20$| $\sum {{f_i}{d_i} = 80}$

Once we have founded all the desired values, let us put them in the formula- $\bar X = A + \dfrac{{\sum {{f_i}{d_i}} }}{{\sum {{f_1}} }}$.
We already know that,
$A = 25,$ $\sum {{f_i}{d_i} = 80}$ and $\sum {{f_i} = 102}$.
Substituting them in the formula-
$\Rightarrow \bar X = 25 + \dfrac{{80}}{{102}}$
$\Rightarrow \bar X = 25 + 0.78 = 25.78$

Hence, the average number of marks is $25.78$.

Note: If you find it difficult to remember all the formulas, you can use the most basic formula and method to find mean.
$\bar X = \dfrac{{{\text{Sum of observations}}}}{{{\text{No}}{\text{. of observations}}}}$ $= \dfrac{{\sum {{f_i}{x_i}} }}{{\sum {{f_i}} }}$
Let us find ${f_i}{x_i}$.

Marks $({x_i})$| $15$| $20$| $22$| $24$| $25$| $30$| $33$| $28$| $45$|
---|---|---|---|---|---|---|---|---|---|---
Frequency $({f_i})$| $5$| $8$| $11$| $20$| $23$| $18$| $13$| $3$| $1$| $\sum {{f_i} = 102}$
${f_i}{x_i}$| $75$| $160$| $242$| $480$| $575$| $540$| $429$| $84$| $45$| $\sum {{f_i}{x_i} = 2630}$

Now, let us simply put the values in the formula-
$\Rightarrow \dfrac{{\sum {{f_i}{x_i}} }}{{\sum {{f_i}} }} = \dfrac{{2630}}{{102}}$
On simplifying we will get,
$\Rightarrow \bar X = 25.78$