Question

Find the mean, median and mode of the following data:

Marks obtained	Number of students (Frequency)
20	6
25	20
28	24
29	28
33	15
38	4
42	2
43	1
Total	100

Answer 1

In this question, we are given discrete data and we need to find mean, median and mode of the data. We will use following steps involved in calculating mean:
(i) Indicate various marks as X and number of students (frequencies) as f.
(ii) Multiply each X with its f and then add all the fX obtained to get $\sum{fX}$.
(iii) Add frequencies to get $\sum{f}$.
(iv) Divide $\sum{fX}$ by $\sum{f}$ to evaluate the mean $\overline{X}$.
We will use following step involved in calculating median:
(i) Arrange data in ascending order.
(ii) Find cumulative frequency by adding all preceding frequencies for a row.
(iii) Determine ${{\left( \dfrac{N+1}{2} \right)}^{th}}$ item of series where N is $\sum{f}$ (sum of frequencies).
(iv) Value corresponding to ${{\left( \dfrac{N+1}{2} \right)}^{th}}$ item will be our median.
For finding mode, we will just inspect marks having the highest frequency. That value will be our mode.

Complete step-by-step answer:
Here, we are given data of marks of 100 students. We need to find mean, median and mode of the data.
First let us find the meaning.
Let us make a table where marks are devoted by X, number of students as f. Then, we will add a column in the table containing products of each X with its f labeled as fX. Our table looks like this,

Marks obtained (X)	Number of students (f)	fX
20	6	120
25	20	500
28	24	672
29	28	812
33	15	495
38	4	152
42	2	84
43	1	43
Total	100	2878

Now let us find the sum of all fX. We get:
$\sum{fX}=120+500+672+812+495+152+84+43=2878$.
Also, we know that, number of students is 100 therefore, the sum of frequencies is 100 and hence, $\sum{f}=100$.
As we know, mean of discrete series is calculated using the formula $\overline{X}=\dfrac{\sum{fX}}{\sum{f}}$ where $\overline{X}$ is the mean.
Hence, we get $\overline{X}=\dfrac{2878}{100}=28.78$.
Hence, the mean of the data is 28.78.
Now, let us calculate the median of the data. For this, we first need to arrange data in ascending order. Since data is already in ascending order, so let us find the cumulative frequency and draw table. Cumulative frequency of a row is calculated by adding all preceding frequencies from that row.
Our table looks like this:

Marks obtained (X)	Number of students (f)	Cumulative frequency (c.f)
20	6	6
25	20	6+20=26
28	24	6+20+24=50
29	28	6+20+24+28=78
33	15	6+20+24+28+15=93
38	4	6+20+24+28+15+4=97
42	2	6+20+24+28+15+4+2=99
43	1	6+20+24+28+15+4+2+1=100
100

Since the sum of all frequencies is 100. So,
Now, ${{\left( \dfrac{N+1}{2} \right)}^{th}}$ term will be ${{\left( \dfrac{100+1}{2} \right)}^{th}}={{\left( \dfrac{101}{2} \right)}^{th}}={{50.5}^{th}}$.
Since, ${{50.5}^{th}}$ term corresponds to the c.f. 78 therefore, 29 is the required median.
For mode let us observe the table and check the highest frequency. Since, 28 is the highest frequency and value corresponding to 28 is 29.
Hence, 29 is the required mode.
Therefore, 28.78, 29, 29 are mean, median and mode of the given data respectively.

Note: Students should note that, while selecting cumulative frequency we take c.f which is equal to or just after the found ${{\left( \dfrac{N+1}{2} \right)}^{th}}$ term. Students should keep in mind all the formulas for calculating mean, median and mode. For mean, students can also use shortcut method or step deviation method. If the highest frequencies are the same for two or more values then mode cannot be calculated by inspection, we need to use a grouping method.