Question

Find the median for the following frequency distribution table:

Class interval	0-5	5-10	10-15	15-20	20-25	25-30
Frequency	5	3	9	10	8	5

Answer 1

Hint: From the question, it is mentioned to find the median for the given frequency distribution table where the class intervals and their respective frequencies are given. The given distribution is a grouped frequency distribution.

Complete step-by-step solution:
Median of grouped frequency distribution
Median = \[l+\dfrac{\dfrac{N}{2}-C}{f}\times h\]
l= lower limit of median class interval
C = cumulative frequency preceding to the median class frequency
f = frequency of the class interval to which median belongs
h = width of the class interval
N = \[{{f}_{1}}+{{f}_{2}}+{{f}_{3}}.......+{{f}_{n}}\]
So here first we will fill up the known information into a table containing less than type cumulative frequency using the frequencies given

Class interval	Frequency	Less than type cumulative frequency
0-5	5	5
5-10	3	\[5+3=8\]
10-15	9	\[8+9=17\]
15-20	10	\[17+10=27\]
20-25	8	\[27+8=35\]
25-30	5	\[35+5=40\]

So here the cumulative frequency N=40
Now here let’s find
\[\dfrac{N}{2}=\dfrac{40}{2}=20\]
Now by observing the values in the table clearly the cumulative frequency of 20 will lie in the class interval of 15-20
So our median class is 15-20
lower limit of the median class = l = 15
width of the class interval= h= \[20-15=5\]
Cumulative frequency preceding median class frequency= C= 17
Frequency of median class =10
Substituting the above values in the formula of median that is
\[\begin{align}
  & Median=l+\dfrac{\dfrac{N}{2}-C}{f}\times h \\
 & Median=15+\dfrac{\dfrac{40}{2}-17}{10}\times 5=15+\dfrac{20-17}{10}\times 5 \\
 & Median=15+\dfrac{3}{2}=\dfrac{33}{2}=16.5 \\
\end{align}\]
Therefore the median for the given frequency distribution table is 16.5.

Note: Students must be clear with the two different methods to find the median of grouped and ungrouped data. Here in this question, we have been given a median for grouped frequency distribution so we should use the method of cumulative frequencies to find the median.