Question

Calculate the median for the following distribution class.

Class	0-10	10-20	20-30	30-40	40-50	50-60
Frequency	54	10	20	7	8	5

Answer 1

Hint: In this question, we are given a continuous series and we have to find the median of the data. For this, we will find the sum of frequencies (N) and then use it to locate the class interval in which our median lies. Then we will find the cumulative frequency of the data. Using cumulative frequency and the sum of frequency, we will find the median class which will be the class containing ${{\left( \dfrac{N}{2} \right)}^{th}}$ item. At last, we will apply the formula given below to find the median.
\[\text{Median}={{l}_{1}}+\dfrac{\dfrac{N}{2}-c.f}{f}\times i\]
Where, ${{l}_{1}}$ is the lower limit of median class, c.f. is cumulative frequency of class preceding the median class, f is the simple frequency of median class and i is the class interval of the median class.

Complete step-by-step solution:
Here, we are given continuous series hence, the median cannot be located straight forward. In this case, the median lies in between the lower and upper limits of a class interval. To get the exact value of the median, we have to interpolate the median with the help of the formula given as
\[\text{Me}={{l}_{1}}+\dfrac{\dfrac{N}{2}-c.f}{f}\times i\]
Where, “Me” is the median, ${{l}_{1}}$ is the lower limit of median class, c.f. is cumulative frequency of class preceding the median class, f is the simple frequency of median class, N is the sum of frequencies and i is the class interval of the median class. Let us move step by step for finding all the required terms for the formula to calculate the median.
Step 1: Let us calculate cumulative frequencies and make a frequency distribution table. The cumulative frequency for any row is the sum of preceding all the frequencies. Hence, we get the table below:

Class	Frequency (f)	Cumulative Frequency (c.f.)
0-10	5	5
10-20	10	5+10=15
20-30	20	5+10+20=35
30-40	7	5+10+20+7=42
40-50	8	5+10+20+7+8=50
50-60	5	5+10+20+7+8+5=55

Now, the sum of frequencies will be given by the last cumulative frequency. Hence, N = 55.
Step 2: Now, we will find the median item which will be ${{\left( \dfrac{N}{2} \right)}^{th}}$ item.
As N = 55, therefore, ${{\left( \dfrac{N}{2} \right)}^{th}}$ item = ${{\left( \dfrac{55}{2} \right)}^{th}}$ item ${{27.5}^{th}}$ item
Hence, \[\dfrac{N}{2}=27.5\cdots \cdots \cdots \cdots \cdots \left( 1 \right)\]
Step 3: Now, we will find the median class by inspecting cumulative frequencies. We will take the class of the row in which cumulative frequency is either equal to or just greater than ${{\left( \dfrac{N}{2} \right)}^{th}}$ item. As we can see, 35 is the cumulative frequency just greater than 27.5. Hence, we will take the class 20-30 as the median class.
Step 4: Let us calculate the class interval now. As we know, class interval = upper-class limit - lower class limit. Therefore, class interval = 30-20 = 10.
Hence, \[i=10\cdots \cdots \cdots \cdots \cdots \left( 2 \right)\]
Step 5: Now using median class, let us find remaining terms for formula. As ${{l}_{1}}$ is the lower limit of median class, therefore,
\[{{l}_{1}}=20\cdots \cdots \cdots \cdots \cdots \left( 3 \right)\]
As c.f. is the cumulative frequency of class preceding median class, hence, for 20-30 as a median class,
\[c.f.=15\cdots \cdots \cdots \cdots \cdots \left( 4 \right)\]
Now, f is the frequency of median class, therefore,
\[f=20\cdots \cdots \cdots \cdots \cdots \left( 5 \right)\]
Step 6: Formula for calculating median is given as:
\[\text{Median}={{l}_{1}}+\dfrac{\dfrac{N}{2}-c.f}{f}\times i\]
Using (1), (2), (3), (4) and (5) let us put all terms in above formula, we get:
\[\begin{align}
  & \text{Median}=20+\dfrac{27.5-15}{20}\times 10 \\
 & \Rightarrow 20+\dfrac{12.5}{2} \\
 & \Rightarrow 20+6.25 \\
 & \Rightarrow 26.25 \\
\end{align}\]
Hence, the median of given data is 26.25.

Note: Students should carefully apply the formula for calculating the median. As there are a lot of terms in the formula, try to do the question step by step to avoid mistakes. Students should note that we have to take cumulative frequency of class preceding median class while using the formula and while determining median class, cumulative frequency should be equal to or just greater than ${{\left( \dfrac{N}{2} \right)}^{th}}$ item.