Question

Find the sum of the lower limit of median class and a modal class of the following distribution

$\begin{array}{|cccccc|}\hline \mathrm{Distribution}& 0-5& 5-10& 10-15& 15-20& 20-25\\ \mathrm{Frequency}& 10& 15& 12& 20& 9\\ \hline\end{array}$

Answer 1

Step 1: Calculate cumulative frequency

1. Cumulative frequency is calculated by adding the cumulative frequency of previous class to the frequency of the current class
2. The cumulative frequency of the first class is its frequency itself

Thus,

$$\begin{gathered} \begin{array}{|cccccc|}\hline \mathrm{Distribution}& 0-5& 5-10& 10-15& 15-20& 20-25\\ \mathrm{Frequency}& 10& 15& 12& 20& 9\\ \mathrm{Cumulative} \\ \mathrm{Frequency}& 10& 25& 37& 57& 66\\ \hline\end{array} \end{gathered}$$

Step 2: Calculate Lower limit of median class

$N$ is the total observations.

Since $N$ is even here, the median is the mean of the ${\left(\frac{N}{2}\right)}^{th}$ observation.

Thus, median is,

$\frac{N}{2}=\frac{66}{2}=33$

The class that contains the median observation is the median class.

The class $10-15$ contains the median, it the median class.

Thus, the lower limit of the median class is $10$

Step 3: Lower limits of the modal class

Modal class is the class with the most frequency

Thus, the modal class is the $15-20$ class which has $20$ observations

Thus the lower limit of the modal class is $15$

Step 4: Sum of lower limits of median and modal class

Lower limit of median class $=10$

Lower limit of modal class $=15$

Thus their sum $=10+15=25$

Therefore, the sum of the lower limit of median class and modal class of the given distribution is $25$