**Q) Vocational training complements traditional education by providing practical skills and hands-on experience. While education equips ****individuals with a broad knowledge base, vocational training focuses on job-specific skills, enhancing employability thus making the student self reliant. Keeping this in view, a teacher made the following table giving the frequency distribution of students/adults undergoing vocational ****training from the training institute.**

**From the above answer the following questions :****(i)** **What is the lower limit of the modal class of the above data? ****(ii) (a) Find the median class of the above data. ****OR****(b) Find the number of participants of age less than 50 years who undergo vocational training.****(iii) Give the empirical relationship between mean, median and mode.**

**Ans:**

By observing the grouped data, we can see that these are inclusive data groups i.e 15-19, 20-24,…

We need to convert this inclusive data groups into exclusive data groups.

We will do this by adjusting the limits of each group i.e. by subtracting 0.5 from all lower limits and adding 0.5 to all upper limits.

Hence, our re-organized data groups will be:

**(i) Lower limit of modal class:**

Since the modal class is the class with the highest frequency.

In the given question, class “19.5 – 24.5” has 132 frequency which is the highest frequency among all other classes.

Hence, the modal class is “19.5 – 24.5”

**Therefore, the lower limit of modal class is 19.5.**

**(ii) (a) Median Class:**

Let’s re-organize the data in the frequency table to find out median:

To find the median, we need to take following steps:

- First, we need to find the cumulative frequency in the frequency table to find the median. Its shown in last column.
- Next, Total number of participants or Sum of the frequencies = 365. It shown in the last row of middle column.
- Next, we need to identify Median Class. Since the Median class is the class where the cumulative frequency crosses 50% of the half the total number of participants,
- Here in the table, Cumulative frequency of 194 is crossing 50% of frequency i.e. 182.5, at class “19.5 – 24.5”.

**Therefore, our Median class = 19.5 – 24.5.**

**(ii) (b): Participants less than 50 years age:**

From the above table of re-arranged data, we can see that only last row has participants of 50 years age and above.

Hence, all the participants in all other rows are below 50 years age and total number of participants is 361 (as shown in last column).

**Therefore, there are 361 participants of age less than 50 years.**

**(iii) Empirical relationship between mean, median and mode:**

The empirical Relationship among mean, median and mode of any grouped data requires that the difference between mean and mode is almost equal to three times the difference between the mean and median.

(Mean – Mode) = 3 (Mean – Median)

**3 Median = Mode + 2 Mean**

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