Question

Using empirical formula, calculate the mode of the following data –
$17,16,25,23,22,23,28,25,25,23$.
$$\begin{array}{rl}& (a)14.61\\ & (b)19.6\\ & (c)23.6\\ & (d)28.3\end{array}$$

Answer 1

Hint: In this question, we have to find the mode of given data using empirical formula. Generally mode is the term which occurs most frequently in the data, but here we have to use empirical formula for finding mode. Empirical formula states that $$mode=3median-2mean$$. This means we have to calculate mean and median of the given data and then use the formula to find mode. For finding the mean, we will add all the terms and then divide by total number of terms. For median, we will arrange the data in ascending order, and then find the middle term. If the number of terms are odd, then ${\displaystyle \frac{n}{2}}th$ term is the middle term. If the number of terms are even, then average of ${\displaystyle \frac{n}{2}}th$ and $$({\displaystyle \frac{n}{2}}+1)th$$ is the middle term where $n$ is the number of terms.

Complete step-by-step answer:
We are given the data as $17,16,25,23,22,23,28,25,25,23$.
As we can see that the total number of terms is $10$. Let us find the meaning first. For this, we have to find the sum of the terms first. Hence,
$17+16+25+23+22+23+28+25+25+23=227$
Now, mean is given by $={\displaystyle \frac{Sumofterms}{No.ofterms}}={\displaystyle \frac{227}{10}}=22.7$
Hence, the mean of the given data is $22.7$.
$Mean=22.7...(1)$
Let us calculate the median now. As the number of terms of given data is $10$ which is even, so we will get two middle terms of which average has to be taken. But first let us arrange the data in the ascending order. Data becomes as –
$16,17,22,23,23,23,25,25,25,28$
Let us check ${\displaystyle \frac{n}{2}}th$ and $$({\displaystyle \frac{n}{2}}+1)th$$ from the rearranged data. We get $5th$ and $6th$ terms which are –
$\begin{array}{rl}& 5th=23\\ & 6th=23\end{array}$
Now let us take the average of them to find our required median.
Median $=\left({\displaystyle \frac{23+23}{3}}\right)=23$
$Mean=23...(2)$
As we know that the empirical formula is given by –
$$mode=3median-2mean$$ Putting values from (1) and (2), we get –
$$\begin{array}{rl}& mode=3\left(23\right)-2\left(22.7\right)\\ & =69-45.4\\ & =23.6\end{array}$$

Hence, mode of given data is $23.6$.

So, the correct answer is “Option (c)”.

Note: Students should not try to find the mode by direct method in this type of data. As we can see that two terms are most frequent and this is a bimodular data and so mode cannot be found directly. For bimodular data, only empirical formulas should be used to calculate the mode. Students should not forget to rearrange the data in ascending order before calculating the median.