Question

Ages of players in a cricket team are $25,26,25,27,28,30,31,27,33,27,29$. (i) Find the mean and mode of the data. (ii) Find the minimum number of players to be added to the above team so that mode of the data changes and what must be their ages.
$\begin{align}
  & A(i)Mean=25,Mode=36 \\
 & (ii)3players~of~age~25~years~each \\
 & B(i)Mean=23,Mode=32 \\
 & (ii)4players~of~age~25~years~each \\
 & C(i)Mean=28,Mode=27 \\
 & (ii)2players~of~age~25~years~each \\
 & D(i)Mean=30,Mode=25 \\
 & (ii)1players~of~age~25~years~each \\
\end{align}$

Answer 1

Hint: In this question, we are given a set of data for which we have to find mean and mode for (i). As we know, mean is just the average of the data which means we will find the sum of terms in the data and then divide it by total number of terms to calculate mean. Also, as mode is the number which occurs most frequently in the series so we will analyze the series to find the term which occurs most frequently. For (ii), we have to change mode. Hence, we will find the term with the second most frequent number and make that as mode by adding the number of terms one more than the frequency of original mode value.

Complete step-by-step answer:
Let us solve (i).
We have to find mean and mode of the following data –
$25,26,25,27,28,30,31,27,33,27,29$
Since, $mean=\dfrac{sum~of~terms}{number~of~terms}$, so from data we can see that there are $11$ terms.
Hence, number of terms $=11$.
Sum of terms can be calculated as $25+26+25+27+28+30+31+27+33+27+29=308$
Therefore, $mean=\dfrac{308}{11}=28$
Now, mode is the term which occurs most frequently. As we can see $27$ is repeated three times, therefore the mode is $27$.
$\therefore Mean=28,Mode=27$

Let us solve (ii).
For this, let us analyze the data and find the second most frequent term. As we can see that $25$ occurs two times in the data which is just one less than the number of times original mode $27$ occurs.
To change mode, we need to have frequency $25$ more than the frequency of $27$. So we must have $25$ four times for mode to be changed. Since $25$ is repeated two times already, so we have to add $25$ two more times for mode to be changed.
Hence, minimum players of $25$ years which should be added are $2$.

Hence, $\begin{align}
  & (i)Mean=28,Mode=27 \\
 & (ii)2players~of~age~25~years~each \\
\end{align}$.

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

Note: Students should remember that there can’t be two terms with highest frequencies, because then it will become a bimodal series and can only be solved using empirical formula. When we want to change mode, we should always add one more term than frequency of already existing mode. Don’t make mistakes while adding so many terms. Students can also use empirical formula for calculating mode.