Question

For the data 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, …., 9, 9, the product of the mean and mode equals
(a) 9
(b) 45
(c) 57
(d) 285

Answer 1

Hint: To solve this question, we will first define mean and mode. The formula of mean is given by $$\text{Mean}={\displaystyle \frac{\text{Sum of observations}}{\text{Number of observations}}}$$ and mode is the number that occurs out of the most number of terms. The sum of n terms is given by $${\displaystyle \frac{n(n+1)}{2}}$$ and that of the square of n terms is given by $${\displaystyle \frac{n(n+1)(2n+1)}{6}}$$ to get the result. To solve this question, let us first define the mean of the terms and mode of the terms.

Complete step by step answer:
Let us first define mean and its formula. Mean is also called the Arithmetic Mean n the average of the numbers also called a central value of a set of numbers. To calculate mean, we will first add all the numbers and then divide by the number of terms. The formula is given by
$$\text{Mean}={\displaystyle \frac{\text{Sum of observations}}{\text{Number of observations}}}$$
And let us define mode now. The mode is the value that appears most frequently in a data set. A set of data may have one mode or more than one mode. We are given the terms as
$$1,2,2,3,3,3,4,4,4,4,......,9,9$$
Observing this, we see that 1 occurs 1 time, 2 occurs 2 times, 3 occurs 3 times and so on. We can say that the digits occur according to their value. Then the sum of the observations will be
$$1+2+2+3+3+3+.....+9+9$$
$$\Rightarrow 1+4+9+16.....+81$$
$$\Rightarrow 1+2^2+3^2+4^2.....+9^2$$
So, the sum of observations is $$1+2^2+3^2+4^2.....+9^2.$$ And also the number of observations is also 1 + 2 + 3 + 4 + ….. + 9, as 1 occurs once, 2 occurs twice and similarly for all others.
Therefore, the number of observations is $$1+2+3+4......+9.$$
The formula of the sum of n terms is given by $${\displaystyle \frac{n(n+1)}{2}}$$ and the formula of the square of the sum of the n terms is given by
$$1^2+2^2+....+n^2={\displaystyle \frac{n(n+1)(2n+1)}{6}}$$
Here, we have, n = 9.
Therefore, the number of observations will be
$$\Rightarrow {\displaystyle \frac{n(n+1)}{2}}$$
$$\Rightarrow {\displaystyle \frac{9\left(10\right)}{2}}$$
$$\Rightarrow 45$$
And the sum of the observations will be
$$\Rightarrow {\displaystyle \frac{n(n+1)(2n+1)}{6}}$$
$$\Rightarrow {\displaystyle \frac{9\left(10\right)(18+1)}{6}}$$
$$\Rightarrow {\displaystyle \frac{3\times 10\times 19}{2}}$$
$$\Rightarrow 15\times 19$$
$$\Rightarrow 285$$
Therefore, we will get the mean as
$$\text{Mean}={\displaystyle \frac{285}{45}}$$
$$\Rightarrow \text{Mean}={\displaystyle \frac{19}{3}}$$
So, the mean is given by $${\displaystyle \frac{19}{3}}.$$
And the mode is the value which occurs the maximum number of times. Because 9 is occurring 9 times, the mode will be 9. Therefore, the product of mean and mode will be
$$\Rightarrow {\displaystyle \frac{19}{3}}\times 9=57$$
Hence, the product of the mean and mode of the given terms is given by 57.

So, the correct answer is “Option C”.

Note: Another method to calculate mean can be directly substituting $${\displaystyle \frac{n(n+1)(2n+1)}{6}}$$ and $${\displaystyle \frac{n(n+1)}{2}}$$ in the numerator and denominator respectively and then solving.
$$\Rightarrow \text{Mean}={\displaystyle \frac{{\displaystyle \frac{n(n+1)(2n+1)}{6}}}{{\displaystyle \frac{n(n+1)}{2}}}}$$
$$\Rightarrow \text{Mean}={\displaystyle \frac{2n+1}{3}}$$
Here, n = 9. Therefore, we get the mean as
$$\Rightarrow \text{Mean}={\displaystyle \frac{18+1}{3}}={\displaystyle \frac{19}{3}}$$
Therefore, the answer of the mean is the same.