Question

The mean and median of the data are respectively 20 and 22. The value of mode is:
(a) 20
(b) 26
(c) 22
(d) 21

Answer 1

Hint: In the above problem, we have given the mean and median of the data as 20 and 22 respectively and we are asked to find the mode of the data. We know that there is a relationship between mean, median, and mode which is equal to $Mode=3Median-2Mean$. Now, in the formula, we know the value of mean and median so we are going to substitute the value of a mean as 20 and median as 22 in this formula and will get the value of mode.

Complete step-by-step solution:
 We have given the value of the mean of the data as 20 and the value of the median of the data as 22. We are asked to determine the value of the mode of the data.
Let us discuss the mean, median, and mode.
Let us assume a data in which n observations are there which are:
${{x}_{1}},{{x}_{2}},{{x}_{2}},{{x}_{3}}......{{x}_{n}}$
Now, the mean of the above data is the addition of all the observations divided by the total number of observations.
The median of the data is the middle observation among n observations.
The mode of the data is the observation which occurs the highest number of times. For instance, the observations have given above ${{x}_{2}}$ has occurred twice and the remaining observations have occurred only once so ${{x}_{2}}$ is the mode of the given data.
In regards to statistics, the empirical relation between mean, median and mode is given as:
$Mode=3Median-2Mean$
Substituting the value of mean as 20 and median as 22 in the above formula we get the value of mode:
$\begin{align}
  & Mode=3\left( 22 \right)-2\left( 20 \right) \\
 & \Rightarrow Mode=66-40 \\
 & \Rightarrow Mode=26 \\
\end{align}$
Hence, we got the value of mode of the data as 26 so the correct option is (b).

Note: You can check whether the value of mode that you are getting is correct or not substituting the value of mode in the empirical relation between mean, median, and mode.
$Mode=3Median-2Mean$
Substituting mode as 26 and let us substitute any one value from median and mean. Lets us substitute the median as 22 in the above we get,
$\begin{align}
  & 26=3\left( 22 \right)-2Mean \\
 & \Rightarrow 26=66-2Mean \\
 & \Rightarrow 26-66=-2Mean \\
 & \Rightarrow -40=-2Mean \\
\end{align}$
Dividing -2 on both the sides we get,
$\begin{align}
  & \dfrac{-40}{-2}=Mean \\
 & \Rightarrow 20=Mean \\
\end{align}$
As we are getting the value of a mean as 20 which is given in the question so the mode that we have calculated above is correct.