Question

Choose the correct option from the provided options to the following question:
If the difference of mode and median of a data is \[24\] then the difference of median and mean is
A. \[12\]
B. \[24\]
C. \[8\]
D. \[36\]

Answer 1

Hint: We have the relation between the difference of mode and median. Now we substitute this relation in the formula of mode from the basic statistics. By rearranging the values and getting it in terms of the asked question, we get the relation, i.e., the difference between median and mean.

Complete step-by-step solution:
Given that,
The difference between the mode and median \[ = 24\]
\[ \Rightarrow \]Mode \[ - \] Median \[ = 24\]
Now, the difference between median and mode should be calculated.
According to the basic rules of statistics;
Mode of a given data is equal to the difference of thrice of the median and twice of the mode. So,
Mode = \[3 \times \]Median \[ - \] \[2 \times \] Mode
Now,
We can also write \[3 \times \] Median as Median \[ + \]\[2 \times \] Median
\[ \Rightarrow \]\[3 \times \] Median \[ = \] Median \[ + \]\[2 \times \] Median
Now, we can substitute the above equation in the previous equation of mode.
\[ \Rightarrow \] Mode\[ = \] Median \[ + \]\[2 \times \] Median \[ - \] \[2 \times \]Mode
Rearranging the above terms, we get;
\[ \Rightarrow \] Mode\[ - \] Median\[ = \]\[2\][Median\[ - \] Mode]
We have the value as Mode \[ - \] Median \[ = 24\]
That implies, we can substitute the value of Mode \[ - \] Median \[ = 24\] in the equation above. Then we get;
\[ \Rightarrow 24 = \]\[2\][Median\[ - \] Mode]
Rearranging the terms, we get;
\[ \Rightarrow \][Median\[ - \] Mode] \[ = \dfrac{{24}}{2}\]
\[ \Rightarrow \] Median\[ - \] Mode\[ = 12\]
Hence, we have the value of Median\[ - \] Mode\[ = 12\]

$\therefore $ The correct option is A.

Note: Median is the middle value among the observed set of values and is calculated by arranging all the values in either ascending or descending order. We choose the middle value, terming it as the median. Mode is the data set which has the highest frequency which is calculated by counting the number of times each data value occurs. The highest count is the mode.