Question

If the difference between mean and mode is$63$, the difference between mean and median is?
A. $189$
B. $21$
C. $31.5$
D. $48.5$

Answer 1

Hint:
If we are given the value of mean and median then we can find the value of mode by the formula
${\text{mode}} = 3{\text{median}} - 2{\text{mean}}$

Complete step by step solution:
Here we must have the knowledge of what mean, median and mode means. Mean is the average of the numbers given to us which is the ratio of sum of all the numbers and the total number of values present there or we call it the frequency of the items whose mean is to be found.
Median is the central number of the given set and for finding it we need to first arrange the numbers of the distribution in the ascending or descending order and then it depends on whether the number of observations in the set is odd or even. So in this way we can find the median
Mode is the value which appears most of the time in the set of observations given whose mode is to be found. For example: if we have the set of observations like $1, 2, 2, 2, 5, 5, 4, 6, 6, 6, 6$
Here we see that $6$ appears most of the time, that is its frequency is maximum so we can say that it is the mode of the given set of observations.
So now we know what the meaning of mean, median and mode is.
Now we need to solve this question by using the formula mentioned above which is
${\text{mode}} = 3{\text{median}} - 2{\text{mean}}$$ - - - - (1)$
So we are given that
${\text{mean}} - {\text{mode}} = 63$$ - - - - (2)$
Now subtracting mean on both sides of the equation (1)
$
  {\text{mode}} - {\text{mean}} = 3{\text{median}} - 2{\text{mean}} - {\text{mean}} \\
  {\text{mode}} - {\text{mean}} = {\text{3median}} - {\text{3mean}} \\
  {\text{mode}} - {\text{mean}} = {\text{3(median}} - {\text{mean)}} \\
 $
Now putting the value of the equation (2) in the above formula we get that
$
  {\text{mode}} - {\text{mean}} = {\text{3(median}} - {\text{mean)}} \\
  {\text{mean}} - {\text{mode}} = {\text{3(mean}} - {\text{median)}} \\
 $
$
  63 = 3({\text{mean}} - {\text{median)}} \\
  {\text{mean}} - {\text{median}} = {\text{21}} \\
 $

So we get the difference between mean and median as $21$.

Note:
If we are given the set of observations like $1, 2, 2, 3, 4$
Mean is the average of five numbers$ = \dfrac{{1 + 2 + 2 + 3 + 4}}{5}$
Median is the middle number which is the third number which is $2$
Mode is the maximum frequency term which is $2$
So again we get all the values. So we can again apply the same formula and get the required term we need.