If empirical relationship between mean, median and mode is expressed as ${\text{mean}} = k\left( {3{\text{median - mode}}} \right)$, then find the value of $k$.

Hint: We will use the relation between mean, median and mode, that is ${\text{mean - mode}} = 3\left( {{\text{mean - median}}} \right)$ to solve this question. We will express mean in terms of median and mode and then compare it with the given relation to find the value of $k$.

Complete step-by-step answer:
We are given that the empirical relationship between mean, mode and median is given as,
${\text{mean}} = k\left( {3{\text{median - mode}}} \right)$ eqn. (1)
We also know that the relation between mean, median and mode is given as
${\text{mean - mode}} = 3\left( {{\text{mean - median}}} \right)$ which is also equivalent to
$\Rightarrow$ ${\text{mean - mode}} = 3{\text{mean}} - 3{\text{median}}$
We will determine the value of mean from the above expression.
$\Rightarrow$ ${\text{2mean}} = 3{\text{median}} - {\text{mode}}$
Divide by 2 on both sides.
$\Rightarrow$ ${\text{mean}} = \dfrac{1}{2}\left( {3{\text{median}} - {\text{mode}}} \right)$ eqn. (2)
On comparing equation (1) and (2), we will get, $k = \dfrac{1}{2}$.

Note: Mean, median and mode are statistical terms and are three different types of “averages”. Mean is calculated by dividing the sum of all observations by the total number of observations. Median is referred to as mid-value or central value of the data. Mode is the value of the data which is repeated a maximum number of times.