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Hint: First we will use the formula of the relationship between mean, median and mode is \[{\text{Mode}} = 3{\text{Median}} - 2{\text{Mean}}\]. Then simplify the given relationship by adding and subtracting Mean on both sides to find the required value.
Complete step-by-step answer:
We know that the formula of the relationship between mean, median and mode is \[{\text{Mode}} = 3{\text{Median}} - 2{\text{Mean}}\].
Subtracting and adding the value of mean in the above formula of relationship to find the empirical relationship, we get
\[
\Rightarrow {\text{Mode}} - {\text{Mean}} = 3{\text{Median}} - 2{\text{Mean}} - {\text{Mean}} \\
\Rightarrow {\text{Mode}} - {\text{Mean}} = 3{\text{Median}} - 3{\text{Mean}} \\
\]
Taking the 3 common from the right hand side of the above equation, we get
\[ \Rightarrow {\text{Mode}} - {\text{Mean}} = 3\left( {{\text{Median}} - {\text{Mean}}} \right)\]
Therefore, the required value is \[{\text{Mode}} - {\text{Mean}} = 3\left( {{\text{Median}} - {\text{Mean}}} \right)\].
Hence, option C is correct.
Note: We know that the mean or an average of a data set is found by adding all numbers in the data set and then dividing by the number of values in the set. The median is the middle value when a data set is ordered from least to greatest. The mode is the number that occurs most often in a data set.
Complete step-by-step answer:
We know that the formula of the relationship between mean, median and mode is \[{\text{Mode}} = 3{\text{Median}} - 2{\text{Mean}}\].
Subtracting and adding the value of mean in the above formula of relationship to find the empirical relationship, we get
\[
\Rightarrow {\text{Mode}} - {\text{Mean}} = 3{\text{Median}} - 2{\text{Mean}} - {\text{Mean}} \\
\Rightarrow {\text{Mode}} - {\text{Mean}} = 3{\text{Median}} - 3{\text{Mean}} \\
\]
Taking the 3 common from the right hand side of the above equation, we get
\[ \Rightarrow {\text{Mode}} - {\text{Mean}} = 3\left( {{\text{Median}} - {\text{Mean}}} \right)\]
Therefore, the required value is \[{\text{Mode}} - {\text{Mean}} = 3\left( {{\text{Median}} - {\text{Mean}}} \right)\].
Hence, option C is correct.
Note: We know that the mean or an average of a data set is found by adding all numbers in the data set and then dividing by the number of values in the set. The median is the middle value when a data set is ordered from least to greatest. The mode is the number that occurs most often in a data set.
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