Question

The median and mode of a distribution are $25.2$ and $26.1$ respectively. The mean of the distribution is :

• A. $24.75$
• B. $24.25$
• C. $24.3$
• D. $25.5$

Hint:

A handy way to remember the empirical formula is by ordering the words alphabetically:
Mean, Median, Mode.
The formula is:
\[ \text{Mode} = 3 \text{ Median} - 2 \text{ Mean} \] Think of the coefficients: 3 (larger word 'Median' with 6 letters) and 2 (smaller word 'Mean' with 4 letters).
This simple mnemonic prevents you from mixing up the coefficients!

Answer 1

The Correct Option is A

Step 1: Rearrange the empirical formula symbolically for the Mean first, before plugging in any numbers.
The empirical relationship between mean, median and mode is:
\[ \text{Mode} = 3 \times \text{Median} - 2 \times \text{Mean} \]
Rearranging this to isolate Mean on one side, purely in symbols:
\[ 2 \times \text{Mean} = 3 \times \text{Median} - \text{Mode} \]
\[ \text{Mean} = \frac{3 \times \text{Median} - \text{Mode}}{2} \]

Step 2: Now substitute the given numerical values into this ready-made formula.
We are given Median $= 25.2$ and Mode $= 26.1$.
\[ \text{Mean} = \frac{3(25.2) - 26.1}{2} \]

Step 3: Compute the numerator first.
\[ 3 \times 25.2 = 75.6 \]
\[ 75.6 - 26.1 = 49.5 \]

Step 4: Divide by 2 to get the mean.
\[ \text{Mean} = \frac{49.5}{2} = 24.75 \]

Final Answer:
The mean of the distribution is $24.75$, which matches Option (A). \[ \boxed{\text{Mean} = 24.75} \]