Question

In a moderately asymmetrical distribution, the mode and mean are 51 and 55.5 respectively. Find the median.

• A. 106.5
• B. 54
• C. 53.25
• D. 108

Hint:

To memorize Karl Pearson's empirical formula easily, think of the words alphabetically backwards: "Mean, Median, Mode" and assign the coefficients 1, 3, and 2 respectively: 1 $\times$ Mode = 3 $\times$ Median - 2 $\times$ Mean.

Answer 1

The Correct Option is B

Step 1: Concept:
This problem requires calculating the median of a statistical dataset when the mean and mode are known. Since the distribution is explicitly described as "moderately asymmetrical" (skewed), we can apply Karl Pearson's empirical relationship to find the missing central tendency value.

Step 2: Key Formulas and approach:

• Karl Pearson's empirical formula relating the three measures of central tendency for a moderately skewed distribution is: \[ \text{Mode} = 3 \times \text{Median} - 2 \times \text{Mean} \]

Step 3: Step-by-step Explanation:

• Given variables from the question:
Mode = 51
Mean = 55.5

• Substitute the known numerical values directly into Pearson's empirical formula:
\[ 51 = 3 \times \text{Median} - 2 \times (55.5) \]
• Calculate the multiplication product on the right side of the equation:
\[ 2 \times 55.5 = 111 \]
• Update the equation with this new value:
\[ 51 = 3 \times \text{Median} - 111 \]
• Isolate the Median term by algebraically adding 111 to both sides of the equation:
\[ 51 + 111 = 3 \times \text{Median} \] \[ 162 = 3 \times \text{Median} \]
• Solve for the final Median by dividing the total by 3:
\[ \text{Median} = \frac{162}{3} \] \[ \text{Median} = 54 \]

Step 4: Final Answer:
The calculated median of the distribution is 54, which exactly corresponds to option (B).