Question

Assertion (A) : If the value of mode and mean for a distribution is 50 and 56 respectively, then the value of median is 54.
Reason (R) : Median = \(\frac{1}{3}\) (Mode - 2 Mean)

• A. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
• B. Both Assertion (A) and Reason (R) are true, but Reason (R) is not correct explanation of Assertion (A).
• C. Assertion (A) is true, but Reason (R) is false.
• D. Assertion (A) is false, but Reason (R) is true.

Hint:

An easy way to memorize the empirical relation is by counting the syllables/letters or matching coefficients:
\[ \text{Mode} = 3\ \text{Median} - 2\ \text{Mean} \]
Think of it as "3 Medians minus 2 Means equals the Mode".
Note that the number 3 is paired with "Median" (the longer word) and the number 2 is paired with "Mean" (the shorter word).

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The question consists of an Assertion (A) and a Reason (R). We need to determine the mathematical validity of both statements and establish if the Reason correctly explains the Assertion.

Step 2: Key Formula or Approach:
The empirical relationship between the three measures of central tendency (Mean, Median, and Mode) is:
\[ \text{Mode} = 3\ \text{Median} - 2\ \text{Mean} \]
This formula can be rearranged to express the Median as:
\[ 3\ \text{Median} = \text{Mode} + 2\ \text{Mean} \implies \text{Median} = \frac{1}{3} (\text{Mode} + 2\ \text{Mean}) \]

Step 3: Detailed Explanation:
1.

Evaluate Assertion (A):
We are given:
\[ \text{Mode} = 50 \]
\[ \text{Mean} = 56 \]
Using the empirical relationship, let us compute the Median:
\[ \text{Median} = \frac{1}{3} (\text{Mode} + 2\ \text{Mean}) \]
Substitute the given values:
\[ \text{Median} = \frac{1}{3} (50 + 2 \times 56) \]
\[ \text{Median} = \frac{1}{3} (50 + 112) \]
\[ \text{Median} = \frac{1}{3} (162) = 54 \]
The calculated Median is 54, which matches the value stated in the assertion. Therefore, Assertion (A) is true.

2.

Evaluate Reason (R):
The Reason states that:
\[ \text{Median} = \frac{1}{3} (\text{Mode} - 2\ \text{Mean}) \]
As derived above, the correct rearranged empirical formula is actually:
\[ \text{Median} = \frac{1}{3} (\text{Mode} + 2\ \text{Mean}) \]
The formula stated in Reason (R) has a minus sign instead of a plus sign, which is incorrect. Thus, Reason (R) is false.

3. Combining our findings: Assertion (A) is true, but Reason (R) is false.

Step 4: Final Answer:
The correct option is (C) as the Assertion is true but the Reason is false.