Question

Given below are two statements: one is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A): 4 is mode, median and mean of the following data: 1, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5, 6, 6, 7
Reason (R): The mean, mode and median are always equal for any data.
In the light of the above statements, choose the most appropriate answer from the options given below:

• A. Both (A) and (R) are correct and (R) is the correct explanation of (A)
• B. Both (A) and (R) are correct but (R) is not the correct explanation of (A)
• C. (A) is correct but (R) is not correct
• D. (A) is not correct but (R) is correct

Hint:

Any statement containing absolute words like "always" or "never" in mathematics is highly likely to be false.
Knowing this allows you to immediately identify Reason (R) as false without further analysis!

Answer 1

The Correct Option is C

•

Step 1: Understanding the Question: This is an Assertion-Reasoning question based on the measures of central tendency (Mean, Median, and Mode).
We need to calculate these parameters for the given dataset and evaluate the validity of both statements.

•

Step 2: Key Formula or Approach:
- Mean (\(\mu\)) = \(\frac{\sum x_i}{N}\)
- Median = The middle value of the sorted data (or the average of the two middle values if \(N\) is even).
- Mode = The value that appears with the highest frequency.
- Evaluate Reason (R): Decide if the mean, median, and mode are always equal for any dataset.

•

Step 3: Detailed Explanation:
-
Step 3.1: Let us analyze Assertion (A):
The given data is already sorted:
\[ 1, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5, 6, 6, 7 \]
Total number of observations, \(N = 14\).
1. Mean Calculation:
Sum of observations:
\[ \sum x = 1 + 2 + 2 + 3 + 3 + 4 + 4 + 4 + 4 + 5 + 5 + 6 + 6 + 7 = 56 \]
\[ \text{Mean} = \frac{56}{14} = 4 \]
2. Median Calculation:
Since \(N = 14\) is even, the median is the average of the \(7^{\text{th}}\) and \(8^{\text{th}}\) terms:
\(7^{\text{th}}\) term = 4
\(8^{\text{th}}\) term = 4
\[ \text{Median} = \frac{4 + 4}{2} = 4 \]
3. Mode Calculation:
The frequency of each number is:
- 1 appears 1 time
- 2 appears 2 times
- 3 appears 2 times
- 4 appears 4 times
- 5 appears 2 times
- 6 appears 2 times
- 7 appears 1 time
Since 4 appears most frequently (4 times), the mode is 4.
Since Mean = 4, Median = 4, and Mode = 4, Assertion (A) is correct.
-
Step 3.2: Let us analyze Reason (R):
The mean, median, and mode are only equal for perfectly symmetric, unimodal distributions.
For asymmetric (skewed) data, these three measures are generally different.
Therefore, the statement "The mean, mode and median are always equal for any data" is false.

•

Step 4: Final Answer: (A) is correct but (R) is not correct, which corresponds to option (C).