Question

In a dataset of 10 numbers, if the mean and median are both 20, which of the following statements is/are true?
(A). The dataset contains no outliers.
(B). The dataset contains exactly one outlier.
(C). The dataset contains multiple outliers.
(D). The dataset has no mode.
Choose the correct answer from the options given below:

• A. Only A
• B. Only B
• C. (A), (C) and (D) only
• D. (A), (B) and (D) only

Hint:

Outliers skew the mean but have little to no effect on the median:
- Positive Outlier \(\rightarrow\) \(\text{Mean} > \text{Median}\).
- Negative Outlier \(\rightarrow\) \(\text{Mean} < \text{Median}\).
- Symmetrical distribution No Outliers \(\rightarrow\) \(\text{Mean} \approx \text{Median}\).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The mean and median are both measures of central tendency, but they respond differently to extreme values (outliers).
The mean is highly sensitive to outliers because it incorporates all values in the dataset.
The median is resistant to outliers because it only represents the middle position.

Step 2: Detailed Explanation:
When a dataset is perfectly symmetric, the mean and median are equal.
An outlier is an extreme value that lies far away from the rest of the distribution.
If a dataset contains a single outlier (e.g., a very large positive value), the mean will be pulled significantly in the direction of the outlier (skewed to the right), making the mean larger than the median.
Conversely, a single extremely small value would pull the mean down, making it smaller than the median.
Therefore:
- Statement (B) cannot be true because a single outlier would disrupt the equality of the mean and median.
- Statement (D) is not necessarily true because a dataset with equal mean and median can have a mode (for example, the set \(\{20, 20, 20, 20, 20, 20, 20, 20, 20, 20\}\) has mean = 20, median = 20, and a mode of 20).
In simple, symmetric distributions without heavy tails, the equality of mean and median indicates a balanced, uniform distribution around the center, suggesting there are no outliers pulling the mean.
Hence, Statement (A) is the most reasonable true statement under typical conditions.

Step 3: Final Answer:
The only true statement under standard conditions is Statement (A). This corresponds to "Only A".