Question

Which of the following conditions make 'Mean' misrepresentative of data ?
• [A.] Presence of outlier in the data
• [B.] Ordinal variable
• [C.] Bell shaped data distribution
• [D.] Skewed data
Choose the correct answer from the options given below :

• A. A, B and C only
• B. A and B only
• C. A, B and D only
• D. B and C only

Hint:

Mean is highly sensitive to outliers and skewed distributions. For ordinal or highly skewed data, median is often a better measure of central tendency.

Answer 1

The Correct Option is C

Concept: Mean is one of the most widely used measures of central tendency. It is calculated as: \[ \text{Mean} = \frac{\sum x}{n} \] Although mean is mathematically convenient, it is not always the best representative value for a dataset. Certain conditions distort the mean and make it misleading. The mean works best when:
• Data are numerical,
• Distribution is approximately symmetrical,
• Extreme values are absent.

Step 1: Analyzing option A : Presence of outlier in the data. An outlier is an extremely large or extremely small value compared to the remaining observations. For example: \[ 5,\;6,\;7,\;8,\;100 \] Mean: \[ = \frac{5+6+7+8+100}{5} \] \[ = \frac{126}{5} \] \[ = 25.2 \] Clearly: \[ 25.2 \] does not represent the actual central tendency of most observations. Thus: \[ \boxed{\text{Outliers distort the mean}} \] Hence option A is correct.

Step 2: Analyzing option B : Ordinal variable. Ordinal data represent rankings or ordered categories such as: \[ \text{Poor, Average, Good, Excellent} \] or ranks like: \[ 1^{st}, 2^{nd}, 3^{rd} \] Since intervals between categories are not equal, arithmetic operations like averaging are not meaningful. Therefore: \[ \boxed{\text{Mean is not appropriate for ordinal data}} \] Hence option B is correct.

Step 3: Analyzing option C : Bell shaped data distribution. Bell shaped distribution refers to: \[ \text{Normal distribution} \] In a normal distribution: \[ \text{Mean} = \text{Median} = \text{Mode} \] and the mean represents the data very effectively. Thus: \[ \boxed{\text{Bell shaped distribution does not misrepresent the mean}} \] Hence option C is incorrect.

Step 4: Analyzing option D : Skewed data. Skewed data are asymmetrical distributions where values are stretched more toward one side. In positively skewed distributions: \[ \text{Mean} > \text{Median} \] In negatively skewed distributions: \[ \text{Mean} < \text{Median} \] Because the mean gets pulled toward the long tail: \[ \boxed{\text{Mean becomes less representative}} \] Hence option D is correct.

Step 5: Final conclusion. The conditions which make mean misrepresentative are: \[ A,\;B,\;D \] Hence, the correct answer is: \[ \boxed{\text{(C) A, B and D only}} \]