Question

Which one of the following is INCORRECT about normal distribution?
(1) It is bell-shaped and symmetrical about its mean.
(2) Its mean, median and mode are same.
(3) Mean deviation from the mean is $\frac{4}{5}$ times standard deviation.
(4) Area under the normal distribution within three standard deviation from mean is 95.5%.

• A. 1
• B. 2
• C. 3
• D. 4

Hint:

Memorize the 68-95-99.7 rule! It tells you exactly how much data falls within 1, 2, and 3 standard deviations of the mean in any normal distribution.

Answer 1

The Correct Option is D

Concept: The Normal (Gaussian) distribution is a continuous probability distribution defined by its bell-shaped curve. It is characterized by specific mathematical properties regarding its symmetry and the spread of data relative to the mean.

Step 1: Evaluate General Properties (Statements 1 and 2).
Statement (1) is correct. The normal distribution is perfectly symmetrical around its center, creating the classic "bell curve." Statement (2) is also correct; in a perfectly normal distribution, the peak of the curve occurs at the mean, which is also the middle value (median) and the most frequent value (mode).

Step 2: Evaluate the Mean Deviation (Statement 3).
For a normal distribution, the Mean Deviation ($M.D.$) is approximately $0.7979\sigma$, which is very close to $\frac{4}{5}\sigma$ (or $0.8\sigma$). Thus, statement (3) is considered a correct standard approximation in statistics.

Step 3: Evaluate Area under the Curve (Statement 4).
The Empirical Rule (68-95-99.7 rule) defines the area under the curve:
• Within 1$\sigma$: $\approx$ 68.2%
• Within 2$\sigma$: $\approx$ 95.4% (or 95.5% in some texts)
• Within 3$\sigma$: $\approx$ 99.7% Statement (4) claims the area within $3\sigma$ is 95.5%, which is incorrect because 95.5% corresponds to the area within $2\sigma$.