Question

Consider the following statements:
(A). In symmetrical distribution mean, median and mode coincide
(B). The normal distribution has a Pearson's Coefficient (\(\beta_2\)) value equal to zero
(C). If Pearson's Coefficient (\(\beta_2\)) is less than 3, the distribution is said to be leptokurtic
(D). For a positive skewed distribution mean \(>\) median
(E). Skewness and kurtosis are measures of dispersion
Choose the correct answer from the options given below:

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

Hint:

Associate "Platykurtic" with "flat like a plate" (\(\beta_2 < 3\)), "Leptokurtic" with "leaping high" (\(\beta_2 > 3\)), and "Mesokurtic" with "middle/normal" (\(\beta_2 = 3\)). This visual association helps avoid confusion during exams.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
To evaluate these statements, we must analyze the mathematical and statistical properties of probability distributions, including shape parameters such as symmetry, skewness, and kurtosis, and distinguish them from measures of central tendency and dispersion.

Step 2: Detailed Explanation:
Let us analyze each statement systematically:

Statement (A): In a perfectly symmetrical distribution, the shape of the curve on the left side of the central peak is a mirror image of the right side.
The mean is the arithmetic center, the median is the value dividing the area into two equal halves, and the mode is the point of maximum frequency.
For a single-peaked symmetrical distribution, all three metrics lie at the exact center of symmetry, meaning they coincide.
Hence, Statement (A) istrue.

Statement (B): Pearson's coefficient of kurtosis is defined as \(\beta_2 = \mu_4 / \mu_2^2\), where \(\mu_4\) and \(\mu_2\) are the fourth and second central moments.
For a standard normal distribution, the value of \(\beta_2\) is mathematically equal to 3.
The coefficient that equals zero for a normal distribution is the excess kurtosis, defined as \(\gamma_2 = \beta_2 - 3\).
Therefore, \(\beta_2 = 0\) is incorrect, making Statement (B)false.

Statement (C): Kurtosis measures the relative peakedness or flatness of a distribution curve compared to a normal distribution.
If \(\beta_2 > 3\), the distribution is leptokurtic (highly peaked with fat tails).
If \(\beta_2 < 3\), the distribution is platykurtic (flat-topped with thin tails).
Since the statement claims a distribution with \(\beta_2 < 3\) is leptokurtic, Statement (C) isfalse.

Statement (D): Skewness measures the degree of asymmetry of a distribution.
In a positively skewed distribution, the tail on the right side of the probability density function is longer or fatter than the left side.
The extreme high values pull the mean (which is sensitive to extreme values) to the right, whereas the median remains relatively unaffected.
This results in the standard inequality: \(\text{mean} > \text{median} > \text{mode}\).
Hence, Statement (D) istrue.

Statement (E): Measures of dispersion (such as range, mean deviation, variance, and standard deviation) describe the spread or variability of the data.
Skewness and kurtosis are measures of the shape of the distribution, describing asymmetry and peakedness rather than the scale of spread.
Hence, Statement (E) isfalse.
Combining our findings, only statements (A) and (D) are correct.

Step 3: Final Answer:
The correct option is (B), representing (A) and (D) only.