Relation between Quartile Deviation and Standard Deviation

Quartile deviation (Q.D.) and standard deviation (S.D.) are two measures of dispersion used in statistics. A moderately skewed distribution is a distribution that is not perfectly symmetrical, but has a skewness between -1 and -0.5, or between 0.5 and 1. In such a distribution, the quartile deviation and the standard deviation are related by the formula:

S.D. = 3/2 Q.D.

Explanation:

- Quartile deviation (Q.D.) is the difference between the third quartile (Q3) and the first quartile (Q1), divided by 2. Mathematically, Q.D. = (Q3 - Q1) / 2.
- Standard deviation (S.D.) is a measure of how spread out the data is from the mean. It is calculated as the square root of the variance. Mathematically, S.D. = √(∑(x-μ)²/n), where x is the value of each data point, μ is the mean, and n is the number of data points.
- For a moderately skewed distribution, the S.D. is related to the Q.D. by the formula S.D. = 3/2 Q.D. This means that the standard deviation is 1.5 times larger than the quartile deviation.

Example:

Suppose we have a moderately skewed distribution of exam scores for a class of 30 students. The quartile deviation for the scores is 8. To find the standard deviation using the formula S.D. = 3/2 Q.D., we can substitute the value of Q.D. and get:

S.D. = 3/2 Q.D.
S.D. = 3/2 x 8
S.D. = 12

Therefore, the standard deviation for the exam scores is 12.

Conclusion:

In summary, for a moderately skewed distribution, quartile deviation and standard deviation are related by the formula S.D. = 3/2 Q.D. This formula can be used to find one measure of dispersion when the other is known.