Mean Deviation and Quartile Deviation

Mean deviation and quartile deviation are both measures of dispersion or variation in a set of data. Mean deviation measures the average distance of each data point from the mean of the data set, while quartile deviation measures the spread of the middle 50% of the data set.

Calculating Quartile Deviation from Mean Deviation

There is a relationship between mean deviation and quartile deviation for normal distributions. Specifically, quartile deviation is approximately equal to mean deviation divided by 0.6745.

Using this relationship, we can calculate the quartile deviation for a normal variable with a mean deviation of 16:

Quartile deviation ≈ Mean deviation / 0.6745
Quartile deviation ≈ 16 / 0.6745
Quartile deviation ≈ 23.733

However, quartile deviation is defined as the difference between the upper and lower quartiles, which are the 75th and 25th percentiles of the data set, respectively. In a normal distribution, these quartiles are located at a distance of approximately 0.6745 standard deviations from the mean.

Therefore, we can also calculate the quartile deviation as:

Quartile deviation ≈ 0.6745 x standard deviation
Standard deviation ≈ Mean deviation / 0.8

Plugging in the value of mean deviation, we get:

Standard deviation ≈ 16 / 0.8
Standard deviation ≈ 20

And using the formula for quartile deviation, we get:

Quartile deviation ≈ 0.6745 x 20
Quartile deviation ≈ 13.49

Therefore, the correct answer is option (b) 13.50, which is closest to our calculated value of 13.49.