Calculating Quartile Deviation from Mean Deviation

The question asks us to find the value of the quartile deviation given the mean deviation of a set of observations. While there isn't a single fixed mathematical formula that connects mean deviation and quartile deviation for all types of distributions, there is an empirical relationship that is often used, especially for distributions with moderate skewness, like the normal distribution.

Empirical Relationship Between Measures of Dispersion

For a normal distribution or distributions that are close to normal, there are approximate relationships between the standard deviation ($\sigma$), mean deviation (MD), and quartile deviation (QD).

The commonly used empirical relationships are:

• Mean Deviation $\approx \frac{4}{5} \times$ Standard Deviation
• Quartile Deviation $\approx \frac{2}{3} \times$ Standard Deviation

Using these two relationships, we can derive an approximate relationship between mean deviation and quartile deviation.

From the first relationship, we have Standard Deviation $\approx \frac{5}{4} \times$ Mean Deviation.

Substituting this into the second relationship:

Quartile Deviation $\approx \frac{2}{3} \times \left(\frac{5}{4} \times \text{Mean Deviation}\right)$

Quartile Deviation $\approx \frac{10}{12} \times \text{Mean Deviation}$

Quartile Deviation $\approx \frac{5}{6} \times \text{Mean Deviation}

Applying the Empirical Formula

We are given that the mean deviation of the set of observations is 15.

Using the derived empirical relationship:

Quartile Deviation $\approx \frac{5}{6} \times \text{Mean Deviation}

Substitute the given value:

Quartile Deviation $\approx \frac{5}{6} \times 15

Quartile Deviation $\approx \frac{5 \times 15}{6}

Quartile Deviation $\approx \frac{75}{6}

Now, we calculate the value:

$\frac{75}{6} = \frac{25 \times 3}{2 \times 3} = \frac{25}{2} = 12.5$

So, the value of the quartile deviation is approximately 12.5 based on this empirical relationship.

Summary of Calculation

Based on the commonly used empirical relationship between mean deviation and quartile deviation, a mean deviation of 15 corresponds to a quartile deviation of approximately 12.5.

Revision Table: Measures of Dispersion

Additional Information on Measures of Dispersion

Measures of dispersion are statistical values that describe the spread or variability of data points in a distribution. Key measures include range, quartile deviation, mean deviation, variance, and standard deviation.

• Quartile Deviation: It is based on the middle 50% of the data. It is not affected by extreme values, making it a good measure for skewed distributions or when extreme values are present.
• Mean Deviation: It measures the average absolute difference between each data point and the central value (mean or median). It is less affected by extreme values than standard deviation but uses absolute values, which makes further mathematical analysis difficult.
• Standard Deviation: The most widely used measure of dispersion. It considers the square of deviations from the mean, giving more weight to larger deviations. It is sensitive to extreme values but is mathematically tractable and forms the basis for many statistical tests.

The empirical relationships discussed are approximations that hold best for symmetrical distributions like the normal distribution. For distributions that are highly skewed or have unusual shapes, these relationships may not be accurate.