Introduction to Quartile Deviation
Quartile Deviation (QD), also known as the semi-interquartile range, is a measure of statistical dispersion. It tells us about the spread of the middle 50% of the data. Unlike the range, it is not sensitive to extreme outliers, making it a robust measure for skewed distributions.
The Formula
The Quartile Deviation is calculated using the first quartile ($Q_1$) and the third quartile ($Q_3$):
$$QD = \frac{Q_3 - Q_1}{2}$$
Step-by-Step Solution
Given the data: 15, 7, 25, 12, 4, 22, 19, 10
Step 1: Arrange the data in ascending order
Sorted data: 4, 7, 10, 12, 15, 19, 22, 25 Total number of observations ($n$) = 8
Step 2: Find the Quartiles
For an even number of observations ($n=8$):
$Q_1$ (First Quartile): Position = $\frac{n+1}{4} = \frac{9}{4} = 2.25$. This falls between the 2nd (7) and 3rd (10) values. Linear interpolation: $7 + 0.25 \times (10 - 7) = 7 + 0.75 = 7.75$
$Q_3$ (Third Quartile): Position = $\frac{3(n+1)}{4} = \frac{27}{4} = 6.75$. This falls between the 6th (19) and 7th (22) values. Linear interpolation: $19 + 0.75 \times (22 - 19) = 19 + 2.25 = 21.25$
Step 3: Calculate the Quartile Deviation
$$QD = \frac{21.25 - 7.75}{2}$$ $$QD = \frac{13.5}{2} = 6.75$$
Conclusion
The quartile deviation for the given dataset is 6.75. This value provides a reliable estimate of the typical variation around the median of your data.