Quartile Deviation (QD) is a measure of dispersion that captures the spread of the middle 50% of data, calculated as half the difference between the third and first quartiles.
## Core concept
Quartile Deviation measures variability using the interquartile range (IQR). It focuses on the central portion of data, making it resistant to extreme values compared to range or standard deviation.
- Q1 (First Quartile): The value below which 25% of observations lie
- Q3 (Third Quartile): The value below which 75% of observations lie
- Quartile Deviation: QD = (Q3 − Q1) ÷ 2
The middle 50% of data lies between Q1 and Q3. Quartile Deviation is half this spread.
## Formula and calculation
Quartile Deviation: $$QD = \frac{Q_3 - Q_1}{2}$$
Coefficient of Quartile Deviation (CQD): $$CQD = \frac{Q_3 - Q_1}{Q_3 + Q_1}$$
Use CQD to compare dispersion across datasets with different units or scales (a dimensionless measure).
Finding Q1 and Q3 for ungrouped data: - Arrange data in ascending order - Q1 position = (n + 1) ÷ 4 - Q3 position = 3(n + 1) ÷ 4 - Interpolate if position is not a whole number
For grouped data: $$Q_1 = L + \left(\frac{n}{4} - CF\right) \frac{h}{f}$$
$$Q_3 = L + \left(\frac{3n}{4} - CF\right) \frac{h}{f}$$
Where: L = lower class boundary, CF = cumulative frequency before the class, h = class width, f = frequency of the class, n = total frequency
## Worked example
Ungrouped data: 12, 18, 22, 25, 28, 32, 35, 40, 48
n = 9
- Q1 position = (9 + 1) ÷ 4 = 2.5 → between 2nd and 3rd values
- Q1 = 18 + 0.5(22 − 18) = 20
- Q3 position = 3(9 + 1) ÷ 4 = 7.5 → between 7th and 8th values
- Q3 = 35 + 0.5(40 − 35) = 37.5
QD = (37.5 − 20) ÷ 2 = 8.75
CQD = (37.5 − 20) ÷ (37.5 + 20) = 17.5 ÷ 57.5 = 0.304
## Common exam applications
- Comparing variability: Use CQD when datasets have different units (e.g., wages in rupees vs. height in cm)
- Skewed distributions: QD is more reliable than standard deviation when data contains outliers
- Box plots: QD defines the box width in a box-and-whisker plot
- Industry applications: Quality control, income distribution analysis, performance metrics
## Common mistakes
- Confusing QD with IQR: IQR = Q3 − Q1; QD is half of IQR
- Wrong position formula: Use (n + 1) ÷ 4 for ungrouped data, not n ÷ 4
- Forgetting interpolation: When quartile position is decimal, always interpolate between adjacent values
- Misapplying CQD: Use CQD only for comparative analysis; it's not a stand-alone dispersion measure
- Incorrect class identification: In grouped data, ensure CF is cumulative frequency *before* the quartile class
## Key distinction from siblings
Unlike Range (max − min), QD ignores extremes. Unlike Standard Deviation, QD is easier to compute and unaffected by outliers. Unlike Variance, QD is in original units (not squared). In presence of Skewness, QD is preferred over SD.