Range and Interquartile Range

Overview

Range and IQR are measures of spread that describe how dispersed the data is. IQR is more robust to outliers than range.

Range

Definition

$$\text{Range} = \text{Maximum} - \text{Minimum}$$

Properties

• Simplest measure of spread
• Uses only two values (extremes)
• Very sensitive to outliers
• Increases with sample size

Example

Data: 3, 7, 8, 12, 15, 18, 22

$$\text{Range} = 22 - 3 = 19$$

Interquartile Range (IQR)

Definition

$$\text{IQR} = Q_3 - Q_1$$

Where:

• $Q_1$ = 25th percentile (first quartile)
• $Q_3$ = 75th percentile (third quartile)

Properties

• Measures spread of middle 50% of data
• Resistant to outliers
• Used in boxplots
• More stable than range

Calculating Quartiles

1. Order data from lowest to highest
2. Find the median ($Q_2$)
3. $Q_1$ = median of lower half
4. $Q_3$ = median of upper half

Five-Number Summary

A complete summary of data distribution:

Value	Description
Minimum	Smallest value
$Q_1$	25th percentile
Median ($Q_2$)	50th percentile
$Q_3$	75th percentile
Maximum	Largest value

Outlier Detection

Using IQR Method

Lower fence:

$$Q_1 - 1.5 \times \text{IQR}$$

Upper fence:

$$Q_3 + 1.5 \times \text{IQR}$$

• Values below lower fence or above upper fence are considered outliers
• For extreme outliers: use $3.0 \times \text{IQR}$

Boxplot Components

                    ┌───────────────┐
    ─────────────── │      │        │ ───────────────
         Min       Q₁     Q₂       Q₃            Max
                    └───────────────┘
                         IQR

Part	Represents
Box	Middle 50% (IQR)
Line in box	Median
Whiskers	Data within 1.5×IQR
Points	Outliers

Examples

Example 1: Calculating IQR

Data: 2, 3, 5, 6, 7, 8, 9, 11, 12

$$n = 9, \quad \text{Median} = 7$$

Lower half: 2, 3, 5, 6 → $Q_1 = \frac{3+5}{2} = 4$

Upper half: 8, 9, 11, 12 → $Q_3 = \frac{9+11}{2} = 10$

$$\text{IQR} = 10 - 4 = 6$$

Example 2: Outlier Detection

Using Example 1: $Q_1 = 4$, $Q_3 = 10$, $\text{IQR} = 6$

$$\text{Lower fence} = 4 - 1.5(6) = 4 - 9 = -5$$ $$\text{Upper fence} = 10 + 1.5(6) = 10 + 9 = 19$$

Any value below -5 or above 19 is an outlier.

Example 3: With Outliers

Data: 2, 3, 5, 6, 7, 8, 9, 11, 12, 50

• Range = 50 - 2 = 48 (heavily affected by outlier)
• $Q_1 = 4.5$, $Q_3 = 10.5$, IQR = 6 (minimally affected)
• Upper fence = 10.5 + 9 = 19.5
• 50 > 19.5, so 50 is an outlier

Comparison: Range vs IQR

Aspect	Range	IQR
Formula	Max - Min	$Q_3 - Q_1$
Uses	Extreme values	Middle 50%
Outlier resistance	None	High
Information used	2 points	All data
Best for	Quick overview	Robust analysis

Semi-Interquartile Range

Also called quartile deviation:

$$\text{SIQR} = \frac{Q_3 - Q_1}{2} = \frac{\text{IQR}}{2}$$