Range, Variance & IQR - Measuring Data Spread Guide

Two classes both average 75% on an exam. But in Class A, everyone scored 70-80. In Class B, scores ranged from 30 to 100. Same mean, completely different picture.

Measures of spread tell you how much variability exists in your data. Here's how to calculate and interpret each one.

The Four Measures of Spread

| Measure | What It Tells You | Pros | Cons | |---------|------------------|------|------| | Range | Max − Min | Simple | Affected by outliers | | Variance (σ²) | Average squared deviation | Mathematically useful | In squared units | | Std Dev (σ) | √Variance | Same units as data | Affected by outliers | | IQR | Q3 − Q1 | Resistant to outliers | Ignores 50% of data |

Range — The Simplest Measure

Range = Maximum − Minimum

Example: 12, 15, 18, 22, 25, 28, 30, 35

Range = 35 − 12 = 23

Problem: Add an outlier (100): 12, 15, 18, 22, 25, 28, 30, 35, 100

Range = 100 − 12 = 88 — tripled by one data point!

Range is useful for a quick check but unreliable when outliers exist.

Variance — The Foundation

Population Variance: σ² = Σ(x − μ)² / n

Sample Variance: s² = Σ(x − μ)² / (n − 1)

Step-by-Step Calculation

Data: 4, 8, 6, 5, 3, 2, 8, 9, 2, 5

1. Mean = 52/10 = 5.2
2. Deviations and squared deviations:

| x | x − μ | (x − μ)² | |---|-------|----------| | 4 | −1.2 | 1.44 | | 8 | 2.8 | 7.84 | | 6 | 0.8 | 0.64 | | 5 | −0.2 | 0.04 | | 3 | −2.2 | 4.84 | | 2 | −3.2 | 10.24 | | 8 | 2.8 | 7.84 | | 9 | 3.8 | 14.44 | | 2 | −3.2 | 10.24 | | 5 | −0.2 | 0.04 |

3. Sum of squared deviations = 57.6
4. Population variance = 57.6/10 = 5.76
5. Sample variance = 57.6/9 = 6.4

Standard Deviation — The Practical Measure

σ = √Variance

• Population: σ = √5.76 = 2.4
• Sample: s = √6.4 = 2.53

Standard deviation is in the same units as the original data, making it interpretable.

Interpretation: "The typical data point is about 2.4 units away from the mean."

IQR — The Outlier-Resistant Measure

IQR = Q3 − Q1

Step-by-Step

Data (sorted): 2, 2, 3, 4, 5, 5, 6, 8, 8, 9

1. Q1 (25th percentile) = 3
2. Q3 (75th percentile) = 8
3. IQR = 8 − 3 = 5

Outlier Detection with IQR

Fences:

• Lower fence = Q1 − 1.5 × IQR = 3 − 7.5 = −4.5
• Upper fence = Q3 + 1.5 × IQR = 8 + 7.5 = 15.5

Any value outside [−4.5, 15.5] is an outlier. None in this dataset.

Which Measure Should You Use?

| Situation | Best Measure | |-----------|-------------| | Quick overview, no outliers | Range | | Normally distributed data | Standard deviation | | Skewed data or outliers present | IQR | | Further statistical calculations | Variance | | Comparing different datasets | Coefficient of Variation (CV) |

The Trench Truth: The IQR is the most underused statistic in school. Students default to range because it's easy, but IQR is far more reliable. A single outlier can make the range meaningless, while IQR stays stable. For real data analysis, always report IQR alongside the median.

Calculate all spread measures with our statistics calculator.

Related: Standard Deviation Calculator · Derivative Calculator · Square Root Calculator