Understanding Range, IQR, Variance & Standard Deviation for IB, A Level & AP Math

🎯 Why Study Measures of Dispersion?

Averages tell only half the story.

Two data sets may have the same mean but wildly different spread. That’s why understanding dispersion — how data varies — is essential for interpreting data accurately.

Whether you’re taking IB Math, A Level Statistics, or AP Stats, these tools will help you analyze data more meaningfully.

🧠 Learning Objectives

By the end of this post, you’ll:

Understand and calculate the range, interquartile range (IQR), variance, and standard deviation
Know which measure is most appropriate in different contexts
Learn how examiners expect you to apply these concepts
See how dispersion links to boxplots, histograms, and normal distributions

📦 1. What Are Measures of Dispersion?

These are statistical tools used to describe the spread or variability in a dataset.

Measure	Description
Range	Difference between highest and lowest values
IQR	Range of the middle 50% of the data
Variance	Average of squared differences from the mean
Standard Deviation	Square root of the variance

🔹 2. Range — The Simplest Measure

Formula: Range=Maximum−Minimum\text{Range} = \text{Maximum} – \text{Minimum}Range=Maximum−Minimum

📌 Example:

Data: 12, 15, 18, 22, 25
Range = 25 − 12 = 13

✅ Quick and easy
❌ Very sensitive to outliers

🔹 3. Interquartile Range (IQR)

The IQR shows the range of the middle 50% of values — it’s not affected by extreme values, which makes it more robust than the range.

Formula: IQR=Q3−Q1\text{IQR} = Q_3 – Q_1IQR=Q3−Q1

Where:

Q1Q_1Q1 = 25th percentile
Q3Q_3Q3 = 75th percentile

📌 Example:

Data: 10, 12, 15, 18, 20, 25, 30
Q1 = 12, Q3 = 25 → IQR = 25 − 12 = 13

✅ Best for skewed data or box plots
✅ Used heavily in IB, AP, and A Level visual analysis

🔹 4. Variance — Squared Spread

Variance is the mean of the squared deviations from the mean.

Formula (for sample): s2=1n−1∑(xi−xˉ)2s^2 = \frac{1}{n-1} \sum (x_i – \bar{x})^2s2=n−11∑(xi−xˉ)2

Where:

xix_ixi = each data point
xˉ\bar{x}xˉ = mean
nnn = number of data points

This gives a sense of how “spread out” the values are from the mean — in squared units.

🔹 5. Standard Deviation — The Star Player

Standard deviation (SD or sss) is the square root of the variance, and the most commonly used measure of spread.

Formula: s=1n−1∑(xi−xˉ)2s = \sqrt{ \frac{1}{n – 1} \sum (x_i – \bar{x})^2 }s=n−11∑(xi−xˉ)2

📌 Example:

Data: 2, 4, 6, 8
Mean = 5
Squared deviations: (9, 1, 1, 9) → variance = 20/3
SD = √(20/3) ≈ 2.58

✅ Widely used in statistics, probability, and data science
✅ Interpreted in real units, unlike variance

📊 Visual Comparison

(Insert a visual showing two data sets with same mean, different spreads — normal curves or dot plots)

🧩 6. Which Measure Should You Use?

Scenario	Best Measure
Rough estimate	Range
Boxplots or skewed data	IQR
In-depth analysis	SD or variance
Normal distribution	SD (essential)

💡 Real-Life Example

Two classes both scored a mean of 70 on a test.

Class A: All students scored between 68–72 → low SD
Class B: Scores ranged from 40 to 100 → high SD

Even with the same mean, Class B’s performance is more inconsistent — that’s what SD tells us.

📚 Exam Relevance

IB Math AA/AI HL: Box plots, comparing dispersion, calculating SD with GDC, interpreting variability
A Level S1: Full manual calculation of variance & SD from raw/grouped data, comparison of spread
AP Stats: Using technology to compute SD, comparing distributions, discussing variability

📝 Practice Problem

Question:
Data: 10, 15, 15, 20, 25, 30
Find:

Range
IQR
Mean
Standard Deviation (using calculator)

✅ Try this yourself or book a session for detailed walkthroughs.

🔔 Ready to Master Statistics with Confidence?

💡 Struggling with topics like standard deviation, boxplots, or deeper statistical reasoning?

🎓 Book a personalized 1-on-1 session with me — Rishabh Kumar,
Elite Private International Tutor and Founder of Mathematics Elevate Academy.

I specialize in helping IB, A Level, IGCSE, AP, and global students build deep conceptual clarity, boost exam scores, and approach mathematics with confidence — one topic at a time.

📍 Visit: clientwork4.shivamdev.in/
📩 Or DM me directly on LinkedIn: linkedin.com/in/rishabh-kumar-iitg-isi

🎯 Aiming for Ivy League or Oxbridge?
Preparing for MAT, STEP, TMUA, AEA, or any advanced university entrance exam?

Join Math by Rishabh for personalized, high-quality math lessons with:

Structured study plans
Rigorous problem-solving
Deep understanding of mathematical concepts
Exam-focused practice and study materials