📊 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.

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🧠 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

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📦 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

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🔹 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

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🔹 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

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🔹 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.

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🔹 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

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📊 Visual Comparison

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

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🧩 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)

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💡 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.

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📚 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

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📝 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: www.mathematicselevateacademy.com
📩 Or DM me directly on LinkedIn: linkedin.com/in/rishabh-kumar-iitg-isi

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🪪 Math by Rishabh
🎓 Elite Private International Tutor
🌍 Founder, Mathematics Elevate Academy
📚 Specializing in IB | IGCSE | A Level | AP | SAT | JEE Advanced | Olympiads | University Entrance Exams