📈 Intro to Statistics · Lesson 3 of 7
Spread: range, IQR and standard deviation
How spread out the data are, and how to read a box plot.
Why spread matters
Two classes can both average 75 on an exam: one where everyone scored 70–80, and one where scores ran from 40 to 100. The center is the same; the spread tells the real story.
Quartiles and IQR
| Q1 | 25% of the data are below it |
|---|---|
| Median (Q2) | 50% |
| Q3 | 75% |
| IQR = Q3 − Q1 | The range of the middle half |
| Outlier rule | Below Q1 − 1.5·IQR or above Q3 + 1.5·IQR |
Standard deviation, step by step
- Data: 4, 6, 8, 10, 12. Mean = 8
- Deviations from the mean: −4, −2, 0, 2, 4
- Square them: 16, 4, 0, 4, 16 → sum = 40
- Divide by n − 1 = 4 → variance = 10
- Square root → s = √10 ≈ 3.16
Reading it
The standard deviation is roughly the typical distance of values from the mean. Bigger s = more spread. Use mean with standard deviation for symmetric data, and median with IQR for skewed data. In a spreadsheet: =STDEV.S(range) for a sample.
Tip
💡 A box plot shows five numbers: min, Q1, median, Q3, max (with outliers as dots). It’s the fastest way to compare spread between groups.
✏️ Practice: your turn
- Compute the standard deviation of 3, 5, 7, 9, 11 by hand
- Check it with =STDEV.S in the online spreadsheet
- Find Q1, Q3 and the IQR of 15 exam scores
- Use the 1.5·IQR rule to test for outliers
✏️ Scientific calculator with a statistics tab (mean, median, standard deviation)
🧠 Quick quiz
Answer all 4 questions correctly to mark this lesson as done.
0 of 4 correct
All lessons in Intro to Statistics
- 🎲 1. Data, variables and samplingCategorical vs. quantitative, populations, samples and bias
- 🎯 2. Mean, median and modeMeasures of center, and which one to use when data are skewed
- ↔️ 3. Spread: range, IQR and standard deviationHow spread out the data are, and how to read a box plot
- 🔔 4. The normal curve and z-scoresThe 68–95–99.7 rule and comparing values on different scales
- 📉 5. Correlation and regressionScatterplots, r, the line of best fit — and causation
- 🎯 6. Confidence intervalsEstimate ± margin of error, and what “95% confident” means
- ⚗️ 7. Hypothesis tests and p-valuesNull and alternative, the p-value, significance and errors