## What You Will Learn

By the end of this page, you will be able to find the five-number summary of a data set, draw a box-and-whisker plot, calculate the interquartile range (IQR), and interpret what a box plot reveals about the shape and spread of data.

## Theory

### The Five-Number Summary

A **box plot** (box-and-whisker plot) is built from five key values:

1. **Minimum** — smallest value in the data set
2. **Q1** (Lower Quartile) — median of the lower half of the data (25th percentile)
3. **Median (Q2)** — middle value of the whole data set (50th percentile)
4. **Q3** (Upper Quartile) — median of the upper half of the data (75th percentile)
5. **Maximum** — largest value in the data set

### How to Find Q1, Median, Q3

1. **Sort** the data from smallest to largest
2. **Find the median:** the middle value (or average of two middle values for even-sized sets)
3. **Find Q1:** the median of all values _below_ the overall median
4. **Find Q3:** the median of all values _above_ the overall median

### Structure of a Box Plot

- **Whiskers:** lines from the minimum to Q1, and from Q3 to the maximum
- **Box:** rectangle from Q1 to Q3
- **Line inside the box:** the median (Q2)

The box covers the **middle 50%** of the data.

### Interquartile Range (IQR)

IQR=Q3−Q1

IQR measures the spread of the middle half of the data. It is more reliable than range for describing spread because it ignores extreme outliers.

## Worked Examples

### Example 1: Finding the Five-Number Summary

**Problem:** Find the five-number summary for this data set:
4, 8, 3, 14, 7, 9, 11, 6, 12

**Step 1:** Sort the data.
3,4,6,7,8,9,11,12,14

**Step 2:** Find the median. There are 9 values — the middle (5th) value is **8**.

**Step 3:** Lower half (values below the median): 3, 4, 6, 7. Q1 = average of 4 and 6.
Q1=4+62=5

**Step 4:** Upper half (values above the median): 9, 11, 12, 14. Q3 = average of 11 and 12.
Q3=11+122=11.5

**Five-number summary:**

- Minimum = 3, Q1 = 5, Median = 8, Q3 = 11.5, Maximum = 14

**IQR** = 11.5−5=6.5

### Example 2: Drawing the Box Plot

**Problem:** Draw the box plot for the data from Example 1.

**Step 1:** Draw a number line covering the range (3 to 14).

**Step 2:** Mark the five values above the number line:
- Whisker start at 3 (minimum)
- Left edge of box at Q1 = 5
- Vertical line at median = 8
- Right edge of box at Q3 = 11.5
- Whisker end at 14 (maximum)

**Step 3:** Draw the box from Q1 to Q3, the median line, and the two whiskers.

### Example 3: Interpreting a Box Plot

**Problem:** A box plot for exam scores shows:

- Minimum = 42, Q1 = 58, Median = 70, Q3 = 80, Maximum = 95

Answer: (a) What is the IQR? (b) What fraction of students scored between 58 and 80? (c) Was the distribution skewed?

**(a)** IQR = 80−58=22 marks.

**(b)** The box spans Q1 to Q3, which contains **50%** of students — half the class scored between 58 and 80.

**(c)** The median (70) is closer to Q3 (80) than to Q1 (58), and the lower whisker is longer. The data is **left-skewed** (skewed towards lower scores).

### Example 4: Comparing Two Box Plots

**Problem:** Class A has median 65, IQR 10. Class B has median 72, IQR 25. What conclusions can you draw?

**Conclusion 1:** Class B has a higher median score (72 vs 65) — Class B performed better overall.

**Conclusion 2:** Class A's IQR is much smaller (10 vs 25) — Class A's results are more consistent; Class B has a wider spread of abilities.

## Common Mistakes

**Mistake 1: Including the Median in Both Halves When Finding Q1 and Q3**

❌ For odd-numbered data sets, including the median value itself in both the lower and upper halves.

✅ For an odd number of values, exclude the median when splitting into halves. Q1 and Q3 are medians of the lower and upper halves only.

**Mistake 2: Confusing IQR with Range**

❌ IQR = Maximum − Minimum.

✅ IQR = Q3 − Q1 (middle 50%). Range = Maximum − Minimum (total spread). They measure different things.

**Mistake 3: Drawing the Whiskers to the Wrong Values**

❌ Drawing whiskers to Q1 and Q3 (making the box disappear).

✅ The **whiskers** go from the minimum to Q1, and from Q3 to the maximum. The **box** spans from Q1 to Q3.

## Practice Problems

Try these on your own before checking the answers:

1. Find the five-number summary for: 12, 5, 20, 8, 15, 10, 18, 3, 7.
2. Using your answer from Problem 1, what is the IQR?
3. A box plot shows Q1 = 30 and Q3 = 54. What percentage of data lies between 30 and 54?
4. Two basketball players' point scores are compared. Player A: median = 18, IQR = 4. Player B: median = 15, IQR = 12. Which player is more consistent?
5. A data set has minimum 10, maximum 90, median 40. The lower whisker is much longer than the upper whisker. Is the data left-skewed or right-skewed?

## Summary

- The five-number summary: Minimum, Q1, Median (Q2), Q3, Maximum.
- Build the box from Q1 to Q3, with the median line inside; whiskers extend to min and max.
- IQR = Q3 − Q1 measures the spread of the middle 50% of data.
- A longer whisker or box section on one side indicates skew in that direction.
- Box plots are ideal for comparing two or more data sets side by side.