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Box Plots — How to Make and Read a Box-and-Whisker Plot

Grade: 6–7 | Topic: Statistics

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\text{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. 3,4,6,7,8,9,11,12,143, 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=5Q1 = \frac{4 + 6}{2} = 5

Step 4: Upper half (values above the median): 9, 11, 12, 14. Q3 = average of 11 and 12. Q3=11+122=11.5Q3 = \frac{11 + 12}{2} = 11.5

Five-number summary:

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

IQR =11.5−5=6.5= 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=2280 - 58 = \mathbf{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?
Click to see answers
  1. Sorted: 3, 5, 7, 8, 10, 12, 15, 18, 20. Median = 10. Lower half: 3, 5, 7, 8 → Q1 = (5+7)/2 = 6. Upper half: 12, 15, 18, 20 → Q3 = (15+18)/2 = 16.5. Five-number summary: 3, 6, 10, 16.5, 20
  2. IQR = 16.5 − 6 = 10.5
  3. The box (Q1 to Q3) always contains 50% of the data.
  4. Player A is more consistent (smaller IQR = 4 vs 12), meaning their scores vary less from game to game.
  5. A longer lower whisker means values are spread further on the left side — the data is left-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.

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