Five-number summary and box plots
The five numbers: Minimum, Q1 (first quartile), Median (Q2), Q3 (third quartile), Maximum. These divide data into four equal parts.
| Number | Position | Meaning |
|---|---|---|
| Minimum | Lowest 25% | Smallest value |
| Q1 | 25th percentile | Quarter of data below |
| Median | 50th percentile | Middle value |
| Q3 | 75th percentile | Three-quarters of data below |
| Maximum | Highest 25% | Largest value |
Interquartile range (IQR): IQR = Q3 - Q1. Shows spread of middle 50% of data.
Drawing and reading box plots
Worked example: draw box plot
Data: 2, 5, 7, 8, 9, 12, 15. Find five-number summary and draw box plot.
Solution
- Sort: 2, 5, 7, 8, 9, 12, 15 (n=7)
- Min=2, Max=15
- Median (middle): position=(7+1)/2=4, so Median=8
- Q1 (lower half): 2,5,7 -> median=5
- Q3 (upper half): 9,12,15 -> median=12
Final answer
Five-number summary: 2, 5, 8, 12, 15. IQR=12-5=7.
Box plot shape: Symmetric box = symmetric data. Long whiskers = outliers possible.
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Comparing distributions
Comparing box plots: Place box plots side by side to compare median (center line), spread (box width=IQR), range (whisker length), and skewness (box position).
Worked example: interpret comparison
Dataset A: box plot with IQR=5, Dataset B: box plot with IQR=10. Which is more consistent?
Solution
- Smaller IQR = tighter middle 50% of data
- Dataset A: IQR=5 is smaller
- Dataset A shows more consistent values
Final answer
Dataset A is more consistent. Smaller IQR means less variation in middle data.
Skewness: Box left of median = right-skewed. Box right of median = left-skewed. Centered box = symmetric.
Outliers and modified box plots
Identifying outliers: Any data point outside the fences is considered an outlier.
Worked example
From earlier: Q1=5, Q3=12, IQR=7. Check if any data points 2, 5, 7, 8, 9, 12, 15, 30 are outliers.
Solution
- Lower fence = 5 - 1.5(7) = 5 - 10.5 = -5.5
- Upper fence = 12 + 1.5(7) = 12 + 10.5 = 22.5
- All points between -5.5 and 22.5 are normal
- The value 30 > 22.5, so 30 is an outlier
Final answer
30 is an outlier. Shown as dot beyond whisker in modified box plot.