What Is a 5-Number Summary?
The five-number summary is a set of descriptive statistics that provides a concise overview of a data set's distribution. It consists of five values: the minimum, first quartile (Q1), median (Q2), third quartile (Q3), and maximum. Together, these values divide the data into four equal parts, each containing 25% of the observations.
The five-number summary is the foundation for box plots (box-and-whisker plots) and provides a robust way to understand data spread, central tendency, and skewness without being overly influenced by outliers.
How to Calculate the 5-Number Summary
Worked Example
| Step | Calculation | Result |
|---|---|---|
| Data | 3, 7, 8, 5, 12, 14, 21, 15, 18 | -- |
| Sorted | 3, 5, 7, 8, 12, 14, 15, 18, 21 | -- |
| Minimum | First value | 3 |
| Q1 | Median of {3, 5, 7, 8} | 6 |
| Median | Middle value | 12 |
| Q3 | Median of {14, 15, 18, 21} | 16.5 |
| Maximum | Last value | 21 |
Interpreting Results
- IQR (Q3 - Q1): The interquartile range measures the spread of the middle 50% of data.
- Skewness: If the median is closer to Q1 than Q3, the data is right-skewed, and vice versa.
- Outliers: Values below Q1 - 1.5*IQR or above Q3 + 1.5*IQR are potential outliers.
- Range: Max - Min gives the total spread of the data.
Frequently Asked Questions
What is the difference between Q1 and Q3?
Q1 (first quartile) is the median of the lower half of the data, representing the 25th percentile. Q3 (third quartile) is the median of the upper half, representing the 75th percentile. The difference Q3 - Q1 is the interquartile range (IQR).
How do you find the median for an even number of data points?
When the data set has an even number of values, the median is the average of the two middle values. For example, in {2, 4, 6, 8}, the median is (4+6)/2 = 5.
Why is the 5-number summary useful?
It provides a quick, robust summary of a data set's distribution that is resistant to extreme values. It forms the basis for box plots and helps identify skewness, spread, and potential outliers without requiring complex calculations.