How to find the five-number summary
Example: 6, 7, 15, 36, 39, 40, 41, 42, 43, 47, 49 (11 values, already sorted).
- Minimum and maximum: 6 and 49.
- Median: the middle (6th) value is 40.
- Q1: the lower half (without the median) is 6, 7, 15, 36, 39, so Q1 = 15.
- Q3: the upper half is 41, 42, 43, 47, 49, so Q3 = 43.
Five-number summary: 6, 15, 40, 43, 49. The IQR is 43 − 15 = 28. Load “Textbook example” above to see each step and the box plot.
Quartile methods compared
| Method | Used by | Q1 and Q3 for the example |
|---|---|---|
| Median of halves | TI-83/84, most textbooks | 15 and 43 |
| Inclusive | Excel QUARTILE.INC, Google Sheets, R default | 25.5 and 42.5 |
| Exclusive | Excel QUARTILE.EXC, Minitab, SPSS | 15 and 43 |
Same data, different answers, and all of them “correct”. So always check which method your course or software expects.
From summary to picture
The five-number summary is exactly what a box plot shows. Use the box plot maker to compare several groups, or the IQR and outlier calculator to focus on unusual values.
From five numbers to a box plot
Worked example with an even number of values
Data (sorted): 3, 5, 7, 8, 12, 13, 14, 18, 21, 25 (10 values).
- Minimum 3, maximum 25.
- Median: the average of the 5th and 6th values, (12 + 13) ÷ 2 = 12.5.
- Lower half: 3, 5, 7, 8, 12, so Q1 = 7. Upper half: 13, 14, 18, 21, 25, so Q3 = 18.
Five-number summary: 3, 7, 12.5, 18, 25. Excel’s QUARTILE.INC gives Q1 = 7.25 and Q3 = 17 for the same data, and QUARTILE.EXC gives 6.5 and 18.75: same data, three methods.
How to read the summary
- Spread of the middle half: the IQR, Q3 − Q1 = 11 above.
- Skew: compare median − Q1 (5.5) with Q3 − median (5.5). Equal gaps suggest symmetry; a bigger upper gap suggests right skew.
- Outliers: anything beyond Q1 − 1.5 × IQR or Q3 + 1.5 × IQR. Here the fences are −9.5 and 34.5, so there are none.
Common mistakes to avoid
- Including the median in both halves. For an odd number of values, the TI-84 and most textbooks leave the median out when finding Q1 and Q3.
- Not sorting first. Quartiles are positions in the sorted list.
- Mixing up the summary order. It is always minimum, Q1, median, Q3, maximum.
- Comparing quartiles from different methods. State the method, especially with small data sets.
Five-number summary in other software
| Software | How |
|---|---|
| Excel | =MIN(r), =QUARTILE.INC(r, 1), =MEDIAN(r), =QUARTILE.INC(r, 3), =MAX(r) (or QUARTILE.EXC). |
| Google Sheets | =QUARTILE(r, 0) to =QUARTILE(r, 4) give all five numbers (inclusive method). |
| TI-84 | STAT › CALC › 1-Var Stats, then scroll down to minX, Q1, Med, Q3 and maxX. |
| Python | numpy.percentile(data, [0, 25, 50, 75, 100]) (inclusive method by default). |
Related tools: Box Plot Maker, IQR & Outlier Calculator, Descriptive Statistics Calculator, Stem-and-Leaf Plot Maker and Standard Deviation Calculator.