Interquartile Range Calculator
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The interquartile range (IQR) is the spread of the middle half of a data set: IQR = Q₃ − Q₁. For 1, 3, 5, 7, 9, 11, 13, 15, the lower half has median Q₁ = 4 and the upper half Q₃ = 12, so IQR = 8. Values beyond 1.5 × IQR from the quartiles are outliers.
Box plot: the box spans Q₁ to Q₃ (the IQR), the line is the median, whiskers reach the most extreme values inside the fences, and circles are outliers.
| Five-number summary | |
| Minimum | 1 |
| Q₁ (first quartile) | 4 |
| Median | 8 |
| Q₃ (third quartile) | 12 |
| Maximum | 15 |
| Spread | |
| Interquartile range (IQR) | 8 |
| Range | 14 |
| Semi-interquartile range | 4 |
| Lower fence | −8 |
| Upper fence | 24 |
| Outliers | none |
| Count n | 8 |
| Other quartile methods | |
| Inclusive (Excel QUARTILE.INC) | Q₁ 4.5, Q₃ 11.5, IQR 7 |
| Exclusive (Excel QUARTILE.EXC) | Q₁ 3.5, Q₃ 12.5, IQR 9 |
Show steps
- Sort the data: 1, 3, 5, 7, 9, 11, 13, 15
- The median is 8. n = 8 is even, so split the data into two halves of 4.
- Lower half: median Q₁ = 4. Upper half: median Q₃ = 12.
- IQR = Q₃ − Q₁ = 12 − 4 = 8
- Outlier fences: Q₁ − 1.5·IQR = −8 and Q₃ + 1.5·IQR = 24. No value is outside the fences, so there are no outliers.
How to use the IQR calculator
- Paste or type the data, separated by commas, spaces or new lines.
- Choose a quartile method; the default (median of each half) is the one most textbooks and TI calculators use.
- Press Calculate to get the IQR, the five-number summary, outlier fences and a box plot.
How to find the interquartile range
- Sort the data from smallest to largest.
- Find the median. It splits the data into a lower half and an upper half (leave the median out of both halves when n is odd).
- Q₁ is the median of the lower half; Q₃ is the median of the upper half.
- IQR = Q₃ − Q₁.
Example: 2, 4, 4, 5, 6, 7, 8, 9, 30. The median is 6. Lower half 2, 4, 4, 5 gives Q₁ = 4; upper half 7, 8, 9, 30 gives Q₃ = 8.5. IQR = 4.5.
Outliers and the 1.5 × IQR rule
Lower fence = Q₁ − 1.5 × IQR and upper fence = Q₃ + 1.5 × IQR. In the example, the fences are −2.75 and 15.25, so 30 is an outlier. Because the IQR ignores the most extreme values, it is not distorted by outliers the way the range and standard deviation are.
Why quartile methods differ
| Method | Used by | Q₁, Q₃ for 1, 3, …, 15 |
|---|---|---|
| Median of halves | TI-83/84, most textbooks | 4, 12 |
| Inclusive | Excel QUARTILE.INC, Google Sheets | 4.5, 11.5 |
| Exclusive | Excel QUARTILE.EXC, Minitab | 3.5, 12.5 |
All are correct definitions; use the one your class or software uses. The calculator shows all three.
Interquartile range practice quiz: 10 questions
Test yourself with 10 multiple-choice questions. Choose an answer and press Submit to see if you are right and why. Your score appears at the end.
Quiz complete
Frequently asked questions
How do you calculate the interquartile range?
Sort the data, find the first quartile Q₁ (median of the lower half) and the third quartile Q₃ (median of the upper half), then subtract: IQR = Q₃ − Q₁.
What does the IQR tell you?
How spread out the middle 50% of the data is. A small IQR means the central values are close together.
How do you find outliers with the IQR?
Values below Q₁ − 1.5 × IQR or above Q₃ + 1.5 × IQR are usually called outliers.
Why does Excel give a different IQR?
Excel’s QUARTILE.INC and QUARTILE.EXC interpolate between data values, while many textbooks use medians of halves. Choose the matching method in the calculator.
Is the IQR the same as the range?
No. The range is maximum minus minimum and uses the two most extreme values; the IQR uses only the middle half.
Sources
- NIST/SEMATECH e-Handbook: Percentiles
- NIST/SEMATECH e-Handbook: Box plot
- OpenStax Introductory Statistics 2e: Measures of the Location of the Data
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