The interquartile range measures how spread out the central half of a data set is. It is the distance between the first and third quartiles, so it ignores the lowest and highest quarters of the data entirely. That makes it a robust measure of spread: one extreme salary or a mistyped reading changes the range and the standard deviation a lot, but barely moves the IQR.
The IQR is used to compare the variability of groups, to draw box plots, to scale data robustly and – through the 1.5 × IQR rule – to define outliers. For roughly normal data the IQR is about 1.35 standard deviations, so IQR ÷ 1.35 is a quick robust estimate of σ.
This calculator defaults to the (n + 1) position method, which interpolates Q1 at rank (n + 1)/4 and Q3 at rank 3(n + 1)/4. If your class finds quartiles as medians of halves, or uses Excel’s QUARTILE.INC, switch the method and the IQR updates.
How to use the Interquartile Range Calculator
- Data set
- Quartile method
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How it works
Sort the data, locate Q1 and Q3 with the chosen method (interpolating between neighbouring values when the position is fractional) and subtract: IQR = Q3 − Q1.
Worked example
Data: 3, 7, 8, 12, 13, 14, 18, 21, 25, 30 (n = 10). With the (n + 1) method Q1 sits at position 11/4 = 2.75: 7 + 0.75 × (8 − 7) = 7.75. Q3 sits at position 33/4 = 8.25: 21 + 0.25 × (25 − 21) = 22. IQR = 22 − 7.75 = 14.25. The textbook halves method gives Q1 = 8, Q3 = 21 and IQR = 13; Excel’s QUARTILE.INC gives 9 and 20.25, so IQR = 11.25.
IQR versus other measures of spread
| Measure | Uses | Sensitive to outliers? |
|---|---|---|
| Range | min and max | Very |
| Standard deviation | every value, squared | Yes |
| Mean absolute deviation | every value | Somewhat |
| IQR | Q1 and Q3 | No |
| Median absolute deviation | the median of deviations | No |
Quartile methods compared
Statistics packages do not agree on one definition of a quartile. For the 10 values 68, 72, 75, 79, 81, 84, 86, 90, 93, 97 the four methods offered here give:
| Method | Used by | Q1 | Q3 | IQR |
|---|---|---|---|---|
| Median of halves (median excluded) | TI-84, most school textbooks | 75 | 90 | 15 |
| Linear interpolation, position 1 + (n − 1)p | Excel QUARTILE.INC, R type 7, Google Sheets | 76 | 89 | 13 |
| (n + 1)p positions | Excel QUARTILE.EXC, Minitab | 74.25 | 90.75 | 16.5 |
| Median of halves (median included) | Tukey’s hinges | 75 | 90 | 15 |
With an even number of values the two “halves” methods coincide; they differ only when n is odd. The differences shrink as the data set grows, so with a few hundred values the choice rarely matters.
Formula
IQR = Q3 − Q1 · Q1 at position (n + 1)/4, Q3 at 3(n + 1)/4 (interpolated) · quartile deviation = IQR ÷ 2
Worked example
Inputs — Data set: 3, 7, 8, 12, 13, 14, 18, 21, 25, 30; Quartile method: (n + 1) positions (Excel QUARTILE.EXC, Minitab).
Results — Interquartile range (IQR = Q3 − Q1) = 14.25; First quartile (Q1) = 7.75; Third quartile (Q3) = 22; Median (Q2) = 13.5; Quartile deviation (IQR ÷ 2) = 7.125; Lower fence (Q1 − k × IQR) = -13.625.
Key terms
- Interquartile range (IQR)
- Q3 − Q1.
- Quartile deviation
- Half the IQR.
- Robust statistic
- A measure that changes little when a few values are extreme.
- 1.5 × IQR rule
- Values more than 1.5 IQR beyond the quartiles are flagged as outliers.
Frequently asked questions
What is the interquartile range?
Q3 − Q1, the width of the range that holds the middle half of the data.
How do I calculate the IQR?
Sort the data, find Q1 and Q3 and subtract. For 3, 7, 8, 12, 13, 14, 18, 21, 25, 30 with the (n + 1) method, Q1 = 7.75 and Q3 = 22, so IQR = 14.25.
Why is the IQR better than the range?
The range depends only on the two most extreme values; the IQR ignores the outer quarters and is not distorted by outliers.
What does a large IQR mean?
The middle half of the data are widely spread. Compare it with the median: an IQR of 14.25 around a median of 13.5 indicates a lot of variability.
How is the IQR related to the standard deviation?
For normal data IQR ≈ 1.349σ, so σ ≈ IQR ÷ 1.349.
How do I find the IQR in Excel?
=QUARTILE.EXC(range,3) − QUARTILE.EXC(range,1) matches the default method here; QUARTILE.INC gives 11.25 for the example.
Can the IQR be zero?
Yes, if at least half of the values are identical in the middle of the data.