Definition · Plain-language
Interquartile range
The interquartile range (IQR) measures the spread of the middle 50% of a dataset, calculated as the third quartile minus the first quartile: IQR = Q3 − Q1.
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The middle 50% of the data
The interquartile range captures the spread of the central half of a dataset. Quartiles divide ordered data into four equal parts: the first quartile (Q1) is the 25th percentile, with a quarter of the data below it, and the third quartile (Q3) is the 75th percentile, with three-quarters below it. The IQR is the distance between them, IQR = Q3 − Q1, and so spans the middle 50% of observations. To compute it, order the data, find Q1 and Q3, and subtract. A larger IQR indicates the central values are widely spread; a smaller one indicates they are tightly clustered.
A robust measure of spread
Unlike the range, which depends entirely on the single smallest and largest values, or the standard deviation, which uses every value and is pulled by extremes, the interquartile range deliberately discards the lowest and highest quarters of the data. This makes it a robust, or resistant, measure of spread: a few extreme values cannot inflate it. For that reason the IQR is the natural companion to the median, and the two together describe the centre and spread of skewed or outlier-prone data more faithfully than the mean and standard deviation. It is the spread statistic of choice when a distribution is not symmetric.
The 1.5 × IQR outlier rule and the boxplot
The interquartile range provides a widely used rule for flagging outliers. The rule defines fences at 1.5 × IQR beyond each quartile: a lower fence at Q1 − 1.5 × IQR and an upper fence at Q3 + 1.5 × IQR. Any value falling outside these fences is treated as a potential outlier worth examining. This same rule governs the standard boxplot: the box spans Q1 to Q3 (its length is the IQR), the whiskers extend to the furthest data points still inside the fences, and points beyond the fences are drawn individually. The IQR is therefore both the box’s width and the basis for the plot’s outlier detection.
Key facts
At a glance
- Definition: the spread of the middle 50% of a dataset
- Formula: IQR = Q3 − Q1
- Quartiles: Q1 is the 25th percentile, Q3 the 75th percentile
- Robustness: resistant to outliers, unlike the range or standard deviation
- Outlier rule: values beyond Q1 − 1.5 × IQR or Q3 + 1.5 × IQR are flagged
- Boxplot link: the IQR is the length of the box in a boxplot
Common misconceptions
What people often get wrong
Often heard: The interquartile range is the difference between the maximum and minimum values.
Actually: That is the range. The IQR is Q3 − Q1, the spread of the middle 50% of the data, deliberately excluding the lowest and highest quarters.
Often heard: Any value outside the interquartile range is an outlier.
Actually: Half the data lie outside the IQR by definition. The outlier rule uses fences at 1.5 × IQR beyond Q1 and Q3, not the edges of the IQR itself.
Often heard: The interquartile range is heavily affected by extreme values.
Actually: It is the opposite: by ignoring the top and bottom quarters, the IQR is robust to outliers, which is why it pairs with the median for skewed data.
