Formula & Calculator
Interquartile Range (IQR)
Measures the spread of the middle 50% of a dataset, a robust measure of variability not affected by extreme outliers.
Interpretation
IQR = Q3 − Q1. The range of the middle 50% of data. Robust measure of spread, resistant to outliers. Used in boxplots and outlier detection.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| IQR | Interquartile range | |
| Q3 | Third quartile (75th percentile) | |
| Q1 | First quartile (25th percentile) |
What it means
The interquartile range (IQR) is the difference between the third quartile (Q3, 75th percentile) and the first quartile (Q1, 25th percentile). It represents the spread of the central 50% of the data and is a robust measure of dispersion that is not influenced by outliers. The IQR is used to construct boxplots, to identify outliers (data points beyond 1.5×IQR from the quartiles), and to compare variability across groups. It is also used in non‑parametric statistics. Understanding the IQR is important for exploring data and for selecting appropriate measures of spread when the data are skewed or contain outliers.
Worked example
Interquartile Range – Two Examples
Real‑World| Parameter | Value |
|---|---|
| Q1 | 25 |
| Q3 | 75 |
| Parameter | Value |
|---|---|
| Q1 | 15000 |
| Q3 | 35000 |
Common mistakes
- Quartiles: Q1 (25th percentile) and Q3 (75th percentile) – different methods may give slightly different values.
- Ordering: Data must be sorted before calculating quartiles.
- IQR: Is a measure of spread – resistant to outliers.
- Outlier detection: Values below Q1 − 1.5·IQR or above Q3 + 1.5·IQR are often considered outliers.
- Units: IQR has the same units as the data.
Applications
The interquartile range (IQR) is the difference between the third and first quartiles, covering the middle 50% of the data. It is a robust measure of spread, unaffected by outliers. Engineers use IQR to identify outliers, to describe skewed distributions, and to construct box plots. In quality control, it helps assess process dispersion. In healthcare, it summarises patient data with extreme values. By using IQR, professionals can focus on the central bulk of the data, reducing the influence of extreme observations. It is often combined with the median to provide a robust summary. Understanding IQR is crucial for exploratory data analysis and for communicating data variability in a clear, interpretable manner.
- Identification of outliers in datasets
- Exploratory data analysis and box plot construction
- Healthcare and clinical data summarisation
- Quality control – robust process dispersion measure
- Educational assessment – distribution of scores
Frequently Asked Questions
IQR = Q3 – Q1, where Q1 is the 25th percentile (first quartile) and Q3 is the 75th percentile (third quartile). It measures the spread of the middle 50% of the data.
Because it is not affected by outliers or extreme values – it only considers the central half of the data. This makes it ideal for skewed distributions.
Outliers are often defined as values below Q1 – 1.5×IQR or above Q3 + 1.5×IQR. This is the standard rule in boxplot construction.
Both measure spread. For a normal distribution, IQR ≈ 1.35 × σ. The IQR is more robust, while σ is more efficient for symmetric data.
There are different methods. One common approach: after sorting, the median splits the data; Q1 is the median of the lower half, Q3 is the median of the upper half. Interpolation may be used for even splits.
It is often reported alongside the median in the five‑number summary (min, Q1, median, Q3, max). It gives a quick picture of the spread without being swayed by outliers.
If you add a constant b, the IQR is unchanged. If you multiply by a, the IQR is multiplied by |a|.
Yes, if Q1 = Q3, meaning all values in the middle 50% are identical (or the dataset has little variability).
It is useful for comparing the variability of different groups, especially when distributions are skewed. Boxplots visually display IQR for each group.
The range covers all data (from min to max) and is sensitive to outliers. The IQR covers only the middle 50% and is robust. IQR is often preferred for skewed data.