Statistics · BEGINNER LESSON
Interquartile range
The interquartile range, often shortened to IQR, is a measure of spread in numerical data. It is the difference between the third quartile (Q3) and the first quartile (Q1), so it describes the middle 50% of the observations.
What it is and what it measures
The interquartile range (IQR) is a way to measure variability, meaning how spread out numerical data are. It focuses on the middle half of a dataset rather than the smallest and largest values. Quartiles divide ordered data into four equal parts; the first quartile (Q1) marks the point below which 25% of the data fall, and the third quartile (Q3) marks the point below which 75% of the data fall. The IQR is Q3 minus Q1, so it measures the distance between those two points and shows where the central 50% lies. [0,1,2]
[1] Statistics review 1: Presenting and summarising data - PMC [2] Measures of dispersion - PMC - NIH [3] Biostatistics and Epidemiology for the Toxicologist - PMC - NIH
How it is used
Because it is based on the middle half of the data, the IQR is less affected by unusually high or low values, called outliers. That makes it useful when data are skewed, meaning the values are not evenly balanced around the center. In those cases, the median, which is the middle value, is often paired with the IQR because both describe the center and spread without being pulled strongly by extreme values. The IQR can also be helpful when extreme values are not recorded exactly. [0,1,2]
[1] Statistics review 1: Presenting and summarising data - PMC [2] Measures of dispersion - PMC - NIH [3] Biostatistics and Epidemiology for the Toxicologist - PMC - NIH
How to interpret it
A larger IQR means the middle 50% of values are spread farther apart. A smaller IQR means those middle values are clustered more tightly. For example, if Q1 is 8.7 and Q3 is 10.8, then the IQR is 2.1, and the central half of the observations lies between 8.7 and 10.8. This does not tell you the full range of the data, only the width of the middle portion. [0]
[1] Statistics review 1: Presenting and summarising data - PMC
Limitations
The IQR ignores the lower 25% and upper 25% of observations, so it does not show how far the most extreme values are from the center. It also is not convenient for some mathematical manipulations, so other measures may be chosen when exact algebraic handling is needed. Like other summary statistics, it should be used with the type of data it describes: it is meant for numerical data, not categories such as labels or groups. [1,2]
[2] Measures of dispersion - PMC - NIH [3] Biostatistics and Epidemiology for the Toxicologist - PMC - NIH
SEE THE IDEA
How the interquartile range is found
Arrange numerical values from smallest to largest.
Q1 is the 25th percentile and Q3 is the 75th percentile.
Subtract Q1 from Q3 to get the spread of the middle 50%.
A wider IQR means the central values are more spread out.
Each step builds on the one before it: ordering data allows quartiles, and quartiles give the IQR.
The IQR is the distance between the two quartiles that bound the middle half of the data. [0,1,2]
TRY IT YOURSELF
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Sources and further reading
This general lesson was generated from public source material. It explains a concept; it does not interpret a private paper or provide medical advice.