Concept Architecture
Concept
Theoretically, Interquartile Range is a robust measure of statistical dispersion that quantifies the spread of the central 50% of an ordered distribution. It is founded on order statistics and quantile theory and is defined by the distance between the first and third quartiles. The concept exists to describe variability while limiting the influence of extreme observations and skewed tails.
Mathematically, the interquartile range is calculated as the difference between the 75th percentile and the 25th percentile. The quartiles are estimated from ranked observations using a recognised sample quantile algorithm, with interpolation applied when a required quartile position falls between observed values. The resulting measure is expressed in the same units as the original variable.
In practice, the interquartile range is calculated after ordering the data and estimating the first and third quartiles. It is commonly reported alongside the median for skewed distributions and used to identify potential outliers through lower and upper fences. In health economics, it is used to summarise healthcare costs, resource use, length of stay, utility values and other variables that may be asymmetrically distributed.
Purpose
Used to quantify the spread of the central half of a distribution, provide a robust summary of variability and identify potential outliers in skewed health economic data.
Mathematical Formulae
Primary Formula
IQR = Q? ? Q?
where:
- Q? = first quartile or 25th percentile
- Q? = third quartile or 75th percentile
Supporting Formulae
Lower outlier fence:
LF = Q? ? 1.5 ? IQR
Upper outlier fence:
UF = Q? + 1.5 ? IQR
Semi-interquartile range:
SIQR = IQR / 2
Quartile coefficient of dispersion:
CQD = (Q? ? Q?) / (Q? + Q?)
Related Mathematical Methods
- Quartiles
- Percentiles
- Quantiles
- Median
- Order Statistics
- Box Plot
- Tukey Outlier Fences
- Quantile Regression
Example
Annual healthcare costs for nine patients are ordered as follows:
�800, �1,000, �1,200, �1,500, �1,900, �2,400, �3,100, �4,500, �9,000
The median is �1,900. Excluding the median when dividing the data into halves:
Q? = (�1,000 + �1,200) / 2 = �1,100
Q? = (�3,100 + �4,500) / 2 = �3,800
Therefore:
IQR = �3,800 ? �1,100
IQR = �2,700
The central 50% of annual healthcare costs spans �2,700.
The upper outlier fence is:
UF = �3,800 + 1.5 ? �2,700
UF = �7,850
The observation of �9,000 exceeds the upper fence and is identified as a potential outlier.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| QUARTILE.INC | =QUARTILE.INC(A2:A101,3)-QUARTILE.INC(A2:A101,1) | Calculates the interquartile range of healthcare costs, utility values or resource use. |
| QUARTILE.EXC | =QUARTILE.EXC(A2:A101,3)-QUARTILE.EXC(A2:A101,1) | Calculates the interquartile range using the exclusive quartile definition. |
| PERCENTILE.INC | =PERCENTILE.INC(A2:A101,0.75)-PERCENTILE.INC(A2:A101,0.25) | Calculates the interquartile range directly from percentiles. |
| IF | =IF(OR(A2<$D$2-1.5*$D$4,A2>$D$3+1.5*$D$4),"Potential outlier","Within fences") | Identifies healthcare observations outside Tukey's outlier fences. |
| MEDIAN | =MEDIAN(A2:A101) | Reports the median alongside the interquartile range for skewed data. |
VBA (Optional)
Automate calculation of quartiles, interquartile ranges and Tukey outlier fences across multiple health economic variables.
Sources
- Tukey JW. Exploratory Data Analysis.
- Hyndman RJ, Fan Y. Sample Quantiles in Statistical Packages. The American Statistician. 1996.
- Conover WJ. Practical Nonparametric Statistics.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation.
- NICE. Health Technology Evaluation Manual.
Related Concepts (2)
Library
Publications
1
Statistical Analysis of Cost-Effectiveness Data — Willan & Briggs, 1st Edition ed., 2006 (John Wiley & Sons)
A synthesis of statistical methods for analysing cost-effectiveness data, including net-benefit regression, confidence intervals for the ICER, cost-effectiveness acceptability curves, and covariate adjustment. Part of the Wiley Statistics in Practice series.
BookView source →
Frequently Asked Questions (6)
What is the interquartile range?
A dispersion measure calculated as the difference between the seventy-fifth and twenty-fifth percentiles, the range containing the middle half of observations.
Source: Casella G, Berger RL. Statistical Inference. 2nd ed. Duxbury; 2002.
What span of the data does the interquartile range describe?
The interquartile range describes the spread of the middle half of a dataset, the gap between the twenty-fifth and seventy-fifth percentiles once the values are ordered. Because it ignores the lowest and highest quarters, it is barely affected by extreme values, which makes it a stable measure of dispersion for skewed data where the standard deviation would be distorted. It is naturally reported alongside the median, which marks the middle. Capturing the spread of the central half is its purpose. Kirkwood and Sterne (2003) describe this measure.
Source: Kirkwood & Sterne 2003
How is the interquartile range calculated?
The interquartile range is calculated by finding the first quartile, the value below which a quarter of the ordered data lie, and the third quartile, below which three quarters lie, and taking the difference between them. So the interquartile range is calculated as the third quartile minus the first quartile, giving the span of the middle half of the data, and because it depends only on these two quartiles and not on the extreme values, it is straightforward to compute from ordered data and provides a measure of spread that is not distorted by the smallest and largest observations.
Source: Casella & Berger 2002
Why is the interquartile range robust to outliers?
The interquartile range is robust to outliers because it is based only on the first and third quartiles, which mark the middle half of the data, and ignores the lowest and highest quarters where extreme values lie, so outliers do not affect it. This contrasts with the range or standard deviation, which extreme values can inflate. So the interquartile range is robust because it excludes the tails of the distribution, depending only on the central portion, which makes it a stable measure of spread for data containing outliers or with skewed distributions, where measures using all the values would be distorted by the extremes.
Source: Casella & Berger 2002
When is the interquartile range used?
The interquartile range is used to describe the spread of data when the distribution is skewed or contains outliers, situations where the standard deviation would be misleading, and it is commonly reported alongside the median as a robust summary. It also underlies box plots and the identification of outliers. So the interquartile range is used for skewed or outlier-prone data, providing a resistant measure of dispersion that pairs with the median, which is why it is preferred over the standard deviation for such data and is a standard component of box plots and of common rules for flagging potential outliers.
Source: Casella & Berger 2002
How does the interquartile range relate to the median and quartiles?
The interquartile range relates to the median and quartiles as part of a set of robust summaries: the median is the second quartile, the middle value, and the interquartile range is the distance between the first and third quartiles, which straddle the median. Together they describe the centre and spread of the data resistantly. So the interquartile range, the median, and the quartiles form a coherent set of order-based summaries, with the median giving the centre and the interquartile range the spread of the middle half, which is why they are reported together and displayed in box plots as a robust description of a distribution.
Source: Casella & Berger 2002
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 16 Dec 2025
Content version: 1.0.0
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- Persistent URI
- https://healtheconomics.wiki/concept/interquartile-range
- Term code
- HE-ES-SA-086
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