When a data set contains unusually high or low values, simple measures like the range can give a distorted picture. This is why the interquartile range (IQR) is such an important concept in statistics.
The interquartile range measures the spread of the middle 50% of data, focusing on the values between the lower quartile (Q1) and the upper quartile (Q3).
What Is The Interquartile Range
The interquartile range is the difference between the third quartile (Q3, the 75th percentile) and the first quartile (Q1, the 25th percentile). It measures the spread of the middle 50% of your data.
Formula: IQR = Q3 − Q1
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Because it ignores the top 25% and bottom 25% of values, the IQR is completely unaffected by outliers. This is its key advantage over the range and the standard deviation.
The IQR is the natural companion of the median, just as the standard deviation is the natural companion of the mean. When you report the median, you should almost always report the IQR alongside it.
Understanding Quartiles
To calculate the IQR, you first need to find the quartiles. Quartiles divide an ordered dataset into four equal parts:
| Quartile | Position | What It Represents |
|---|---|---|
| Q1 (Lower Quartile) | 25th percentile | 25% of values are below this point |
| Q2 (Median) | 50th percentile | The middle of the dataset; 50% below, 50% above |
| Q3 (Upper Quartile) | 75th percentile | 75% of values are below this point |
| IQR | Q3 − Q1 | The span of the middle 50% of data |
How To Calculate The IQR – Step By Step
- Step 1: Arrange all the data values in ascending order, from the smallest to the largest.
- Step 2: Find the median (Q2). This middle value divides the dataset into two equal parts: the lower half and the upper half.
- Step 3: Identify Q1 by finding the median of the lower half of the data (excluding Q2 if the number of values is odd).
- Step 4: Identify Q3 by finding the median of the upper half of the data (again excluding Q2 if the number of values is odd).
- Step 5: Calculate the interquartile range using the formula IQR = Q3 − Q1, which measures the spread of the middle 50% of the data.
Example: Dataset: waiting times (days) for a GP appointment across 10 practices: 3, 5, 6, 7, 9, 11, 12, 15, 18, 28. Sorted. Median (Q2) = (9+11)÷2 = 10. Lower half: 3, 5, 6, 7, 9 → Q1 = 6. Upper half: 11, 12, 15, 18, 28 → Q3 = 15. IQR = 15 − 6 = 9 days. Range = 28 − 3 = 25 days. The IQR of 9 days gives a much more representative picture of typical variation than the range of 25 days, which is heavily influenced by the outlier practice with a 28-day wait.
IQR & Outlier Detection
The IQR is also the basis for the most common method of formally identifying outliers, the 1.5 × IQR rule (also called the Tukey fence method):
- Lower fence: Q1 − (1.5 × IQR)
- Upper fence: Q3 + (1.5 × IQR)
- Any value below the lower fence or above the upper fence is flagged as a potential outlier.
Example: Using the GP waiting times data: Q1 = 6, Q3 = 15, IQR = 9. Lower fence = 6 − (1.5 × 9) = 6 − 13.5 = −7.5. Upper fence = 15 + (1.5 × 9) = 15 + 13.5 = 28.5. The value of 28 days is just within the upper fence, it is an extreme value but not technically an outlier by this rule. Any practice with a wait above 28.5 days would be flagged as an outlier.
IQR & The Box Plot
The IQR is the core of the box plot (box-and-whisker plot), one of the most useful charts in statistics. In a box plot:
- The box spans from Q1 to Q3, this is the IQR.
- The line inside the box is the median (Q2).
- The whiskers extend to the smallest and largest values within 1.5 × IQR of the box.
- Points beyond the whiskers are plotted individually as potential outliers.
Box plots are particularly useful for comparing distributions across groups, for example, comparing NHS waiting times across regions, or exam score distributions across different schools.
When To Use IQR Vs Standard Deviation
| Scenario | IQR or SD | Why? |
|---|---|---|
| Skewed data (income, house prices, wait times) | IQR | Resistant to the outliers and skew that distort SD |
| Normally distributed data | SD | More statistically efficient; basis for parametric tests |
| Ordinal data (Likert items, degree classifications) | IQR | SD assumes equal intervals, which ordinal data lacks |
| Data with clear outliers | IQR | SD is heavily inflated by outliers; IQR is not |
| Reporting alongside median | IQR | Natural pairing: median + IQR for non-normal data |
| Reporting alongside mean | SD | Natural pairing: mean + SD for normal data |
Student Tip: In SPSS, you can get the IQR via Analyze > Descriptive Statistics > Explore. In the output, look for the ‘Percentiles’ table which shows Q1 (25th percentile) and Q3 (75th percentile). Subtract Q1 from Q3 to get the IQR. The Explore output also produces a box plot automatically.
Frequently Asked Questions
Not necessarily. A large IQR means the middle 50% of your data is spread across a wide range of values, which reflects genuine diversity in your sample. Whether that is ‘bad’ depends on your research context. In quality control, high IQR might indicate inconsistency. In social research, it might simply reflect real variation in a population.
The semi-interquartile range (SIQR) is simply half the IQR: SIQR = IQR ÷ 2. It represents the average distance of Q1 and Q3 from the median. Some older textbooks use it, but IQR is more standard today.
With very small samples (n < 10), quartiles become unstable because there are too few data points to reliably define the 25th and 75th percentiles. In such cases, report the range and all individual values rather than the IQR.
