Interquartile Range (IQR) Calculator

How to Use This Calculator

1. Enter your data as a comma-separated list.
2. Read IQR, Q1, Median (Q2), and Q3 instantly.
3. Use Five-Number Summary tab for complete min/Q1/median/Q3/max with range and IQR.
4. Use Outlier Detection tab to identify mild (1.5×IQR) and extreme (3×IQR) outliers.
5. Professional tab adds expected IQR for normal distribution, skew direction indicator, and full summary.

Formula

Example

Frequently Asked Questions

• What is the interquartile range (IQR) and why is it useful? ▼
  The interquartile range (IQR) is the difference between the third quartile (Q3, the 75th percentile) and the first quartile (Q1, the 25th percentile): IQR = Q3 − Q1. It measures the spread of the middle 50% of a dataset, making it a robust measure of statistical dispersion. John Tukey popularized the IQR and the five-number summary (min, Q1, median, Q3, max) in his landmark 1977 book 'Exploratory Data Analysis.' The IQR is particularly useful for describing skewed distributions where the mean and standard deviation can be misleading. For example, household income data in any country is heavily right-skewed — a few billionaires inflate the mean dramatically. The median and IQR describe the typical income and spread experienced by most households far more accurately. The IQR is also the basis of box plots (box-and-whisker plots), one of the most information-dense statistical visualizations: the box spans Q1 to Q3, the horizontal line inside the box is the median, and whiskers extend to the most extreme non-outlier values.
• How is IQR more robust to outliers than standard deviation? ▼
  Robustness in statistics means resistance to the influence of extreme values. Standard deviation is not robust: it squares deviations from the mean, so extreme values — which are far from the mean — have their influence amplified quadratically. A single outlier 10 standard deviations from the mean inflates the standard deviation by a factor of √((n×100 + other terms)/n) ≈ 10/√n, which is substantial for small samples. IQR is immune to outliers in the outer 25% of the distribution on each side — you could set the top 24% of values to positive infinity and the IQR would not change at all. Consider [10, 12, 13, 14, 15, 1000]: SD ≈ 394, but IQR = Q3 − Q1 = 14 − 11 = 3. The IQR correctly shows that the middle data is tightly clustered, while SD is distorted by the outlier 1000. In formal statistics terms, the breakdown point of the IQR is 25% — up to 25% of data can be arbitrarily corrupted without affecting the IQR. The breakdown point of the standard deviation is 0% — even one unbounded outlier can make it infinite.
• How do I detect outliers using the 1.5×IQR rule? ▼
  Tukey's fence method defines mild and extreme outliers relative to the IQR. Compute the lower and upper fences: Lower mild fence = Q1 − 1.5 × IQR; Upper mild fence = Q3 + 1.5 × IQR. Any data point outside these fences is a mild (potential) outlier. For extreme outliers: Lower extreme fence = Q1 − 3 × IQR; Upper extreme fence = Q3 + 3 × IQR. Points beyond these are extreme outliers. The 1.5 multiplier was chosen by Tukey based on the normal distribution: for normally distributed data, only 0.7% of values fall outside Q1 − 1.5×IQR to Q3 + 1.5×IQR, making such points genuinely rare. Box plots display these fences as whisker endpoints — dots or asterisks beyond the whiskers represent outliers. Important caveats: the 1.5×IQR rule is a heuristic for identifying values that warrant investigation, not a rule for automatic deletion. An 'outlier' might be a data entry error (delete), a measurement anomaly (investigate), or a genuinely extreme but valid observation (keep and report). Always investigate the cause of outliers before deciding how to handle them.
• What's a five-number summary and how does it relate to IQR? ▼
  The five-number summary consists of: Minimum, Q1 (25th percentile), Median (Q2, 50th percentile), Q3 (75th percentile), and Maximum. Together, these five numbers provide a complete overview of a distribution's center, spread, and extremes without any assumed distributional shape. The IQR = Q3 − Q1 is directly derived from three of these five numbers. The five-number summary is the foundation of a box plot: the box spans Q1 to Q3 (showing IQR), the median line divides the box, and whiskers extend to the min and max (or to the Tukey fences if outliers are flagged). Comparing the five-number summaries across groups is often more informative than comparing means and standard deviations, especially for skewed data. In clinical research, regulatory submissions often require five-number summaries for continuous variables. The five-number summary also reveals distribution shape: if Q2 − Q1 ≈ Q3 − Q2, the data is roughly symmetric; if Q3 − Q2 > Q2 − Q1, the distribution is right-skewed; the opposite indicates left-skew. Tukey 1977 introduced this as a key tool of exploratory data analysis.
• When should I use IQR instead of standard deviation in reporting data? ▼
  Prefer IQR over standard deviation whenever your data is not normally distributed, particularly in these situations. First, skewed distributions: income, reaction times, hospital lengths of stay, and survival times are typically right-skewed. Report median and IQR rather than mean and SD. The mean is pulled toward the tail while the median stays near the bulk of the data. Second, small samples with potential outliers: with n < 30, one outlier can substantially inflate the SD while leaving the IQR unchanged. Third, ordinal data: strictly speaking, SD requires interval-scale data, but IQR can be meaningfully computed for ordinal measurements. Fourth, when data contains natural outliers that are real but extreme — reporting SD implies these are close to impossible, but they happened. Fifth, clinical and epidemiological reporting conventions: many clinical journals require continuous variables to be reported as 'median (IQR)' when the distribution is non-normal, following APA and ICMJE guidelines. The choice between SD and IQR is not arbitrary — it reflects your knowledge of the data's distributional properties and should be disclosed in the methods section.

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