Outlier Calculator — IQR, Z-Score & Modified Z-Score Methods
Dataset and thresholds
Outlier comparison
Your comparison will appear here.
Enter a dataset and calculate.
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Outlier formulas used
| Method | Formula | Default rule and assumptions |
|---|---|---|
| IQR fences | IQR = Q3 − Q1; lower = Q1 − 1.5 × IQR; upper = Q3 + 1.5 × IQR | Flag values strictly outside the fences. Q1 and Q3 use linear inclusive percentiles: position = (n − 1)p, with interpolation. Robust to individual extremes, but sensitive to quartile convention in small samples. |
| Z-score | zᵢ = (xᵢ − x̄) / s | Flag |z| > 3 by default. The selector controls sample SD (divisor n − 1) or population SD (divisor n). Mean and SD can be pulled toward extreme values; interpretation is strongest for roughly symmetric data. |
| Modified z-score | Mᵢ = 0.67448975(xᵢ − median) / MAD | Flag |M| > 3.5 by default. MAD is the median of |xᵢ − median|. Median and MAD resist extreme values, but M is undefined when MAD = 0. |
Boundary rule: equality to a fence or cutoff is not flagged; a value must be strictly beyond it. “Potential outlier” is a screening label, not proof that a value is erroneous or generated by a different process.
How to choose a method
- IQR: useful for quick, distribution-free box-plot screening. It considers rank and spread of the middle half rather than distance from a modeled mean.
- Z-score: useful when the mean and standard deviation appropriately describe the distribution. A single extreme can inflate both, and small samples may be unable to reach a cutoff of 3.
- Modified z-score: useful when you want a distance score based on robust center and scale. Heavy ties can make MAD zero, in which case the formula cannot rank extremeness.
- Agreement: values flagged by several methods deserve attention, but disagreement is information about distribution shape—not a vote that automatically decides removal.
How to use the results
- Confirm the parser found the intended number of observations.
- Review each method's center, scale, cutoffs, and availability.
- Use the row table to locate flagged values by their original input position.
- Check for transcription, unit, sensor, sampling, or process changes.
- Document any correction or exclusion rule before downstream analysis; keep the original data when possible.
Assumptions and limits
- These rules label observations; they do not calculate a formal hypothesis-test p-value or establish that a point is invalid.
- A skewed, multimodal, bounded, seasonal, or heavy-tailed distribution can naturally produce values that a generic cutoff flags. Analyze meaningful groups separately when appropriate.
- Repeated observations are retained as separate rows. Their original positions are included in the CSV.
- Linear inclusive quartiles match the page's stated R-7 convention, but other software may use hinges, nearest ranks, or other percentile estimators and return different IQR fences.
- Values are limited to absolute magnitude 1e150 to keep arithmetic inside the browser's reliable finite range. Display rounding does not change comparisons.
Do not delete by rule alone. Investigate why a point is unusual. It may be a correct, scientifically important observation; changing or excluding it can materially change conclusions.
Frequently asked questions
Which outlier method should I use?
IQR and modified z-scores are robust starting points for skewed or contaminated data. Conventional z-scores are most interpretable when mean and standard deviation fit the data's shape and context. There is no universally best rule.
Why do IQR, z-score, and modified z-score disagree?
They use different centers and scales. Z-scores use the mean and SD, IQR uses quartiles, and modified z-scores use the median and MAD. Disagreement often reflects skew, ties, small sample size, or influence from extreme observations.
What default cutoffs are used?
The defaults are outside Q1 − 1.5 × IQR or Q3 + 1.5 × IQR, |z| > 3, and |modified z| > 3.5. You can change all three thresholds.
Why might z-score flag nothing in a small dataset?
Each observation contributes to the sample mean and SD used in its own score. This limits the maximum possible internally studentized z-score; a cutoff of 3 can be unreachable in small samples.
What happens when SD or MAD is zero?
Division by zero makes that score unavailable, so the calculator labels the method unavailable instead of displaying infinity. IQR screening can still work, even when IQR equals zero.
Should I delete every flagged observation?
No. First check data entry, units, measurement, sampling, grouping, and subject-matter meaning. If you exclude or correct a value, preserve the raw data and document the reason.
Why do quartiles differ from another calculator?
Sample quartiles have multiple accepted definitions. This page uses linear inclusive interpolation at position (n − 1)p, also known as R-7. Other hinges or rank conventions can produce different small-sample fences.
Are duplicate values preserved?
Yes. Every observation remains a separate row with its original one-based input position, including duplicates.
Are my values uploaded or tracked?
No. The calculation stays in your browser, and the values are not sent, stored, or attached to analytics events.
Methodology and references
Last reviewed: August 4, 2026. Fence definitions and interpretation cautions were checked against the NIST/SEMATECH Engineering Statistics Handbook. Its outlier-detection guidance defines conventional and modified z-scores and cites the 3.5 modified-z labeling cutoff. The linear percentile calculation was checked against NumPy's documented default method.