Outlier Detection Calculator

StatisticsLast updated: August 22, 2026

An outlier is an observation that clearly departs from the general pattern of a data set. A single extreme observation can pull the mean, the standard deviation, the correlation, and the regression coefficients in a direction they would not otherwise take; that is why every analysis should begin with a scan for outliers.

This calculator applies three criteria at once: the quartile-based Tukey rule (outside Q1 − 1.5·IQR to Q3 + 1.5·IQR), the z-score based on the mean and the standard deviation (|z| > 3), and the resistant modified z-score built on the median absolute deviation (0.6745·(x − median)/MAD > 3.5). The decision follows the method you select, but the table shows the verdict of all three criteria side by side.

You can separate the numbers with commas, spaces, or line breaks. At least 5 observations are required.
Tukey and MAD are resistant methods; the z-score is itself affected by extreme observations.
Leave it empty to use the standard threshold of the selected method. With Tukey, choosing 3 flags only extreme outliers.

Outlier Criteria

Interquartile range:     IQR = Q3 − Q1
Tukey bounds:            [Q1 − 1.5·IQR ,  Q3 + 1.5·IQR]
Extreme bounds:          [Q1 − 3·IQR ,  Q3 + 3·IQR]
z-score:                 z = (xᵢ − x̄) / s        outlier: |z| > 3
Median absolute dev.:    MAD = median(|xᵢ − median|)
Modified z:              Mᵢ = 0.6745·(xᵢ − median) / MAD     outlier: |Mᵢ| > 3.5
z limit (small n):       |z| ≤ (n − 1) / √n

The constant 0.6745 is the factor that scales MAD to the standard deviation under a normal distribution (Φ⁻¹(0.75) ≈ 0.6745). The modified z-score is the resistant criterion proposed by Iglewicz and Hoaglin (1993).

How to Calculate

  1. Paste your data values into the box; commas, spaces, and line breaks are all accepted.
  2. Choose a method: Tukey or MAD for a resistant result, or the z-score if you trust the normality of the data.
  3. Change the threshold coefficient if you wish; leave it empty and the standard threshold of the method (1.5 / 3 / 3.5) is applied.
  4. Check in the table which observations were flagged and whether the three criteria agree with one another.
  5. Look at the points beyond the whiskers in the box plot and at the extreme points in the order plot.
  6. See how much the mean changes once the outliers are removed to judge how sensitive your result is to these observations.

Worked Examples

Measurement series with a single extreme value (Tukey)

Across 19 measurements, Q1 = 16.50 and Q3 = 22.50, so IQR = 6 and the Tukey upper bound is 22.50 + 1.5·6 = 31.50. The value 48 lies above that bound and is flagged as the only outlier; its z-score is 3.57 and its modified z-score is 6.52, so all three methods reach the same verdict. Removing this observation drops the mean from 20.58 to 19.06 and the standard deviation from 7.68 to 3.98.

Number of outliers: 1 · Flagged values: 48.00 · Outlier rate: 5.3%

A mild extreme value the z-score misses

In this series the extreme observation is milder (the largest value is 34) and the z-score method flags nothing: for 34, z = 2.74, below the threshold of 3. The Tukey rule, however, treats 34 as an outlier because it exceeds the upper bound of 31.50, while the modified z of 3.37 stays just below the 3.5 threshold. This is a classic illustration of how the z-score can mask an extreme observation in small samples.

Number of outliers: 0 · Flagged values: None · Outlier rate: 0.0%

Extreme values at both ends (modified z)

In these 18 observations, with extreme values at both ends, the median is 55 and MAD is 3.50. The modified z-score is -10.21 for the value 2 and 8.29 for the value 98, and both exceed the 3.5 threshold, so both are flagged. The z-score method catches only the value 2 (z = -3.10), because the z-score of 98 stays at 2.55: the two extreme observations inflate the standard deviation together and mask each other.

Number of outliers: 2 · Flagged values: 2.00, 98.00 · Outlier rate: 11.1%

Frequently Asked Questions

Which outlier method should I choose?
If you do not want to assume anything about the distribution, the Tukey rule is a good default and lines up directly with the box plot. If the data is approximately normal and the sample is large, the z-score can be used. When there are several extreme observations, or the sample is small, the mean and the standard deviation break down, so the MAD-based modified z-score is the most reliable choice.
Should I delete an outlier from my data?
Not automatically. Investigate the cause first: a data entry error, a mix-up of units, or a measurement fault should be corrected or removed. If it is a genuine extreme observation (a very high income, an unusual patient) it should stay in the data; in that case the right approach is to use resistant measures such as the median and the IQR, or to report the result both with and without the extreme observation.
Why do some observations look like outliers in the box plot without their z-score passing 3?
The two criteria use different references. The Tukey rule rests on the quartiles and is unaffected by extreme observations; the z-score rests on the mean and the standard deviation, and an extreme observation inflates both of them itself. Because of this "masking" effect, the z-score can miss genuine outliers, especially in small samples.
Where does the constant 0.6745 in the modified z-score come from?
Under a normal distribution the median absolute deviation is about 0.6745 times the standard deviation (the value of Φ⁻¹(0.75)). Scaling MAD by that constant gives you a resistant measure of spread that corresponds to the standard deviation. Iglewicz and Hoaglin proposed the threshold of 3.5 for this criterion.
What happens if MAD comes out as zero?
If more than half of the observations are exactly equal to the median, MAD becomes zero and the formula is undefined. In that situation this calculator derives the scale from the mean absolute deviation instead (MeanAD × 1.2533), so no infinite or undefined score is produced. This usually occurs in discrete data containing many repeated values.
Will I get the same result in SPSS and Excel?
The box plot in the SPSS Explore output uses the same Tukey rule and marks points beyond 1.5·IQR with a circle and those beyond 3·IQR with a star. In Excel you can compute Q1 and Q3 with QUARTILE.INC and build the bounds by hand. For the quartiles this calculator uses linear interpolation (Excel's PERCENTILE.INC); small differences can arise against some of the quartile definitions used by SPSS.
How many outliers are normal?
In normally distributed data the 1.5·IQR rule flags about 0.7% of the observations, so one in a hundred is to be expected. If the rate exceeds 5%, that is usually not an outlier problem but a sign that the distribution is skewed or heavy-tailed. In such cases a transformation (logarithm, square root) or nonparametric methods should be considered.