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Whichever method you pick, it only flags a value. It does not show the value is wrong. The NIST Engineering Statistics Handbook puts it this way: “If there is no reason to believe that the outlying point is in error, it should not be deleted without careful consideration.”
First decide what you are trying to do
NIST separates three tasks that people often blur together:
- Labeling: flagging candidates for further investigation.
- Accommodation: using robust methods that unusual values cannot distort much, without removing anything.
- Identification: formally testing whether specific observations are outliers.
The five approaches below fall under these tasks. Approaches 1 and 2 are mainly for labeling, and they also give you robust summaries. Approaches 3 and 4 are for identification. Approach 5 labels cases that univariate rules cannot see.
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Quick comparison
| Approach | Data | Main assumption | Best for |
|---|---|---|---|
| Modified z-score (median/MAD) | One numeric variable | Few distributional assumptions; needs some spread in the data | Screening that resists extreme points |
| IQR fences | One numeric variable | None formal; a convention | Transparent boxplot-style screening |
| Grubbs’ test | One numeric variable | Approximately normal | Testing one suspected outlier |
| Generalized ESD | One numeric variable | Approximately normal; you set an upper bound on the count | Testing when the number of outliers is uncertain |
| Multivariate / algorithmic (e.g., Isolation Forest, LOF) | Several features | Depends on the method and the feature representation | Unusual combinations or local-density anomalies |
1. Modified z-scores using median and MAD
An ordinary z-score uses the mean and standard deviation. Both are pulled by the very points you are hunting, so a large outlier can inflate the standard deviation and hide itself. The modified z-score uses the median for the center and the median absolute deviation (MAD) for the scale. NIST gives the formula as:
Mᵢ = 0.6745 × (xᵢ − x̃) / MAD
Here x̃ is the sample median and MAD is the median of |xᵢ − x̃|. The 0.6745 multiplier is the normalizing constant NIST uses in this formula.
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Worked example
Take the values 10, 12, 11, 13, 12, 11, 95. The median is 12. The absolute deviations from 12 are 2, 0, 1, 1, 0, 1, 83, so the MAD is 1. The score for 95 is 0.6745 × 83 ≈ 56. The score for 10 is about −1.35. The extreme point stands out sharply and did not distort its own yardstick.
Practical notes
- Pick and state a cutoff. A cutoff of 3.5 in absolute value is a widely used convention (it traces to Iglewicz and Hoaglin, whom NIST cites), but it is a convention, not a universal truth. Report the one you used.
- If more than half your values are identical, the MAD is zero and the score cannot be computed. This is common with discrete or heavily rounded data. Use another method.
- The score is symmetric. For strongly skewed data, a long tail can produce many flags that are simply the normal shape of the distribution.
2. IQR fences (boxplot screening)
Compute IQR = Q3 − Q1. NIST’s boxplot conventions are:
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- Inner fences: Q1 − 1.5 × IQR and Q3 + 1.5 × IQR. Points beyond them are conventionally called mild outliers.
- Outer fences: Q1 − 3 × IQR and Q3 + 3 × IQR. Points beyond them are called extreme outliers.
The scikit-learn documentation notes that the median and interquartile range are less affected by extreme values than the range, the mean and the standard deviation. That makes the fences hard to distort by the outliers themselves, and they are easy to explain to non-statisticians.
IQR or MAD?
Both are robust, and on well-behaved data they often agree. Choose IQR when you want a rule that matches the boxplot your audience already sees. Choose MAD when you want a continuous score, where you can rank points by how extreme they are and tune the cutoff. Quartile definitions differ slightly between software packages, so the fences can shift a little between tools. For skewed data, consider transforming first (for example, taking logs), because fences on the raw scale will flag the long tail.
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3. Grubbs’ test for one suspected outlier
Grubbs’ test answers a narrow question: is the single most extreme point an outlier in an approximately normal sample? The statistic is the largest absolute deviation from the sample mean, in units of the sample standard deviation.
- Normality matters. If the underlying distribution has a heavy or skewed tail, the test can call a legitimate tail value an outlier. Check a probability plot or histogram first.
- It is not for several outliers. Running it repeatedly invites masking, where multiple outliers make the test conclude that none is present. NIST recommends other procedures when more than one outlier may exist.
4. Generalized ESD when the count is unknown but bounded
The generalized extreme studentized deviate (ESD) test handles one or more outliers in approximately normal univariate data. You do not need to know the exact count. You only supply an upper bound on how many might be present. That makes it a better fit than Grubbs’ test when you suspect a few bad points but cannot say how many.
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- Set the upper bound from what is plausible for your data, not from how many points you would like to remove.
- It shares the normality assumption, so a skewed distribution can produce false detections.
- Pair it with a probability plot, boxplot, histogram, or run-sequence plot. NIST recommends these for viewing potential outliers and the shape of the distribution.
5. Multivariate and algorithmic detectors
A person who is 1.85 m tall and one who weighs 55 kg may each be ordinary. Both together may be rare. Univariate fences check each column separately, so they miss this kind of case. They also miss points that are unusual only relative to their local neighborhood.
Options
- Robust multivariate methods: estimate a center and covariance that extreme points cannot dominate, then measure each point’s distance from them.
- Isolation Forest: isolates points with random splits. Points that are separated quickly are treated as more anomalous.
- Local Outlier Factor (LOF): compares a point’s local density with that of its neighbors, which helps when clusters have different densities.
Cautions
- Scikit-learn’s estimator comparison indicates Isolation Forest tends to train faster than LOF on large datasets in the benchmark it shows. That is a computational observation in one context, not an accuracy ranking.
- The same page warns that feature scaling and the presence of outliers can affect the comparison. Scale features deliberately, and check that the result makes sense for your domain.
- These methods have tuning choices, such as the expected contamination rate or neighborhood size. Different choices flag different points, so treat the output as a ranked list of candidates.
How to choose
| Question | What it means for your choice |
|---|---|
| One feature or several? | One feature: approaches 1 to 4. Several, where relationships matter: approach 5. |
| Is approximate normality plausible? | If not, avoid Grubbs and generalized ESD. Use MAD or IQR, possibly after a transformation. |
| Is the number of suspected outliers known? | One: Grubbs. Unknown but bounded: generalized ESD. |
| Flag, test, or tolerate? | Flag with MAD or IQR, test with Grubbs or ESD, or tolerate by using medians and robust estimators for your analysis. |
| Can you check the source of the record? | If yes, investigate coding, units, and measurement before changing anything. |
What to do after a point is flagged
- Plot the data. Look at a boxplot, histogram, probability plot, and (for time-ordered data) a run-sequence plot.
- Check the record. Look for unit mix-ups, typos, sensor faults, or import errors.
- Consider the context. An outlier may be bad data, ordinary random variation, or the most scientifically interesting observation you have.
- If it is a confirmed error, correct it or remove it and document why.
- If it is valid, keep it. Use robust summaries or models, or run the analysis with and without it and report both results.
NIST points readers who want deeper treatment to the books by Iglewicz and Hoaglin and by Barnett and Lewis.
Quick Recap
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