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To find an outlier in statistics, sort the data, calculate the first and third quartiles (Q1 and Q3), compute the interquartile range (IQR = Q3 − Q1), and flag any value below Q1 − 1.5 × IQR or above Q3 + 1.5 × IQR. That is the fence rule used for box plots in the NIST/SEMATECH e-Handbook of Statistical Methods, and it works on any data set without assuming a distribution. The rest of this page shows the arithmetic on a real data set, the one trap that changes the answer (which quartile definition you use), and what to do once a value is flagged.
What is an outlier?
The NIST/SEMATECH e-Handbook defines an outlier as “an observation that lies an abnormal distance from other values in a random sample from a population.” The definition is deliberately loose: whether a point is “abnormal” depends on a rule you choose. NIST also notes that outliers should be investigated carefully, because they often contain valuable information about the process or about how the data were gathered and recorded.
How to find an outlier: the 1.5 × IQR rule
The fence method has two tiers. NIST (e-Handbook section 7.1.6) defines the interquartile range as IQ = Q3 − Q1 and builds four fences from it:
| Fence | Formula | A point beyond it is |
|---|---|---|
| Lower inner fence | Q1 − 1.5 × IQ | a mild outlier |
| Upper inner fence | Q3 + 1.5 × IQ | a mild outlier |
| Lower outer fence | Q1 − 3 × IQ | an extreme outlier |
| Upper outer fence | Q3 + 3 × IQ | an extreme outlier |
Symbols: Q1 is the 25th percentile, Q3 is the 75th percentile, and IQ (usually written IQR) is the spread of the middle half of the data. Because quartiles ignore the extreme tails, a single wild value cannot move the fences much. The same inner fences set the whiskers of a box plot (NIST e-Handbook section 1.3.3.7); see our guides to box plots in R and SPSS for drawing them.
How to calculate an outlier: worked example
Take 15 measurements, already sorted: 12, 13, 14, 15, 15, 16, 17, 18, 19, 20, 21, 23, 26, 34, 60.
- Median. The 8th of 15 values is 18.
- Q1 (inclusive method). Position = 1 + 0.25 × (15 − 1) = 4.5, halfway between the 4th and 5th values (15 and 15), so Q1 = 15.
- Q3 (inclusive method). Position = 1 + 0.75 × 14 = 11.5, halfway between the 11th and 12th values (21 and 23), so Q3 = 22.
- IQR. 22 − 15 = 7.
- Inner fences. Lower = 15 − 1.5 × 7 = 4.5. Upper = 22 + 1.5 × 7 = 32.5.
- Outer fences. Lower = 15 − 3 × 7 = −6. Upper = 22 + 3 × 7 = 43.
- Compare. No value is below 4.5. 34 is above 32.5 but below 43, so it is a mild outlier. 60 is above 43, so it is an extreme outlier. The other 13 values are inside the fences.
For comparison, the mean of this data set is 21.53 and the sample standard deviation is 12.06; the single value 60 pulls both upward, which is why quartile-based fences are preferred over mean-based screens when you suspect outliers are present.
Which quartile method? The same data can give a different answer
Software does not agree on how to compute Q1 and Q3, and for small samples the difference can decide whether a point is flagged. Repeat the example with the exclusive method (position = 0.25 × (n + 1) = 4 and 0.75 × 16 = 12):
| Method | Q1 | Q3 | IQR | Upper inner fence | Flagged |
|---|---|---|---|---|---|
| Inclusive (Excel QUARTILE.INC, R type 7, NumPy default) | 15 | 22 | 7 | 32.5 | 34 and 60 |
| Exclusive (Excel QUARTILE.EXC, R type 6) | 15 | 23 | 8 | 35 | 60 only |
Under the exclusive method the upper inner fence is 23 + 1.5 × 8 = 35, so 34 sits inside it and is no longer flagged. Neither answer is wrong; they follow different quartile definitions. State which one you used. The R boxplot() function uses yet another definition (hinges), so a box plot and a hand calculation can disagree on borderline points; our box plots in R guide shows a concrete case.
How to find outliers in a data set in Excel, R or Python
Excel
With the data in A2:A16, put Q1 in D1 and Q3 in D2, then compute fences and flag each row:
D1: =QUARTILE.INC($A$2:$A$16,1)
D2: =QUARTILE.INC($A$2:$A$16,3)
D3: =D2-D1 (IQR)
D4: =D1-1.5*D3 (lower inner fence)
D5: =D2+1.5*D3 (upper inner fence)
B2: =OR(A2<$D$4,A2>$D$5) (TRUE = outlier; fill down)R
x <- c(12,13,14,15,15,16,17,18,19,20,21,23,26,34,60)
q <- quantile(x, c(.25, .75), type = 7)
iqr <- q[2] - q[1]
x[x < q[1] - 1.5*iqr | x > q[2] + 1.5*iqr] # 34 60Python
import numpy as np
x = np.array([12,13,14,15,15,16,17,18,19,20,21,23,26,34,60])
q1, q3 = np.quantile(x, [0.25, 0.75])
iqr = q3 - q1
print(x[(x < q1 - 1.5*iqr) | (x > q3 + 1.5*iqr)]) # [34 60]How to find the outlier of a data set: reading the result
| Where the value falls | NIST label | Reasonable next step |
|---|---|---|
| Inside both inner fences | Not flagged | No action from this rule |
| Beyond an inner fence, inside the outer fence | Mild outlier | Check the record and the measurement |
| Beyond an outer fence | Extreme outlier | Check the record and the measurement |
A flag is a prompt to look, not a verdict. NIST advises understanding why an outlier appeared before considering elimination. Typical checks are a keying or unit error, an instrument fault, or a valid but unusual case. For what to do afterwards, see winsorizing vs trimming, and for outliers that distort a regression see Cook’s distance and influential observations.
Assumptions and common mistakes
- Unstated quartile method. As shown above, it can change the flagged set.
- Treating a flag as proof of error. The fence rule says a point is far from the middle half of the data, not that it is wrong.
- Using Grubbs’ test on non-normal data. NIST describes Grubbs’ test as detecting a single outlier in a univariate data set that follows an approximately normal distribution; check normality first (see the Shapiro-Wilk test in R). For more than one suspected outlier, NIST recommends the Tietjen-Moore or generalized ESD test instead.
- Dropping values after seeing their effect on the result. Decide the rule before looking at the outcome and apply it to every variable the same way.
Grubbs’ test in brief
The test statistic is G = max|Yi − Ȳ| / s, the largest absolute deviation from the sample mean in units of the sample standard deviation s. The null hypothesis is that there are no outliers; the alternative is exactly one outlier. In the two-sided test, H0 is rejected when G > ((N − 1)/√N) × √( t² / (N − 2 + t²) ), where t is the critical value of the t distribution with N − 2 degrees of freedom at significance level α/(2N).
How to report it
Report the rule, the quartile method, how many values were flagged and what you did. A template sentence in the numeric style typical of APA-formatted results:
Outliers were identified with the 1.5 × IQR rule (inclusive quartile method; Q1 = 15, Q3 = 22, IQR = 7). Two of 15 observations fell outside the fences (34 and 60). Both were checked against the source records and retained, and the analysis was repeated without them as a sensitivity check.
Adapt the last sentence to what you actually did. If you removed or modified values, say so and give the reason.
Frequently asked questions
How to find an outlier in statistics?
Sort the data, compute Q1, Q3 and the IQR, then flag values below Q1 − 1.5 × IQR or above Q3 + 1.5 × IQR. Values beyond 3 × IQR from the quartiles are extreme outliers.
How to calculate an outlier?
You do not calculate an outlier itself; you calculate fences and test each value against them. The step-by-step example above gives fences of 4.5 and 32.5 for the sample data.
How to find the outlier in a data set when there is only one suspect value?
If the data are approximately normal, Grubbs’ test checks a single suspect value formally. Otherwise use the IQR fences, which need no distributional assumption.
Is every outlier a mistake?
No. NIST notes that outliers often carry valuable information about the process or data collection, and that they should be investigated carefully before any decision to remove them.
Why do Excel and R give different outliers?
They can use different quartile definitions, which changes Q1, Q3 and so the fences. Name the method in your report.
Sources
- NIST/SEMATECH e-Handbook of Statistical Methods, section 7.1.6, “What are outliers in the data?” (read 2026-10-29): itl.nist.gov/div898/handbook/prc/section1/prc16.htm
- NIST/SEMATECH e-Handbook, section 1.3.3.7, “Box Plot” (read 2026-10-29): itl.nist.gov/div898/handbook/eda/section3/boxplot.htm
- NIST/SEMATECH e-Handbook, section 1.3.5.17.1, “Grubbs’ Test for Outliers” (read 2026-10-29): itl.nist.gov/div898/handbook/eda/section3/eda35h1.htm
Related on CASRAI: the interquartile range and five-number summary and z-scores in SPSS.
