Outlier Calculator

What an outlier is, and what a flag means

An outlier is a value that sits far enough from the rest of a sample to deserve a second look. The NIST/SEMATECH e-Handbook puts the reason plainly: a suspected outlier may be a recording error, in which case it should be corrected or removed, or it may be a real event that the analysis has to accommodate rather than delete.

Those are two different jobs, and the arithmetic only handles the first half of either. A test says that a number is unusual for the shape the rest of the data describes. It does not say that the number is wrong, and it cannot tell you that a delivery did take 41 minutes because a van broke down.

Every rule on this page assumes one variable measured in comparable units, and the formal tests add an assumption that the data are roughly normal. Check that first. On a strongly skewed column such as household income or file size, the upper tail is long by design, and a fence built for a bell curve will flag ordinary values all day.

IQR fences: where the 1.5 comes from

The quartile rule needs three numbers: the first quartile Q1, the third quartile Q3 and the gap between them, the interquartile range. Half the data sits inside that gap. The fences sit one and a half IQRs beyond each quartile, and anything outside is flagged.

For normal data the quartiles land at 0.6745 standard deviations either side of the mean, so the IQR is 1.349 standard deviations. Add 1.5 IQRs to the upper quartile and the fence sits 0.6745 + 1.5 x 1.349 = 2.698 standard deviations out. On a bell curve that leaves about 0.7 percent of the data outside each fence, which is why 1.5 is a flag and not a verdict. Raise the multiplier to 3 and the fence moves to 0.6745 + 4.047 = 4.72 standard deviations, roughly one point in a million, and only the extreme cases survive.

The quartile itself has more than one definition, and the choice moves the fence; OpenStax Introductory Statistics works the same fences from the box plot side. Linear interpolation, the method behind Excel QUARTILE.INC and the default in most statistics packages, places Q1 and Q3 by position in the sorted list. The median of halves method, the Tukey hinge, splits the data at the median and takes the median of each half, and the NIST box plot page draws them that way. They agree closely on large samples and can disagree by half a gap on small ones.

Both methods flag the same point here, but 22 has 0.75 of room under the first fence and 2.0 under the second. When a value sits that close to a line, say which quartile method produced the line, because the other one may put it on the other side.

The fifteen delivery times, worked through

The sample is 12, 13, 13, 14, 14, 15, 15, 15, 16, 16, 17, 18, 19, 22 and 41 minutes. The sum is 260, so the mean is 17.3333. The median is the eighth value, 15. Q1 is 14 and Q3 is 17.5 by linear interpolation, so the IQR is 3.5, the lower fence is 14 - 5.25 = 8.75 and the upper fence is 17.5 + 5.25 = 22.75.

Only 41 clears a fence. The 22 sits 0.75 below the upper fence, which is inside, and nothing falls below 8.75 because the smallest value is 12. Apply the stricter 3 x IQR rule instead and the fences move to 3.5 and 28: 41 is still outside, and it is the only one.

Now compare the four rules on the two candidate points. The table shows the threshold each rule applies and what it makes of 22 and 41.

Four rules, one answer, which is the comfortable case. Disagreement between them is more informative than agreement, and the next two sections show why.

Z-scores and the masking problem

A z-score is how many standard deviations a value sits from the mean: (x - mean) divided by the standard deviation. For this sample the standard deviation is 7.0373, so 41 scores 3.363 and 22 scores 0.663. A cut-off of 3 flags 41 and nothing else.

The problem is that the outlier helps set the yardstick. Those fifteen numbers have a standard deviation of 7.0373 with 41 in the sample and 2.6777 without it, a factor of 2.63. One point in fifteen nearly tripled the spread, and every other z-score shrank by the same factor. Run the sample without 41 and the mean drops to 15.6429 while 22 climbs from 0.663 to 2.374. Same 22 minutes, a different verdict.

This is what NIST calls masking: an extreme point inflates the standard deviation enough to hide the next candidate. It is also why a fixed cut-off behaves oddly at small sample sizes. With n numbers the largest possible z-score is (n - 1) divided by the square root of n, so nine numbers can never produce a z above 2.67 and a cut-off of 3 will never fire on them, no matter how odd the data look. At fifteen numbers the ceiling is 3.615, and 41 is close to it.

Treat a z-score as a scale-free distance and keep the sample size in view. A z of 3 on 15 points and a z of 3 on 15,000 points are not the same evidence.

The modified z-score that survives masking

The fix is to build the yardstick from pieces that ignore the tail. Take the median instead of the mean, and the median absolute deviation instead of the standard deviation. The MAD is the median of the distances from each value to the median, and unlike the standard deviation it barely moves when one point runs away.

For the delivery times the median is 15. The distances are 3, 2, 2, 1, 1, 0, 0, 0, 1, 1, 2, 3, 4, 7 and 26, whose median is 2. The modified z-score multiplies (x - median) by 0.6745 and divides by the MAD, the form Iglewicz and Hoaglin describe in the NIST handbook. The constant 0.6745 is the value that makes the MAD match a standard deviation on normal data.

That gives 0.6745 x 26 / 2 = 8.768 for the 41 and 0.6745 x 7 / 2 = 2.361 for the 22. The usual cut-off is 3.5, so the 41 is flagged and the 22 is not, and the margin is wide in both directions. Drop the 41 and the MAD and the median stay at 2 and 15, so the 22 does not move at all. That stability is the whole point: the score of a point is not affected by its neighbours in the tail.

One failure mode to know about. If more than half the values are identical, the MAD is zero and the score cannot be formed. A column of mostly 5s with one 9 is the classic case, and the calculator reports it rather than dividing by zero.

The Grubbs test: one outlier, stated as a probability

Grubbs turns the question into a hypothesis test. The statistic is the largest absolute deviation from the mean, divided by the standard deviation: G = max|x - mean| / s. Here that is 23.667 / 7.0373 = 3.3630, and the critical value at 15 numbers and a two-sided 0.05 level is 2.5483, so the null hypothesis of no outliers is rejected.

The critical value is not a look-up constant in the formula, it comes from the t distribution: take (n - 1) over the square root of n, multiply by the square root of t squared over (n - 2 + t squared), and use the t value at a tail probability of alpha divided by 2n with n - 2 degrees of freedom. That is the version the NIST handbook prints, and it is what this calculator evaluates, so any sample size from 3 up works without a printed table.

The worked example in the NIST handbook is eight mass spectrometer readings on a uranium isotope, 199.31 through 202.18 with a 245.57 at the end. The mean is 206.4338, the standard deviation 15.8526 and G is 2.4688 against a one-sided critical value of 2.032 at 0.05, so the 245.57 is an outlier. Judged two-sided, as this calculator does by default, the critical value is 2.1266 and the conclusion is the same. The generalized ESD page in the same handbook covers several outliers at once, and NIST notes that Grubbs tests for a single outlier and that sequential passes are the wrong tool when several are suspected; the generalized ESD test is the one to reach for there.

Deciding what to do with a flagged value

A flag is the start of a decision, not the end of one. Go back to the source first: a timestamp, a meter reading, an order number. If the record shows a typo, the value is wrong and correcting it is not optional. If the record is clean, the number is data and deleting it is a choice with consequences.

Three paths cover most cases. Keep the point and switch to median-based statistics, reporting the median and the MAD instead of the mean and the standard deviation, which is the honest route when the tail is real. Keep the point and change the model, since a count with a long tail may fit a lognormal better than a normal, and the NIST handbook notes that a log transform brings such data back to a shape the tests can handle. Or drop the point and say so in the write-up, with the rule and the threshold named, so a reader can rebuild the analysis.

What not to do is drop it quietly. The mean of the fifteen delivery times is 17.33 minutes and the mean of the fourteen that remain is 15.64, a 1.7 minute shift that would quietly change any service level quoted from the data. The standard deviation calculator and the z-score calculator cover the pieces this page builds on if you want to run them one at a time.

Report the rule with the result. Saying that one value was outside the 1.5 IQR fence is checkable by anyone with the data; saying that one value was an outlier is an opinion wearing a number's clothes.