Outlier Calculator - IQR Fences & Z-Score Method

About the Outlier Calculator

An outlier is a value that lies unusually far from the rest of the data. It might be a typing mistake, a faulty sensor reading or a genuinely exceptional case — a record sale, a very tall person, a house worth ten times its neighbours. Whatever the cause, outliers can distort averages and standard deviations badly, so it pays to find them before analysing data.

This outlier calculator finds outliers by the two most common rules. The IQR rule, proposed by John Tukey, flags values more than 1.5 interquartile ranges beyond the quartiles. The z-score rule flags values more than a chosen number of standard deviations from the mean. The calculator shows the limits, lists every outlier, and shows how the mean and median change when the outliers are left out.

How to Use the Outlier Calculator

Enter your numbers, separated by commas, spaces or new lines.

Choose the IQR rule or the z-score rule.

For the IQR rule, enter the multiplier — 1.5 for ordinary outliers, 3 for extreme ones. For the z-score rule, enter the threshold, usually 2, 2.5 or 3.

The outliers, limits and their effect on the average appear below.

The Two Rules

  IQR rule:      lower fence = Q1 − 1.5 × IQR
                 upper fence = Q3 + 1.5 × IQR
                 outliers lie outside the fences

  z-score rule:  z = (x − mean) ÷ s
                 outliers have |z| above the threshold

The calculator finds quartiles by the textbook median-of-halves method and uses the sample standard deviation for z-scores.

Step-by-Step Example: IQR Rule

The values 2, 3, 5, 7, 9, 120.

  Lower half:   2, 3, 5    → Q1 = 3
  Upper half:   7, 9, 120  → Q3 = 9
  IQR:          9 − 3 = 6
  Lower fence:  3 − 1.5 × 6 = −6
  Upper fence:  9 + 1.5 × 6 = 18

Only 120 lies outside the fences, so it is the one outlier. Without it the mean falls from 24.33 to 5.2, while the median moves only from 6 to 5.

Step-by-Step Example: Z-Score Rule

The same values, with a threshold of 2.

  Mean:          146 ÷ 6 = 24.3333
  Sample SD:     46.9368
  z for 120:     (120 − 24.3333) ÷ 46.9368 = 2.04
  Limits:        24.3333 ± 2 × 46.9368 → −69.54 to 118.21

With a threshold of 2, 120 is flagged — but only just. With a threshold of 3 it would not be flagged at all, even though it is plainly unusual.

Why the Z-Score Rule Can Fail

The z-score rule measures each value against the mean and standard deviation — but an outlier inflates both of them, partly hiding itself. This is called masking. In a sample of n values, no z-score can ever exceed (n − 1) ÷ √n. For six values that limit is 5 ÷ √6 ≈ 2.04, so a threshold of 3 can never flag anything, however extreme. Even with 10 values the limit is only 2.85. The IQR rule does not suffer from this, because quartiles are barely affected by the extremes, which is why it is generally preferred for small or skewed data sets.

Choosing the Multiplier or Threshold

Tukey's 1.5 × IQR identifies values worth a second look; 3 × IQR marks "far out" values that are extreme by any standard. For roughly normal data, the 1.5 × IQR fences correspond to about 2.7 standard deviations from the mean, so only about 0.7 percent of genuine values fall outside them. With z-scores, a threshold of 2 flags about 5 percent of normal data, 2.5 about 1.2 percent and 3 about 0.3 percent. In a large data set even a strict rule will flag some perfectly genuine values.

What to Do With Outliers

Finding an outlier is the start of an investigation, not the end. First check whether the value is an error — a misplaced decimal point, the wrong units, a duplicated entry. Errors should be corrected or removed. If the value is genuine, it is part of the story: a hospital's longest waiting time or a fund's worst month may be exactly what matters. Options include reporting results with and without the outlier, using the median and IQR instead of the mean and standard deviation, or transforming skewed data with logarithms before analysis.

A Real-World Example

Imagine a small shop records daily sales of 410, 385, 432, 398, 4,150 and 420. The value 4,150 stands out immediately, and the IQR rule confirms it: the quartiles are 398 and 432, the IQR is 34, and the upper fence is 432 + 51 = 483. A quick check of the till records might show that someone typed an extra zero into a sale of 415. Left in, that single error would more than double the average daily takings, from 410 to about 1,033, and lead to badly wrong stock orders.

Understanding Your Result

The headline lists the outliers, or says there are none.

The limits line gives the lowest and highest values that are not outliers.

The rule used line shows the quartiles or mean and standard deviation behind the limits.

The effect of removing line compares the mean and median with and without the outliers.

The worth knowing line adds a caution about the chosen rule.

When Should You Use This Calculator?

Use it to screen data for errors before calculating averages.

Use it to identify unusual measurements, sales or response times.

Use it for statistics homework on the 1.5 × IQR rule and z-scores.

Use it to judge how sensitive a result is to extreme values.

Common Mistakes

Deleting outliers automatically. Check each one; genuine extremes are real data.

Using a z threshold of 3 on small samples. It may be mathematically impossible to flag anything.

Forgetting that outliers inflate the SD. This masks them under the z-score rule.

Mixing quartile methods. Different methods move the fences slightly.

Treating skewed data as full of outliers. Long tails are normal for incomes and prices; consider a log scale instead.