Outliers and the fence rule

What you will be able to do

• find the lower and upper fences and list the outliers of a data set;
• draw a box plot whose whiskers stop at the last value inside the fences;
• decide what to do with an outlier, and explain why the median and the IQR resist outliers.

Twelve reaction times

In a made-up class test, 12 students press a button as soon as a light comes on. Their reaction times, in milliseconds (ms) and in order, are

$212, 225, 231, 238, 240$, $244, 249, 253, 258, 262$, $271, 389$.

The last value looks odd. Eleven students took between 212 ms and 271 ms. One took 389 ms. Is that value an outlier? "It looks far away" is not a rule. Two people could disagree. We need a rule that everyone applies in the same way.

The fence rule

1. Find $Q_1$, $Q_3$ and the IQR.
2. Build two fences:

3. A value below the lower fence or above the upper fence is an outlier.

For the reaction times, the halves are the first six and the last six values. So $Q_1 = \dfrac{231 + 238}{2} = 234.5$ ms and $Q_3 = \dfrac{258 + 262}{2} = 260$ ms. The IQR is $25.5$ ms, and $1.5 \times 25.5 = 38.25$ ms.

• Lower fence: $234.5 - 38.25 = 196.25$ ms. No time is below it.
• Upper fence: $260 + 38.25 = 298.25$ ms. The time of 389 ms is above it, so it is an outlier.

Box plots that show outliers

Most box plots today mark outliers on their own. The box is the same, but each whisker stops at the last value inside the fence. Each outlier gets its own mark, often a dot or a cross. Here the right whisker stops at 271 ms, and 389 ms is drawn alone. The fences are only lines of thought: most box plots do not draw them.

Why 1.5? It is a convention, a choice that statisticians agreed on, not a law of nature. Some books also mark values beyond 3 IQR as "far out". For mound-shaped data, the 1.5 IQR fences sit about 2.7 standard deviations from the middle. Then only about 7 values in 1000 fall outside them. So in such data an outlier is truly unusual.

You should find that the amber value becomes an outlier once it passes 298.25 ms, and the whisker then jumps back to 271 ms. With a multiplier of 3, the upper fence moves out to 336.5 ms. With a multiplier of 1, it moves in to 285.5 ms.

What to do with an outlier

An outlier is a question, not a verdict. Work through these steps.

1. Check for a mistake. Was a number typed wrongly, or written in the wrong unit, such as grams instead of kilograms? If so, correct it if you can, and remove it if you cannot.
2. If it is real, keep it. A real outlier is part of the story. The student with 389 ms may have been looking away. A very rich family is still a family.
3. Use resistant measures. The median and the IQR barely notice an outlier. The mean, the range and the standard deviation can change a lot. For the reaction times, the mean is 256 ms with the outlier and about 243.9 ms without it. The median is 246.5 ms with it and 244 ms without it.
4. Report honestly. If you leave a value out, say so and say why. It is often best to give results with and without it.

Never delete a value only because it spoils a neat result. That is how false conclusions are made.

Practise with new numbers

Try it in Python

The program finds the quartiles and the fences of the 12 reaction times, lists the outliers, and prints the mean and the median with and without them.

Common mistakes

• Using 1 IQR instead of 1.5 IQR, or measuring from the median instead of from the quartiles.
• Running the whisker to an outlier. In a box plot that shows outliers, the whisker stops at the last value inside the fence.
• Deleting every outlier. Check first. A real value stays.
• Thinking the rule proves a mistake. It only flags a value for a closer look.

Remember these

• Fences: $Q_1 - 1.5\,\text{IQR}$ and $Q_3 + 1.5\,\text{IQR}$. Beyond them is an outlier.
• Whiskers stop at the last value inside the fences; outliers are marked alone.
• Check an outlier for a mistake. Keep it if it is real, and report what you did.
• The median and the IQR resist outliers; the mean, the range and the standard deviation do not.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A data set has $Q_1 = 30$ and $Q_3 = 42$. Find both fences.

2. With those fences, which of the values 10, 15, 58 and 61 are outliers?

3. In a survey of the heights of adults, one height is written as 17.2 m. What should you do?

4. One very large value is added to a data set. Which changes more, the IQR or the range?

5. Why do box plots that mark outliers stop the whisker at the last value inside the fence?

References

• OpenStax, Introductory Statistics, Chapter 2 (Descriptive Statistics), sections on measures of the location of the data, including outliers, and on box plots.