How to find outliers: the 1.5 x IQR rule

How to find outliers with the 1.5 x IQR rule

The 1.5 IQR rule (also written 1.5 x IQR) flags a data value as an outlier when it sits far below the first quartile or far above the third quartile. You build two cutoff values called fences, and any value outside the fences is an outlier.

Here is how to find outliers, step by step:

1. Sort the data from smallest to largest.
2. Find the first quartile $Q_1$ and the third quartile $Q_3$.
3. Compute the interquartile range $IQR = Q_3 - Q_1$.
4. Compute the lower fence $Q_1 - 1.5 \times IQR$.
5. Compute the upper fence $Q_3 + 1.5 \times IQR$.
6. Any value below the lower fence or above the upper fence is an outlier.

$Q_1$, the first quartile, is the value with about one quarter of the data below it. $Q_3$, the third quartile, is the value with about three quarters of the data below it. The IQR (interquartile range) measures the spread of the middle 50% of the data, so the rule asks how far a point sits beyond that middle band.

You can check your quartiles and IQR with the five-number summary calculator.

How to find the quartiles (median-excluded convention)

Before you can use the rule, you need $Q_1$ and $Q_3$. AP Statistics and TI graphing calculators locate quartiles using the median-excluded method, so that is the convention used here.

Start by finding the median of the whole dataset. Then split the sorted values into a lower half and an upper half.

• When $n$ is even, the median falls between two values. The lower half is every value below that point, and the upper half is every value above it.
• When $n$ is odd, the median is a single data value. Leave that one value out of both halves.

Now $Q_1$ is the median of the lower half and $Q_3$ is the median of the upper half. Some spreadsheet functions use a different rule and can report slightly different quartiles, so on AP problems stay with the median-excluded method.

Why outliers matter: the mean versus the median

Outliers change which summary of center you can trust. The mean $\bar{x}$ (the average of the values) is pulled toward an extreme value, while the median (the middle value) barely moves.

In the worked example below, one large value raises the mean by almost 3 while the median shifts by only half a point. Statisticians call the median resistant because a single extreme value does not change it much. The mean is not resistant.

That is why a skewed dataset with a high outlier usually has a mean larger than its median. If you have to report one number for a skewed distribution, the median often describes it better. For a fuller comparison, see mean vs median.

How outliers show up on a boxplot

A modified boxplot uses the same fences as the 1.5 IQR rule. The whiskers extend only to the most extreme values that are still inside the fences, and any outliers are drawn as separate dots or stars beyond the whiskers.

That means you can read outliers straight off a well-drawn boxplot. The isolated points are exactly the values the 1.5 x IQR rule flags. To build one from scratch, see how to make a boxplot.

Common mistakes to avoid

• Forgetting to sort the data first. Quartiles only make sense on sorted values.
• Multiplying in the wrong place. The rule multiplies $1.5 \times IQR$ and then adds to or subtracts from the quartiles, not $1.5 \times Q_1$.
• Rounding the IQR too early. Keep full precision until you reach the final fence values.
• Treating every outlier as an error. An outlier can be a typo, or it can be a real and interesting value, so investigate before you remove it.
• Assuming outliers only appear on the high side. A value below the lower fence is an outlier too.

Where this appears in AP Statistics

The 1.5 x IQR rule lives in Unit 1 of the Fall 2026 AP Statistics course, under topic 1.7 (summary statistics for one quantitative variable) and topic 1.8 (graphical representations of summary statistics for one quantitative variable, which include boxplots). Unit 1 is 20 to 30% of the multiple-choice section.

Because these topics cover both summary statistics and boxplots, practice the full fence calculation by hand and by reading outliers off a boxplot. The course lists a second outlier method too: a value more than 2 standard deviations from the mean. For the official course description, see AP Central.