Applying the 1.5 IQR Rule for Outliers

From a Boxplot to Outlier Fences

In Five-Number Summary and Boxplot Construction, you used the minimum, \(Q_1\), median, \(Q_3\), and maximum to draw a basic boxplot. A modified boxplot adds a useful feature: it identifies observations that are unusually far from the middle of the data by the 1.5 IQR rule. Those flagged observations are plotted individually, and the whiskers stop at the most extreme observations that are not flagged.

The rule uses the quartiles and the spread of the middle half of the data. It is a consistent way to flag possible outliers, not a test that proves an observation is incorrect or explains why it is unusual. Keep every observation in the data while applying the rule; do not delete a value just because it falls beyond a fence.

Calculate the Fences

The lower fence is \(1.5\) IQR below \(Q_1\), and the upper fence is \(1.5\) IQR above \(Q_3\). Calculate the IQR first, then use it in both fence formulas. The fences are cutoffs for checking the observations; they do not need to be actual data values.

After finding the fences, compare every observation with them. A value less than the lower fence or greater than the upper fence is flagged. A value exactly equal to a fence is not beyond it, so it is not flagged by this rule. The lower and upper fences can be decimals even when all the observations are whole numbers.

For 15 observations, continue to use the quartile positions from the previous tutorial: \(Q_1\) is the 4th ordered value, the median is the 8th, and \(Q_3\) is the 12th. Keep repeated observations in the ordered list as you count positions. Once you have the quartiles, finding fences does not require any new way of locating them.

How the Rule Changes a Boxplot

In the basic boxplot constructed in the previous tutorial, the whiskers extended to the minimum and maximum. In a modified boxplot, the box still extends from \(Q_1\) to \(Q_3\), and the median is still marked inside it. The difference is how the whiskers and possible outliers are shown.

The lower whisker reaches the smallest observation that is not below the lower fence. The upper whisker reaches the largest observation that is not above the upper fence. Any observations beyond either fence are plotted separately, often as individual dots or asterisks. If a fence does not flag any observations at an end, the whisker reaches the minimum or maximum at that end.

The fences are calculated from the quartiles, but a whisker ends at an observed data value, not at a fence. For example, if the upper fence is 43.5 and the largest non-outlier is 31, the upper whisker ends at 31. It does not extend to 43.5. Label the variable and its units, and place all boxplot marks accurately on a numerical scale.

Worked Example: Evening Reading Times

Worked Example: Minutes to Complete a Puzzle

Worked Example: A Low Water-Use Reading

What an Outlier Flag Does—and Does Not—Mean

The 1.5 IQR rule is a descriptive rule for identifying values that are far from the middle half of a distribution. “Possible outlier” is careful wording: the rule marks a value for attention, but it does not establish a cause. A flagged value may be correct and meaningful, or it may reflect a recording or measurement problem. Check the context and the original information before drawing conclusions about it.

Do not automatically remove flagged values before describing the distribution. Removing a real observation changes the data and can change the quartiles, IQR, and fences. If a value is excluded for a justified reason, explain that reason and be clear about which data are being summarized. Also, a data set can contain an unusual-looking value that is not beyond a fence, or have no flagged values at all; the rule uses the quartiles and IQR rather than visual judgment alone.

Common Mistakes and AP Exam Tips

• Using the range instead of the IQR. The rule uses \(Q_3-Q_1\), not the maximum minus the minimum. Show the quartiles and the IQR calculation.
• Adding and subtracting from the wrong quartile. Subtract \(1.5(\text{IQR})\) from \(Q_1\) for the lower fence; add it to \(Q_3\) for the upper fence.
• Flagging a value on a fence. The rule flags values strictly below or strictly above the fences. State the comparison clearly, especially if a value equals a fence.
• Drawing whiskers all the way to the minimum and maximum automatically. In a modified boxplot, a whisker ends at the most extreme non-outlier. Plot flagged observations separately.
• Drawing the whisker to the fence. A fence is a cutoff, not an observed data value. The whisker endpoint is an actual observation within the fence.
• Calling a flagged value an error or deleting it without explanation. The rule identifies a possible outlier; it does not establish why the observation is unusual or whether it should be excluded.
• Leaving out context or units. A complete explanation reports the fences and flagged observation in the variable’s context, and a boxplot should have a labeled scale and units.

For a full-credit response, show \(Q_1\), \(Q_3\), the IQR, and both fence calculations. Identify the observation or observations that fall beyond a fence, or state that none do. When asked to draw a modified boxplot, keep the box and median at their five-number-summary positions, extend whiskers only to the most extreme non-outliers, and plot flagged observations individually. Give the conclusion in context and describe them as possible outliers rather than automatically treating them as mistakes.