Math Core

"Outlier" has meant "a value that looks far from the rest" so far. That's too vague for a statistics course: two people looking at the same graph could disagree. This lesson gives two precise rules for identifying outliers and shows how to build the graph that displays them: the box plot, drawn from the five-number summary.

The five-number summary

Definition

Five-number summary

The five-number summary of a distribution is

\text{minimum}, \quad Q_1, \quad \text{median}, \quad Q_3, \quad \text{maximum}.

These five values split the ordered data into four parts, each holding about $25\%$ of the values.

The five-number summary gives a quick sense of center (the median) and spread (the range and the IQR), and it is the basis for a box plot.

Two rules for outliers

The 1.5 × IQR rule

A value is an outlier if it is more than $1.5 \times \text{IQR}$ below $Q_1$ or above $Q_3$ . That is, an outlier is any value outside the fences

Q_1 - 1.5 \cdot \text{IQR} \qquad \text{and} \qquad Q_3 + 1.5 \cdot \text{IQR}.

This is the standard rule in AP Statistics, and the one used to draw box plots. Because it is built from quartiles, it is resistant: the outliers themselves don't distort the fences.

A second rule uses the mean and standard deviation.

The two-standard-deviation rule

A value may be called an outlier if it is more than $2$ standard deviations from the mean: below $\bar{x} - 2s_x$ or above $\bar{x} + 2s_x$ .

The two rules don't always agree. The $2s_x$ rule works best for roughly symmetric, mound-shaped data; for strongly skewed data, it can flag many ordinary values in the long tail. When a question doesn't specify a rule, use $1.5 \times \text{IQR}$ and say so.

Box plots

A box plot (or boxplot) draws the five-number summary above a number line.

Draw a box from $Q_1$ to $Q_3$ , with a line across it at the median.
Check for outliers with the $1.5 \times \text{IQR}$ rule and plot each one as a separate dot.
Draw whiskers from the box out to the smallest and largest values that are not outliers.

A box plot that shows outliers this way is called a modified box plot, and it is the kind you should always draw in AP Statistics.

Worked example: Building a modified box plot

Fifteen students reported how many hours they spent playing video games last week:

0, \ 2, \ 3, \ 3, \ 4, \ 5, \ 5, \ 6, \ 7, \ 8, \ 8, \ 9, \ 10, \ 12, \ 24

Identify any outliers and make a modified box plot.

Five-number summary. With $n = 15$ , the median is the 8th value, $6$ . The lower half is $0, 2, 3, 3, 4, 5, 5$ , so $Q_1 = 3$ . The upper half is $7, 8, 8, 9, 10, 12, 24$ , so $Q_3 = 9$ . The summary is $0, 3, 6, 9, 24$ .

Outliers. $\text{IQR} = 9 - 3 = 6$ and $1.5 \cdot 6 = 9$ . The fences are $3 - 9 = -6$ and $9 + 9 = 18$ . Only $24$ is outside them, so 24 hours is an outlier.

Draw it. The box runs from $3$ to $9$ with a line at $6$ . The left whisker reaches $0$ . The right whisker stops at $12$ , the largest value that is not an outlier, and $24$ gets its own dot.

Hours of video games last week (15 students)

Reading box plots

Because each section holds about a quarter of the data, a box plot answers "what percent" questions quickly: about $25\%$ of the values are below $Q_1$ , about $50\%$ are between $Q_1$ and $Q_3$ , and about $75\%$ are below $Q_3$ .

A box plot also hints at shape. In the video-game plot, the box and whiskers are balanced, but the high outlier at $24$ stretches the distribution far to the right, a sign of right skew. In general:

Skewed right: the right whisker and the right part of the box tend to be longer.
Skewed left: the left whisker and the left part of the box tend to be longer.
Roughly symmetric: the two halves look about the same.

Worked example: Reading a box plot

The box plot summarizes the ages of the guests at a large wedding.

Ages of wedding guests (years)

Give the five-number summary and the IQR.
About what percent of the guests were older than $26$ ?
Could any guest's age be an outlier?

Solutions.

Minimum $18$ , $Q_1 = 26$ , median $34$ , $Q_3 = 47$ , maximum $71$ . The IQR is $47 - 26 = 21$ years.
$26$ is $Q_1$ , so about $75\%$ of the guests were older than $26$ .
The fences are $26 - 1.5 \cdot 21 = -5.5$ and $47 + 31.5 = 78.5$ . Every age is between them, so there are no outliers, which matches the plot having no separate dots.

Worked example: The two-standard-deviation rule

A snack company weighed a sample of $30$ bags of chips labeled " $28$ grams." The weights had mean $\bar{x} = 28.4$ g and standard deviation $s_x = 0.6$ g. One bag weighed $27.0$ g. Is it an outlier by the $2s_x$ rule?

The cutoffs are $28.4 - 2(0.6) = 27.2$ g and $28.4 + 2(0.6) = 29.6$ g. Since $27.0 < 27.2$ , the bag is an outlier by this rule. It is more than two standard deviations below the mean.

Common mistake

A box plot does not show the number of values, and a longer section doesn't hold more data. Each section always holds about $25\%$ of the values; a long section means those values are spread out. Box plots also hide details such as gaps, clusters and multiple peaks. A bimodal distribution can produce a perfectly ordinary-looking box plot.

Tip

The whisker ends at the most extreme value inside the fences, not at the fence itself. The fences are never drawn on a box plot; they're only a test.

Practice

The next two problems use this box plot of the monthly rent, in hundreds of dollars, of apartments in a neighborhood.

Monthly rent of apartments (hundreds of dollars)

Practice 1

What is the interquartile range of the rents, in hundreds of dollars?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 2

Which statement about the rents is supported by the box plot?

Practice 3

Find the upper fence for outliers for this data set.

4, \ 18, \ 20, \ 21, \ 23, \ 24, \ 25, \ 27, \ 28, \ 30, \ 33, \ 45

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 4

For the data set in the previous problem, how many values are outliers by the $1.5 \times \text{IQR}$ rule?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 5

The scores on a standardized test have mean $\bar{x} = 500$ and standard deviation $s_x = 90$ . Using the two-standard-deviation rule, which score would be considered an outlier?

Practice 6

Two data sets have identical box plots. Which statement must be true?

Practice 7

A data set has $Q_1 = 40$ and $Q_3 = 52$ . In its modified box plot, the lower whisker ends at $27$ and the minimum is plotted as a separate outlier dot. Which value could be the minimum?