When should I use the mean versus the median?

Use the mean and standard deviation for roughly symmetric distributions with no outliers. Use the median and IQR for skewed distributions or when outliers are present, because those statistics are resistant and better represent a typical value.

Do I include the median when finding the quartiles?

When n is odd, exclude the overall median from both halves before finding Q_1 and Q_3. When n is even, the data split evenly into two halves and no value is excluded. Note that some calculators use a slightly different convention, but the AP exam expects this split-the-halves method.

Why do we divide by n-1 instead of n for standard deviation?

Dividing by n-1 (the degrees of freedom) corrects for the fact that sample data tend to underestimate the true variability of the population. This gives an unbiased estimate of variance. On the AP exam, sample standard deviation always uses n-1.

Are the outlier fences ever plotted on a boxplot?

No. The fences from the 1.5×IQR rule are just thresholds you calculate to decide which points are outliers. The whiskers extend only to the most extreme actual data values that are not outliers, and outliers are shown as separate dots.

AP-STATS-1.7-1.8

U1.7 Summary Statistics and Boxplots

Master AP Stats summary statistics: compute mean, median, standard deviation, IQR, range, the 5-number summary, apply the 1.5×IQR outlier rule, and build boxplots.

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A voice-first session with the Crimsora tutor on U1.7 Summary Statistics and Boxplots, then targeted practice and FRQs — with the tutor adapting to where you get stuck.

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What this lesson covers

Every distribution you meet in AP Statistics can be boiled down to a handful of numbers that describe its center and spread. This lesson turns raw data into those numbers: mean, median, standard deviation, IQR, range, and the 5-number summary. You will also learn the exact rule the exam uses to flag outliers and how to translate a 5-number summary into a boxplot.

These tools power almost everything ahead — comparing distributions in U1.9 and reasoning about the Normal model in U1.10 both lean on center and spread. Nail the definitions and the arithmetic now, because free-response graders expect precise, correct calculations, not estimates.

Measures of Center: Mean vs. Median

The two standard measures of center are the mean and the median. The mean is the arithmetic average,

\bar{x} = \frac{\sum x_i}{n}

, the balance point of the distribution. The median is the middle value when data are ordered: if

n

is odd it is the single center value; if

n

is even it is the average of the two middle values.

The critical distinction the exam tests is resistance. The median is resistant — extreme values barely move it. The mean is non-resistant — a single large outlier pulls it toward the tail. This drives a reliable rule about skew:

Shape	Relationship
Symmetric	mean $\approx$ median
Skewed right	mean > median
Skewed left	mean < median

A common misconception is that the mean is always the "best" center. It is not. For skewed distributions or data with outliers, the median better represents a typical value. On the exam, if you are asked which measure to report for a skewed distribution, choose the median and justify it with resistance. Always pair a center with a matching spread: mean with standard deviation, median with IQR.

Measures of Spread: Range, IQR, and Standard Deviation

Spread describes how much the data vary. The range is a single number:

\text{max} - \text{min}

. It is easy but extremely sensitive to outliers.

The interquartile range is

IQR = Q_3 - Q_1

, the spread of the middle 50% of the data.

Q_1

is the median of the lower half;

Q_3

is the median of the upper half. When

n

is odd, do not include the overall median in either half. The IQR is resistant, so it pairs with the median.

The standard deviation measures typical distance from the mean. The sample standard deviation is

s_x = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}.

We divide by

n-1

(degrees of freedom) for sample data. Standard deviation is non-resistant and pairs with the mean. Its square,

s_x^2

, is the variance.

Interpret standard deviation in context: "The values typically fall about

s_x

units away from the mean." Do not say it is the average distance exactly — it is a root-mean-square distance. Larger

s_x

or larger

IQR

means more variability. On multiple-choice items, watch for questions where adding a constant to every value leaves spread unchanged, while multiplying scales spread by that factor.

The 5-Number Summary and the 1.5×IQR Outlier Rule

The 5-number summary is minimum,

Q_1

, median,

Q_3

, maximum. It captures center and spread compactly and is the foundation of a boxplot.

The exam's official outlier definition is the 1.5×IQR rule. Compute the fences:

\text{Lower fence} = Q_1 - 1.5 \times IQR

\text{Upper fence} = Q_3 + 1.5 \times IQR

Any data value below the lower fence or above the upper fence is an outlier. Values inside the fences are not outliers, even if they look far away. The fences themselves are usually not actual data points — they are just thresholds.

A frequent error is comparing to

1.5 \times IQR

alone instead of adding or subtracting from the quartiles. Another is forgetting to check both fences. Show the arithmetic on free-response: state

IQR

, multiply by 1.5, then compute each fence and compare. For example, if

Q_1 = 20

Q_3 = 40

, then

IQR = 20

1.5 \times 20 = 30

, lower fence

= -10

, upper fence

= 70

. A value of 75 would be flagged as a high outlier.

Constructing and Reading Boxplots

A boxplot is a visual of the 5-number summary. The box spans

Q_1

Q_3

, with a line at the median. Whiskers extend from the box to the smallest and largest values that are not outliers. Outliers (by the 1.5×IQR rule) are plotted as separate points, and the whisker stops at the most extreme non-outlier value.

To build one: order the data, find the 5-number summary, check for outliers, draw the box and median line, then extend whiskers to the last non-outlier values and mark any outliers individually.

Boxplot feature	Represents
Left edge of box	$Q_1$
Line in box	Median
Right edge of box	$Q_3$
Box width	$IQR$
Whisker ends	Most extreme non-outliers
Separate dots	Outliers

Boxplots reveal skew: a longer right whisker or a median closer to

Q_1

suggests right skew. A key limitation the exam loves to test: boxplots hide gaps, clusters, and multiple peaks. Two very different distributions can share the same boxplot. When shape detail matters, a histogram or stemplot is better. Boxplots shine for comparing several groups side by side (previewed in U1.9).

Key terms

Mean.: The arithmetic average, $\bar{x} = \frac{\sum x_i}{n}$ ; the balance point of the distribution and non-resistant to outliers.
Median.: The middle value of ordered data; a resistant measure of center unaffected by extreme values.
Standard deviation.: A measure of typical distance from the mean, $s_x = \sqrt{\frac{\sum(x_i-\bar{x})^2}{n-1}}$ ; non-resistant and paired with the mean.
Interquartile range (IQR).: The spread of the middle 50% of data, $IQR = Q_3 - Q_1$ ; a resistant measure of spread paired with the median.
5-number summary.: Minimum, $Q_1$ , median, $Q_3$ , and maximum — the basis for a boxplot.
1.5×IQR rule.: A value is an outlier if it falls below $Q_1 - 1.5\times IQR$ or above $Q_3 + 1.5\times IQR$ .
Resistant.: A statistic is resistant if extreme values do not substantially change it; the median and IQR are resistant, the mean and standard deviation are not.
Boxplot.: A graph of the 5-number summary with a box from $Q_1$ to $Q_3$ , a median line, whiskers to non-outliers, and separate marks for outliers.

Worked example

A student records the number of text messages received in one hour for 11 friends: 3, 5, 6, 8, 8, 10, 12, 14, 15, 20, 42. Find the 5-number summary, the IQR, and determine whether any values are outliers using the 1.5×IQR rule.

The data are already ordered, with

n = 11

. The minimum is 3 and the maximum is 42.

The median is the 6th value (since

\frac{11+1}{2} = 6

), which is 10.

For

Q_1

, take the lower half below the median: 3, 5, 6, 8, 8. Its median is the middle value, 6. So

Q_1 = 6

.

For

Q_3

, take the upper half above the median: 12, 14, 15, 20, 42. Its median is 15. So

Q_3 = 15

.

The 5-number summary is

3, 6, 10, 15, 42

.

Compute

IQR = Q_3 - Q_1 = 15 - 6 = 9

. Then

1.5 \times IQR = 1.5 \times 9 = 13.5

.

Lower fence

= Q_1 - 13.5 = 6 - 13.5 = -7.5

. Upper fence

= Q_3 + 13.5 = 15 + 13.5 = 28.5

.

Compare the data to the fences. Nothing is below

-7.5

. The value 42 exceeds the upper fence of 28.5, so 42 is a high outlier. The whisker on a boxplot would stop at 20 (the largest non-outlier), and 42 would be plotted as a separate point.

Practice questions

A data set has

Q_1 = 30

Q_3 = 50

, and a maximum of 92. Using the 1.5×IQR rule, which statement is correct?

The upper fence is 80, so 92 is an outlier
The upper fence is 65, so 92 is not an outlier
The upper fence is 80, so 92 is not an outlier
The upper fence is 50, so 92 is an outlier

Answer: The upper fence is 80, so 92 is an outlier

First

IQR = 50 - 30 = 20

, so

1.5 \times IQR = 30

. The upper fence is

Q_3 + 30 = 50 + 30 = 80

. Since 92 > 80, the maximum is an outlier. The trap answers use

1.5 \times IQR

alone or forget to add it to

Q_3

Two distributions have the same 5-number summary, but one is clearly bimodal and the other is single-peaked. Explain why their boxplots would look identical and what graph you would use instead to reveal the difference.

Answer: The boxplots would be identical because a boxplot is built entirely from the 5-number summary, which does not record the shape between those five points.

A boxplot only plots minimum,

Q_1

, median,

Q_3

, and maximum, so any two data sets sharing those values produce the same picture. Boxplots cannot show gaps, clusters, or the number of peaks. To reveal bimodality you would use a histogram, dotplot, or stemplot, which display the full shape of the distribution.

A distribution of home prices is strongly skewed to the right. Which pair of statistics best describes its center and spread, and why?

Answer: The median and the IQR, because both are resistant to the high-price outliers that a right-skewed distribution contains.

Right-skewed data have large values in the upper tail that inflate the mean and standard deviation, both of which are non-resistant. The median and IQR are resistant, so they give a more accurate summary of a typical price and the spread of the bulk of the data. Always pair the median with the IQR.

FAQ

When should I use the mean versus the median?: Use the mean and standard deviation for roughly symmetric distributions with no outliers. Use the median and IQR for skewed distributions or when outliers are present, because those statistics are resistant and better represent a typical value.
Do I include the median when finding the quartiles?: When $n$ is odd, exclude the overall median from both halves before finding $Q_1$ and $Q_3$ . When $n$ is even, the data split evenly into two halves and no value is excluded. Note that some calculators use a slightly different convention, but the AP exam expects this split-the-halves method.
Why do we divide by $n-1$ instead of $n$ for standard deviation?: Dividing by $n-1$ (the degrees of freedom) corrects for the fact that sample data tend to underestimate the true variability of the population. This gives an unbiased estimate of variance. On the AP exam, sample standard deviation always uses $n-1$ .
Are the outlier fences ever plotted on a boxplot?: No. The fences from the 1.5×IQR rule are just thresholds you calculate to decide which points are outliers. The whiskers extend only to the most extreme actual data values that are not outliers, and outliers are shown as separate dots.

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