AP Statistics Topic 1.8: Boxplots and Five-Number Summaries

01 · Start here

The concept in 60 seconds

A boxplot displays numerical summaries of one quantitative variable. It is also called a box-and-whisker plot. Read its positions on the numerical axis, rather than treating its lines as bars showing frequencies.

The box

Runs from $Q_{1}$ to $Q_{3}$ , showing the middle half of the ordered data. Its length along the numerical axis is the $\operatorname{IQR}$ .

The line inside

Marks the median. It need not sit halfway between the ends of the box.

Whiskers and points

An ordinary plot reaches the minimum and maximum. A modified plot reaches the extreme non-outliers and marks flagged observations separately.

One rule to remember: a modified boxplot’s whisker ends are observed values. Outlier fences are calculated boundaries used to decide which observations are flagged.

A boxplot makes center and spread easy to see. It does not usually show the mean, sample size, every observation or the number of peaks.

02 · Connect the numbers to a picture

Read the Cedar High travel-time boxplot

Return to the same fictional sample of $20$ Cedar High students. Each value is a student’s one-way home-to-school travel time on the Tuesday being studied, rounded to the nearest minute.

Ordered travel times, minutes:
$8$ , $10$ , $12$ , $14$ , $14$ , $15$ , $16$ , $18$ , $18$ , $18$ ,
$20$ , $22$ , $24$ , $25$ , $26$ , $28$ , $30$ , $32$ , $35$ , $42$

The five-number summary is $8$ , $14.5$ , $19$ , $27$ and $42 \,\mathrm{min}$ . Here is its boxplot:

Cedar High: locate the five summary numbers

Horizontal boxplot · $n = 20$ · $1.5\times\operatorname{IQR}$ rule

On a small screen, scroll the plot sideways to read the full scale.

No observations are flagged by the

\operatorname{IQR}

rule, so the whiskers reach the minimum of

8

and maximum of

42 \,\mathrm{min}

. The box spans

14.5\text{–}27 \,\mathrm{min}

, with median

19

. Invented teaching data.

The median line is at $19 \,\mathrm{min}$ . The box extends from $14.5$ to $27 \,\mathrm{min}$ , so $\operatorname{IQR}=27-14.5=12.5\,\mathrm{min}$ . The smallest and largest recorded times are $8$ and $42 \,\mathrm{min}$ , giving a range of $34 \,\mathrm{min}$ .

Predict: the upper whisker covers a longer interval than the lower whisker. Does that mean the upper part contains more students?

Check your prediction

No. Length along the axis shows how widely values are spread, rather than how many observations are present. Quartiles divide the ordered data into roughly equal quarters; the box covers the middle half. Ties and the quartile convention can affect exact counts in small samples.

Our quartile method takes the median of each half of the sorted data, excluding the overall median from both halves for odd sample sizes. Use the method specified in your question or by your teacher.

03 · Know what each mark represents

The five-number summary and the parts of a boxplot

Minimum · $Q_{1}$ · Median · $Q_{3}$ · Maximum

These are the five-number summary of the full dataset, written in increasing order.

Match a number to its graphical position.
Part	What to read	Common confusion
First quartile, $Q_{1}$	The smaller-value end of the box; roughly the $25^{\mathrm{th}}$ -percentile position.	It is a data position, rather than the lower outlier fence.
Median, $Q_{2}$	The line inside the box; roughly the $50^{\mathrm{th}}$ -percentile position.	It is not automatically the mean or geometric midpoint of the box.
Third quartile, $Q_{3}$	The larger-value end of the box; roughly the $75^{\mathrm{th}}$ -percentile position.	It is not the maximum.
Whisker ends	Minimum and maximum in an ordinary plot; smallest and largest non-outliers in a modified plot.	A modified plot’s whisker end need not be an overall extreme.
Separate outlier symbols	Flagged observations beyond the chosen outlier-rule boundaries.	The observations remain part of the dataset; they have not been deleted.

Proportions and lengths answer different questions

The intervals $\text{minimum}\text{–}Q_1$ , $Q_1\text{–}\text{median}$ , $\text{median}\text{–}Q_3$ and $Q_3\text{–}\text{maximum}$ represent roughly quarters of the ordered data. Those intervals can have very different numerical lengths. A long interval means those values are more spread out.

For Cedar High, the two halves of the box have lengths $19-14.5=4.5\,\mathrm{min}$ and $27-19=8\,\mathrm{min}$ . Each corresponds to roughly a quarter of the ordered sample, although their lengths differ.

Modified-plot caution: the outer quarters include any separately plotted outliers. A shortened whisker by itself does not necessarily account for $25\%$ of the observations. With ties, a closed interval bounded by quartiles can include more observations than the simple percentage guide suggests.

Horizontal or vertical: read the numerical axis

These notes use horizontal boxplots, so numerical length is measured left to right. In a vertical boxplot, the same information runs bottom to top. Box thickness perpendicular to that axis is usually a styling choice; it does not encode frequency unless the graph explicitly says it does.

04 · Build the graph carefully

Construct ordinary and modified boxplots

Use a fictional sample of $8$ visitor waits from Topic 1.7: $2$ , $4$ , $4$ , $6$ , $8$ , $8$ , $10$ and $22 \,\mathrm{min}$ . Its five-number summary is $2$ , $4$ , $7$ , $9$ and $22 \,\mathrm{min}$ .

Same $8$ waits, different plotting conventions

Common minute scale · box $4\text{–}9$ · median $7$

On a small screen, scroll the plot sideways to read the full scale.

The dashed upper fence is shown only as a construction guide. The modified plot keeps the

22

-minute observation, displayed separately. Invented teaching data.

Ordinary boxplot: display the five-number summary

Draw a numerical axis with an even scale, variable name and units.
Draw the box from $Q_{1} = 4$ to $Q_{3} = 9$ .
Draw the median line at $7$ .
Extend whiskers to the minimum of $2$ and maximum of $22$ .

An ordinary plot retains the full endpoints but does not separately identify potential outliers.

Modified boxplot: identify outliers before drawing whiskers

$\operatorname{IQR}=Q_3-Q_1$

$\text{Lower fence}=Q_1-1.5\operatorname{IQR}$

$\text{Upper fence}=Q_3+1.5\operatorname{IQR}$

Find the quartiles from the full dataset. Here $\operatorname{IQR}=9-4=5\,\mathrm{min}$ .
Calculate the fences. $\text{Lower}=4-1.5(5)=-3.5$ ; $\text{upper}=9+1.5(5)=16.5\,\mathrm{min}$ .
Flag observations strictly beyond a fence. Only $22$ is outside these boundaries.
Keep the box and median unchanged. They still come from all $8$ observations.
Draw whiskers to actual non-outlier observations. The smallest is $2$ and the largest is $10$ .
Plot the outlier separately at $22$ . Use a clear point, asterisk or other specified symbol.

The upper whisker ends at $10$ , not $16.5$ . $10\,\mathrm{min}$ is the largest recorded wait inside the fence. The $16.5$ -minute fence is a boundary used for classification, not a recorded wait.

A value exactly at a fence is not beyond it and is not flagged by this rule. If several observations are flagged, plot them all; repeated outlier values may overlap unless shown separately. Do not remove outliers and recalculate the quartiles just to construct the modified plot.

Check your drawing or software output

Verify the quartile convention, ordinary/modified plot setting, numerical axis and all plotted observations. Include enough of the axis to show outlier points. A software default may use a different quartile method, so a small discrepancy is a reason to check the method.

The $1.5\times\operatorname{IQR}$ rule is the convention used for modified boxplots in this lesson. Topic 1.7 also introduced a separate $2$ -standard-deviation rule; do not mix its boundaries into this construction.

05 · Connect location to the tail

Mean, median and clues about shape

The mean uses the size of every observation. Extreme values pull it toward their side more strongly than they pull the median. This creates useful typical relationships:

Relatively symmetric

The mean and median are generally close.

Skewed right

The mean is usually greater than the median.

Skewed left

The mean is usually less than the median.

Three examples: the mean is pulled toward the tail

$1$ dot represents $1$ observation · dashed line marks the median · dotted line marks the calculated mean

On a small screen, scroll the plot sideways to read the full scale.

These dotplots show the underlying values. The added mean lines are explicitly labeled; a standard boxplot does not provide them. Invented teaching data.

What can a boxplot suggest?

A longer extension toward larger values, a median nearer $Q_{1}$ , or high outlier points can suggest right skew. The reverse pattern can suggest left skew. Balanced sides can suggest relative symmetry.

For Cedar High, the upper whisker is $42-27=15\,\mathrm{min}$ long, compared with $14.5-8=6.5\,\mathrm{min}$ for the lower whisker. The upper half of the box is longer too. This supports a description of greater extension toward larger travel times. The mean, calculated from the raw data, is $21.35 \,\mathrm{min}$ , above the median of $19 \,\mathrm{min}$ , consistent with right skew.

Use “suggests” or “consistent with” when appropriate. A boxplot gives shape clues but hides the full distribution. $\text{Mean}>\text{median}$ alone does not prove right skew, and $\text{mean}=\text{median}$ alone does not prove symmetry. Use a dotplot or histogram when the detailed shape matters.

A standard boxplot does not display the mean. The median line supplies the median. Read a mean only if it is provided separately, computed from raw values, or explicitly shown with a labeled additional symbol.

06 · Follow the reasoning

Three worked examples

Example 1: construct Cedar High’s boxplot

Prompt: Construct a modified boxplot for the $20$ travel times and explain whether $42 \,\mathrm{min}$ should be a separate outlier point.

Summaries: minimum $8$ , $Q_{1}$ $14.5$ , median $19$ , $Q_{3}$ $27$ , maximum $42$ .
$\operatorname{IQR}$ : $27-14.5=12.5\,\mathrm{min}$ .
Fences: $14.5-1.5(12.5)=-4.25$ ; $27+1.5(12.5)=45.75\,\mathrm{min}$ .
Outliers: every recorded time is within the fences, including $42$ .
Drawing: box $14.5\text{–}27$ , median line $19$ , whiskers to $8$ and $42$ , no separate outlier symbols.

Model explanation: The $42$ -minute journey is not flagged by the $1.5\times\operatorname{IQR}$ rule because it is below the upper fence of $45.75 \,\mathrm{min}$ . The ordinary and modified plots therefore have the same appearance for these data.

Being the largest observation alone does not make a value an outlier.

Example 2: locate whiskers when a value is flagged

Prompt: For the $8$ visitor waits, give the modified plot’s whisker endpoints and the full data range.

$\operatorname{IQR} = 5 \,\mathrm{min}$ and the fences are $-3.5$ and $16.5 \,\mathrm{min}$ . The $22$ -minute wait is flagged, so the whiskers extend to $2$ and $10 \,\mathrm{min}$ . Plot $22$ as a separate point.

Model response: The modified boxplot has whiskers at $2$ and $10 \,\mathrm{min}$ and a high outlier point at $22 \,\mathrm{min}$ . The full data range is $22-2=20\,\mathrm{min}$ . The $8$ -minute distance between the whisker ends is not the full range because it excludes the flagged observation.

The maximum remains $22$ in the five-number summary. A modified plot changes how it is shown, rather than changing the underlying dataset.

Example 3: read a boxplot and describe it in context

A separate fictional sample of $8$ students has reading-time summaries minimum $12$ , $Q_{1}$ $18$ , median $22$ , $Q_{3}$ $30$ and maximum $45 \,\mathrm{min}$ .

Read the summaries: $8$ recorded reading times

Horizontal boxplot · $n = 8$ · $1.5\times\operatorname{IQR}$ rule

On a small screen, scroll the plot sideways to read the full scale.

The box spans

18\text{–}30 \,\mathrm{min}

and the median is

22 \,\mathrm{min}

. The

\operatorname{IQR}

rule flags no observations, so the whiskers reach

12

and

45 \,\mathrm{min}

. Invented teaching data.

Prompt: Describe center, spread and the visible shape clue. Can you calculate the exact mean from this graph alone?

Center: $\text{median} = 22 \,\mathrm{min}$ .
Spread: $\operatorname{IQR}=30-18=12\,\mathrm{min}$ ; $\text{full range}=45-12=33\,\mathrm{min}$ .
Shape clue: the graph extends farther toward larger reading times. That suggests right skew; it does not reveal the number of peaks.
Limit: the exact mean cannot generally be recovered from the boxplot alone.

Model response: For $\text{these }8\text{ sampled students}$ , the median reading time is $22 \,\mathrm{min}$ and the middle half spans about $18\text{–}30 \,\mathrm{min}$ , an $\operatorname{IQR}$ of $12 \,\mathrm{min}$ . Recorded times extend from $12$ to $45 \,\mathrm{min}$ . The longer extension toward larger values suggests right skew. The standard boxplot does not provide an exact mean.

The figure has no flagged observations under the $1.5\times\operatorname{IQR}$ rule: its fences are $0$ and $48 \,\mathrm{min}$ . “No points drawn” only means no outliers were identified by the plot’s stated rule, rather than proving there are no unusual observations under every possible criterion.

07 · Say what the display supports

Explain a boxplot in context—and know its limits

A useful description names the variable, recorded group, numerical evidence and units. Start with the median and $\operatorname{IQR}$ , then add full extent, flagged observations and cautious shape clues when relevant.

Sentence starter: “For [recorded group], the median [variable] is [value and units], and the middle half extends from [ $Q_{1}$ ] to [ $Q_{3}$ ], giving an $\operatorname{IQR}$ of [difference and units]. The graph also shows [supported feature].”

What a standard boxplot does and does not tell you

Match your claim to the available information.
You can usually read or calculate	You generally cannot recover from the plot alone
Median, quartiles and $\operatorname{IQR}$ .	The exact mean and standard deviation.
Full endpoints if all extreme points are shown; full range using whiskers and any outlier symbols.	Every individual value or exact frequencies within a numerical interval.
Flagged outliers under the stated plotting rule.	Whether a flagged point is an error, or why it occurred.
Shape clues from unequal lengths and isolated points.	The number of peaks, specific internal gaps, clusters or a guarantee of normality.
Approximate relative positions in quarters of the ordered data.	Sample size unless labeled, or an exact percentage at an arbitrary cutoff.

Identical boxes can hide different observations

Dotplot above each ordinary boxplot · common point scale · same five-number summary

On a small screen, scroll the plot sideways to read the full scale.

The values differ while the minimum, quartiles, median and maximum stay the same. The boxes alone do not reveal the difference in means. Invented teaching data.

These $2$ invented sets share the five-number summary $0$ , $2$ , $5$ , $8$ and $10 \,\mathrm{points}$ , yet their observations and means differ. Their ordinary boxplots therefore look identical. A graph of the full observations preserves information that the summary omits.

See the

2

sets and check the means

Set A: $0$ , $1$ , $2$ , $2$ , $4$ , $5$ , $5$ , $6$ , $8$ , $8$ , $9$ , $10$ . $\text{Total} = 60$ ; $\text{mean} = 5 \,\mathrm{points}$ .
Set B: $0$ , $2$ , $2$ , $2$ , $5$ , $5$ , $5$ , $8$ , $8$ , $8$ , $10$ , $10$ . $\text{Total} = 65$ ; $\text{mean} \approx 5.42 \,\mathrm{points}$ .

For both sets, $Q_1=\frac{2+2}{2}=2$ , $\text{median}=\frac{5+5}{2}=5$ , and $Q_3=\frac{8+8}{2}=8$ . The extreme values are also the same.

Can five summary numbers always give a complete modified plot?

Not when an endpoint is flagged and the remaining observations are unknown. The summary lets you locate the box and median and test the minimum and maximum against fences. It does not necessarily give the extreme non-outlier values or all separately plotted outliers.

For example, a summary of $0$ , $10$ , $12$ , $14$ and $30$ has $\operatorname{IQR}$ $4$ and fences $4$ and $20$ . The minimum $0$ and maximum $30$ are flagged, but the modified whisker endpoints require additional data. Do not invent observations at $4$ or $20$ .

If a task supplies a graph rather than exact numbers, respect its scale and use “about” where readings are estimates. Boxplot summaries alone do not establish a cause or justify generalizing from a sample to every member of a population.

08 · Find the wrong turn

Common mistakes and how to fix them

Check the meaning of each line and number.
Mistake	Why it fails	Better approach
Ending a whisker at the fence	A fence is a calculated boundary, not necessarily an observation.	Use the extreme observed value within the fences.
Calling the line inside the box the mean	That line marks the median.	Read a mean only when separately supplied or explicitly labeled.
Saying a longer box segment contains more observations	Length encodes numerical spread.	Use the approximate quarter/half guide and account for ties.
Using whisker-to-whisker distance as the full range in a modified plot	Separate extreme points can be outside the whiskers.	Find the overall minimum and maximum, including flagged points.
Removing outliers before calculating quartiles for the plot	It changes the dataset and the summary.	Use all observations, then display flagged values separately.
Treating a balanced boxplot as proof of symmetry or normality	The summary hides the detailed pattern.	Describe the visible balance cautiously and inspect a fuller graph if needed.
Mixing the $2$ -standard-deviation rule into an $\operatorname{IQR}$ -based boxplot	The rules can flag different observations.	Use the plotting convention stated in the task.

09 · Work first, then check

Six practice questions with hints and solutions

Use the median-of-halves quartile method from this lesson unless a question provides the summary. All data are invented teaching examples.

1. Build a modified boxplot from raw values

$10$ students’ recorded reading times are $6$ , $8$ , $9$ , $10$ , $12$ , $12$ , $14$ , $15$ , $16$ and $28 \,\mathrm{min}$ . Find the five-number summary, fences, whisker endpoints and any outlier points.

Need a hint?

The $2\text{ middle values}$ are both $12$ . Take the median of each $5\text{-value half}$ . Then calculate $1.5\times\operatorname{IQR}$ .

Show the solution

$\text{Minimum} = 6$ , $Q_{1} = 9$ , $\text{median} = 12$ , $Q_{3} = 15$ , $\text{maximum} = 28 \,\mathrm{min}$ . $\operatorname{IQR}=15-9=6\,\mathrm{min}$ . $\text{Lower fence}=9-1.5(6)=0$ and $15+1.5(6)=24\,\mathrm{min}$ .

Only $28$ is flagged. Draw the box from $9$ to $15$ , the median line at $12$ , whiskers to $6$ and $16$ , and a separate point at $28$ on a labeled minute scale.

2. Interpret unequal box segments

A reading-time boxplot has $Q_{1} = 18$ , $\text{median} = 22$ and $Q_{3} = 30 \,\mathrm{min}$ . Find its $\operatorname{IQR}$ and the length of each half of the box. Does the longer half prove that more students are in that part?

Need a hint?

Subtract the endpoints of each interval. Then distinguish numerical length from proportion.

Show the solution

$\operatorname{IQR}=30-18=12\,\mathrm{min}$ . $\text{Lower box segment}=22-18=4\,\mathrm{min}$ ; $\text{upper segment}=30-22=8\,\mathrm{min}$ . Each represents roughly a quarter of the ordered data. The upper segment’s greater length shows greater spread over that quarter, rather than proving a greater number of students. Exact closed-interval counts depend on ties and the quartile convention.

3. Range versus whisker span

A modified wait-time boxplot has whiskers at $2$ and $10 \,\mathrm{min}$ , a box from $4$ to $9$ , median $7$ , and $1\text{ outlier point}$ at $22$ . Calculate the full range and $\operatorname{IQR}$ . What is the dataset’s maximum?

Need a hint?

Keep the separate point when identifying the full extremes.

Show the solution

$\text{Maximum} = 22 \,\mathrm{min}$ ; $\text{minimum} = 2$ . $\text{Full range}=22-2=20\,\mathrm{min}$ . $\operatorname{IQR}=9-4=5\,\mathrm{min}$ . The whisker span $10-2=8\,\mathrm{min}$ excludes the outlier and is not the full range.

4. Mean–median reasoning

A distribution of recorded scores has a long tail toward smaller values. Its median is $23 \,\mathrm{points}$ . Would you usually expect its mean to be above or below $23$ ? Can you find the exact mean from a standard boxplot alone?

Need a hint?

The mean is pulled toward the tail. A typical relationship is different from an exact calculation.

Show the solution

This is a left-skewed pattern, so the mean would usually be below $23 \,\mathrm{points}$ . The lower scores pull the mean toward the left tail. A standard boxplot alone generally does not give the exact mean; raw values or an additional mean summary would be needed.

5. What is missing from a summary?

A dataset has five-number summary $0$ , $10$ , $12$ , $14$ and $30$ . Can you determine a complete modified boxplot using only these five numbers? Explain what you can draw and what additional information you need.

Need a hint?

Calculate the fences, test the minimum and maximum, and ask whether you know the next observation inside each fence.

Show the solution

$\operatorname{IQR} = 4$ , so the fences are $4$ and $20$ . You can draw the box from $10$ to $14$ and the median line at $12$ . The extreme observations $0$ and $30$ are flagged.

You need raw data or additional information identifying the smallest and largest non-outliers and all flagged observations. There might be other outliers between an extreme and its fence. Ending whiskers at $4$ and $20$ would invent endpoints not known to be observed.

6. Diagnose a student’s drawing

For the $8$ visitor waits, a student draws the box at $4\text{–}9$ , places the median line at $6.5$ , extends the upper whisker to $16.5$ and erases the $22$ -minute wait. Identify three errors and correct them.

Need a hint?

The box midpoint is not necessarily the median. A fence is not a whisker endpoint. Flagged observations remain in the display.

Show the solution

The median is $7 \,\mathrm{min}$ , found from $\frac{6+8}{2}$ , rather than the geometric box midpoint $6.5$ .
The upper whisker must end at the largest non-outlier, $10 \,\mathrm{min}$ , rather than the $16.5$ -minute fence.
Keep $22 \,\mathrm{min}$ as a separate outlier point. Do not erase the observation.

The lower whisker is at $2 \,\mathrm{min}$ and the box remains $4\text{–}9 \,\mathrm{min}$ .

10 · Keep the main tools together

Quick revision checklist

Before drawing or interpreting a boxplot.
Check	What to do
Order and method	Sort raw values; use the stated quartile convention.
Five-number summary	Minimum, $Q_{1}$ , median, $Q_{3}$ , maximum—keep the full dataset.
Plot type	Ordinary: whiskers to extremes. Modified: apply the stated rule and show flagged observations separately.
$\operatorname{IQR}$ and fences	$\operatorname{IQR}=Q_3-Q_1$ ; fences $Q_1-1.5\operatorname{IQR}$ and $Q_3+1.5\operatorname{IQR}$ .
Whisker endpoints	Use actual observations; equality at a fence is not beyond it.
Axis and labels	Even scale, variable name, units and all outlier positions.
Interpretation	Median for center; $\operatorname{IQR}$ for middle-half spread; include outliers when finding the full range.
Shape and mean	Use cautious shape clues. Symmetric: $\text{mean}\approx\text{median}$ ; right skew: usually $\text{mean}>\text{median}$ ; left skew: usually $\text{mean}<\text{median}$ .
Limits	A standard boxplot generally does not give the mean, standard deviation, $n$ , individual values, peaks or gaps.

Before submitting: did I place the median correctly, distinguish fences from observations, retain outlier points and explain the graph with context and units?

11 · Put the skills together

Final understanding check

$12$ sampled deliveries have recorded waits of $5$ , $6$ , $7$ , $8$ , $9$ , $10$ , $10$ , $11$ , $12$ , $13$ , $14$ and $30 \,\mathrm{min}$ .

Find the five-number summary and $\operatorname{IQR}$ using the median-of-halves method.
Find the $1.5\times\operatorname{IQR}$ fences and construct a modified boxplot, labeling the axis and outlier points.
Calculate the full range and describe center and spread in context.
The sample mean, calculated from the raw values, is $11.25 \,\mathrm{min}$ . How does it compare with the median, and why is this relationship reasonable here? Does the boxplot alone reveal the exact cause of the long wait?

Show the full solution and completed plot

1. $\text{Minimum} = 5$ ; $Q_1=\frac{7+8}{2}=7.5$ ; $\text{median}=\frac{10+10}{2}=10$ ; $Q_3=\frac{12+13}{2}=12.5$ ; $\text{maximum} = 30 \,\mathrm{min}$ . $\operatorname{IQR}=12.5-7.5=5\,\mathrm{min}$ .

2. $\text{Lower fence}=7.5-1.5(5)=0$ and $12.5+1.5(5)=20\,\mathrm{min}$ . Only $30$ is flagged. Draw the box from $7.5$ to $12.5$ , median line at $10$ , whiskers to $5$ and $14$ , and an outlier point at $30$ .

Completed modified boxplot: delivery waits

$n = 12$ · box $7.5\text{–}12.5$ · median $10$ · fences $0$ and $20$

On a small screen, scroll the plot sideways to read the full scale.

The

30

-minute wait is outside the upper fence. Its separate point remains part of the data; the full range is

25 \,\mathrm{min}

. Invented teaching data.

3. $\text{Full range}=30-5=25\,\mathrm{min}$ . For these $12$ sampled deliveries, the middle recorded wait is $10 \,\mathrm{min}$ , and the middle half spans about $7.5\text{–}12.5 \,\mathrm{min}$ , an $\operatorname{IQR}$ of $5 \,\mathrm{min}$ . The $30$ -minute wait is flagged by the $1.5\times\operatorname{IQR}$ rule.

4. The mean of $11.25 \,\mathrm{min}$ is greater than the median of $10 \,\mathrm{min}$ . This is consistent with the high isolated observation pulling the mean toward larger values. The boxplot does not establish the reason for that wait or prove a cause.

Ready to move on? You should be able to construct and read a modified boxplot, distinguish summaries from observations, explain the typical mean–median relationships and state what a boxplot leaves out.

12 · Connect this lesson to the next

Continue learning

Topic 1.9: Comparisons of the Distributions for One Quantitative Variable →

Bring multiple graphs together to compare distributions and justify claims, then use $z$ -scores to compare relative positions.

Review summary-statistic calculations · Review modified-plot construction · Back to the lesson overview

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Graphical Representations of Summary Statisticsfor One Quantitative Variable

By the end of this lesson, you should be able to:

The concept in 60 seconds

The box

The line inside

Whiskers and points

Read the Cedar High travel-time boxplot

The five-number summary and the parts of a boxplot

Proportions and lengths answer different questions

Horizontal or vertical: read the numerical axis

Construct ordinary and modified boxplots

Ordinary boxplot: display the five-number summary

Modified boxplot: identify outliers before drawing whiskers

Check your drawing or software output

Mean, median and clues about shape

Relatively symmetric

Skewed right

Skewed left

What can a boxplot suggest?

Three worked examples

Example 1: construct Cedar High’s boxplot

Example 2: locate whiskers when a value is flagged

Example 3: read a boxplot and describe it in context

Explain a boxplot in context—and know its limits

What a standard boxplot does and does not tell you

Can five summary numbers always give a complete modified plot?

Common mistakes and how to fix them

Six practice questions with hints and solutions

1. Build a modified boxplot from raw values

2. Interpret unequal box segments

3. Range versus whisker span

4. Mean–median reasoning

5. What is missing from a summary?

6. Diagnose a student’s drawing

Quick revision checklist

Final understanding check

Continue learning