Boxplots and Five-Number Summary: How to Calculate Q1, Q3, IQR and Outliers

 

BOXPLOTS AND THE FIVE-NUMBER SUMMARY: THE STOPS–FENCES–SAY METHOD FOR COMPARING DATA (AP STATISTICS, GCSE, A-LEVEL AND IB MATHS)

A boxplot is one of the most useful statistical graphs for understanding a data distribution. It shows the median, quartiles, interquartile range, spread and potential outliers in a compact visual form.

Learn how to calculate the five-number summary, find Q1 and Q3, construct a box-and-whisker plot, apply the 1.5 × IQR outlier rule, interpret skewness and compare two distributions using worked examples and examination-style questions.

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TWO RUNNING CLUBS, ALMOST THE SAME AVERAGE, BUT DIFFERENT STORIES

Tobias manages a local running club. A rival club has challenged his team to a relay race, and he must decide which runners would make the most consistent training group.

He collects the 5K running times, in minutes, for 15 runners from Riverside Club and 14 runners from Hillside Club.

The average times are:

Riverside: 31.13 minutes

Hillside: 31.43 minutes

At first glance, there is very little difference.

But an average cannot tell the entire story.

Riverside has a runner who took 52 minutes, while Hillside's times are more closely grouped. Their averages are similar, but their distributions are not identical.

This is where a boxplot becomes useful.

A boxplot, also called a box-and-whisker plot, allows us to compare the centre, spread and unusual observations in numerical data without examining every individual value.

The same technique appears in AP Statistics, GCSE Mathematics, A-level Statistics, IB Mathematics and introductory college statistics.

The objective is not simply to draw a box. It is to understand what the box tells us about the data.

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1. WHAT IS A BOXPLOT? UNDERSTANDING THE FIVE-NUMBER SUMMARY

A boxplot is constructed using five important statistical values called the five-number summary.

These are:

1. Minimum: The smallest observation in the data set.

2. First quartile (Q1): The lower quartile, marking approximately the 25th percentile.

3. Median (Q2): The middle value, dividing the ordered observations into two halves.

4. Third quartile (Q3): The upper quartile, marking approximately the 75th percentile.

5. Maximum: The largest observation in the data set.

The five-number summary provides a quick description of the centre and spread of a numerical distribution.

How the five values appear in a boxplot

• The box begins at Q1.

• The box ends at Q3.

• A line inside the box marks the median.

• Whiskers extend towards the extreme observations, according to the type of boxplot being used.

• In a modified boxplot, potential outliers are shown as individual points beyond the whiskers.

The box contains approximately the middle 50% of the observations.

Two additional measures are particularly important.

Range = Maximum − Minimum

The range measures the distance between the smallest and largest observations.

Interquartile Range (IQR) = Q3 − Q1

The IQR measures the spread of the middle 50% of the data.

The range is sensitive to extreme observations. The IQR is generally more resistant because it depends on the two quartiles rather than the endpoints.

An important quartile concept

Quartiles divide an ordered data set into four sections containing approximately equal proportions of observations.

Each section represents approximately 25% of the data.

However, the sections do not need to have equal widths.

For example, if the first quarter of a data set covers only three units while the final quarter covers twenty units, the first section is more concentrated.

A wider section does not contain more observations. It indicates greater spread.

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2. THE STOPS–FENCES–SAY METHOD

Many students understand the definition of a boxplot but lose marks when they have to construct or interpret one.

The problem usually comes down to three tasks:

• Finding the quartiles correctly.

• Identifying potential outliers.

• Comparing distributions using appropriate statistical language.

A simple three-part method makes these tasks easier.

STOPS: Find the five-number summary.

FENCES: Calculate the outlier boundaries using the 1.5 × IQR rule.

SAY: Compare the centre, spread, shape and outliers in the context of the question.

Let us apply this method to Tobias's running-club data.

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3. STOPS: HOW TO CALCULATE THE FIVE-NUMBER SUMMARY

The first requirement is to arrange the observations in ascending order.

Riverside Club: 15 running times

22, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 36, 38, 52

All values are in minutes.

Step 1: Find the minimum and maximum.

Minimum = 22

Maximum = 52

Step 2: Find the median.

There are 15 observations.

For an odd number of observations:

Median position = (n + 1) ÷ 2

= (15 + 1) ÷ 2

= 8th observation

Therefore:

Median = 30

Step 3: Find the lower and upper halves.

When the number of observations is odd, the median is excluded from both halves under the median-of-halves method.

Lower half:

22, 24, 25, 26, 27, 28, 29

Upper half:

31, 32, 33, 34, 36, 38, 52

Step 4: Calculate Q1.

The median of the lower half is its fourth observation.

Q1 = 26

Step 5: Calculate Q3.

The median of the upper half is its fourth observation.

Q3 = 34

Therefore, the five-number summary is:

Minimum = 22

Q1 = 26

Median = 30

Q3 = 34

Maximum = 52

Calculate the IQR:

IQR = Q3 − Q1

= 34 − 26

= 8 minutes

Hillside Club: 14 running times

The ordered observations are:

25, 27, 28, 29, 30, 30, 31, 32, 33, 33, 34, 35, 36, 37

Step 1: Find the minimum and maximum.

Minimum = 25

Maximum = 37

Step 2: Calculate the median.

There are 14 observations, so the median is the average of the seventh and eighth observations.

Median = (31 + 32) ÷ 2

= 31.5

Step 3: Divide the observations into two equal halves.

Lower half:

25, 27, 28, 29, 30, 30, 31

Upper half:

32, 33, 33, 34, 35, 36, 37

Step 4: Calculate the quartiles.

The fourth observation in the lower half is 29.

Q1 = 29

The fourth observation in the upper half is 34.

Q3 = 34

Therefore:

Minimum = 25

Q1 = 29

Median = 31.5

Q3 = 34

Maximum = 37

Calculate the IQR:

IQR = 34 − 29

= 5 minutes

Compare the five-number summaries

Statistic	Riverside	Hillside
Minimum	22	25
Q1	26	29
Median	30	31.5
Q3	34	34
Maximum	52	37
IQR	8	5

Riverside has a wider middle 50% of running times. Hillside's central observations are more closely grouped.

IMPORTANT: DIFFERENT METHODS OF CALCULATING QUARTILES

Quartile calculations can differ depending on the method used.

The median-of-halves method used above is a common classroom approach.

Other methods use percentile-position formulas or interpolation.

For example, one common inclusive percentile-position convention uses:

Q1 position = 1 + 0.25(n − 1)

Q3 position = 1 + 0.75(n − 1)

For Hillside, this gives:

Q1 position = 4.25

Q3 position = 10.75

Interpolating between adjacent observations gives:

Q1 = 28.25

Q3 = 33.75

These differ from the median-of-halves results of 29 and 34.

This is why Excel, Google Sheets, statistical software and calculators can sometimes produce different quartiles for the same observations.

Do not assume that every examination board or software package uses the same convention. Follow the method specified by your teacher, examination board or calculator instructions.

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4. FENCES: HOW TO IDENTIFY OUTLIERS USING THE 1.5 × IQR RULE

An outlier is an observation that is unusually far from the rest of the data.

A common statistical convention for identifying potential outliers is the 1.5 × IQR rule.

It uses two boundaries called fences.

Lower fence = Q1 − 1.5 × IQR

Upper fence = Q3 + 1.5 × IQR

Any observation below the lower fence or above the upper fence is flagged as a potential outlier.

Notice that the fences are calculated boundaries. They are not necessarily observations in the original data.

Riverside outlier calculation

We already know:

Q1 = 26

Q3 = 34

IQR = 8

Calculate:

1.5 × IQR = 1.5 × 8 = 12

Lower fence:

= 26 − 12

= 14

Upper fence:

= 34 + 12

= 46

The smallest observation is 22, which is above the lower fence.

Therefore, there are no lower outliers.

The largest observation is 52, which is greater than the upper fence of 46.

Therefore, 52 is a potential upper outlier.

The largest observation that remains inside the upper fence is 38.

In a modified boxplot, the upper whisker ends at 38, and 52 is drawn separately as a point.

Hillside outlier calculation

Q1 = 29

Q3 = 34

IQR = 5

1.5 × IQR = 7.5

Lower fence:

= 29 − 7.5

= 21.5

Upper fence:

= 34 + 7.5

= 41.5

The minimum is 25, and the maximum is 37.

Both lie within the fences.

Therefore, Hillside has no potential outliers under the 1.5 × IQR rule.

Its whiskers extend to 25 and 37.

Final outlier comparison

Club	Lower fence	Upper fence	Outliers
Riverside	14	46	52
Hillside	21.5	41.5	None

Is every outlier a mistake?

No.

The 1.5 × IQR rule identifies observations that deserve attention. It does not prove that a measurement is incorrect.

A 52-minute running time may be genuine. The runner might have stopped, experienced an injury or simply run more slowly.

An unusual observation should be investigated rather than automatically deleted.

Also remember that not every course uses modified boxplots. Some boxplots extend whiskers directly to the minimum and maximum. Always distinguish the plotting convention being used.

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5. HOW TO DRAW A BOX-AND-WHISKER PLOT

Once the five-number summary and outlier calculations are complete, constructing the graph becomes straightforward.

Follow these steps:

1. Draw a horizontal number line with an appropriate numerical scale.

2. Mark Q1 and Q3.

3. Draw a rectangular box between the two quartiles.

4. Draw a vertical line inside the box at the median.

5. Draw the whiskers to the most extreme observations inside the fences if using a modified boxplot.

6. Plot potential outliers separately.

7. Label the axis and include the measurement units.

For Tobias's data, the important plotting positions are:

Riverside

• Left whisker: 22

• Q1: 26

• Median: 30

• Q3: 34

• Right whisker: 38

• Outlier: 52

Hillside

• Left whisker: 25

• Q1: 29

• Median: 31.5

• Q3: 34

• Right whisker: 37

• Outliers: None

Both plots should use the same horizontal scale to make comparison meaningful.

A boxplot drawn without a labelled scale can be misleading because the visual distances between quartiles and whiskers are essential to interpretation.

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6. HOW TO INTERPRET A BOXPLOT: CENTRE, SPREAD, SHAPE AND OUTLIERS

A boxplot is not simply a collection of lines and rectangles. Each feature communicates something about the distribution.

A. Centre: Read the median

The median is the line inside the box.

It divides the ordered observations into two halves.

A higher median indicates a higher typical numerical value.

Remember that a higher numerical value does not always mean better performance. For running times, a lower median means faster typical performance.

B. Spread: Compare the IQR

The width of the box represents the interquartile range.

A wider box means the middle 50% of the observations are more spread out.

A narrower box means the central observations are more concentrated.

Riverside:

IQR = 8 minutes

Hillside:

IQR = 5 minutes

Therefore, Riverside has greater variability in its middle 50% of running times.

C. Shape: Identify possible skewness

Skewness describes whether a distribution has a longer tail on one side.

Approximately symmetric distribution

The median is roughly central within the box, and the whiskers are approximately equal in length.

Right-skewed distribution (positive skew)

The upper side tends to extend further than the lower side. The upper whisker may be longer, and the median may sit closer to Q1.

Left-skewed distribution (negative skew)

The lower side tends to extend further. The lower whisker may be longer, and the median may sit closer to Q3.

These are visual clues, not definitive proof of distribution shape. A boxplot compresses the data, so a histogram or dot plot can provide additional evidence.

D. Outliers: Examine individual points

A point beyond a modified boxplot's whisker indicates a potential outlier.

Its position tells us whether it is unusually low or unusually high.

For Riverside, 52 is a high outlier.

What do the running-club boxplots tell us?

Riverside's median is 30 minutes, while Hillside's is 31.5 minutes.

The two central distributions are approximately symmetric based on their quartile and whisker positions.

Riverside has a high outlier at 52 minutes, but its central box and whiskers do not show a pronounced right skew.

Hillside has a smaller IQR and no identified outliers.

This distinction matters: an outlier on one side does not automatically prove that the entire distribution is skewed in that direction.

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7. HOW TO COMPARE TWO BOXPLOTS IN AP STATISTICS AND GCSE EXAMINATIONS

This is one of the most important applications of boxplots.

An examination question may show two boxplots and ask:

• Which group has the higher median?

• Which group has greater variation?

• Which distribution is more symmetric?

• Are there any outliers?

• Which group is more consistent?

• What can be concluded from the comparison?

A strong answer should cover four features.

CENTRE: Compare the medians.

SPREAD: Compare the IQRs.

SHAPE: Discuss symmetry or skewness where the graph provides evidence.

OUTLIERS: Identify unusual observations and explain their context.

Worked comparison: Riverside versus Hillside

Consider this model answer:

"The typical Riverside runner completes the 5K in less time than the typical Hillside runner, with median times of 30 and 31.5 minutes respectively. However, Riverside has greater variation in its central running times, with an IQR of 8 minutes compared with Hillside's 5 minutes. Both distributions have approximately symmetric central box-and-whisker structures, although Riverside has one unusually slow runner at 52 minutes, which is a potential outlier under the 1.5 × IQR rule. Hillside's smaller IQR indicates more consistent central running times."

This answer is effective because it:

• Uses actual numerical evidence.

• Compares both groups.

• Identifies the statistical measure being discussed.

• Includes the real-world context.

• Avoids confusing an outlier with skewness.

A reusable boxplot comparison sentence

"The median of Group A is higher/lower than Group B (___ compared with ), while its IQR is larger/smaller ( compared with ___). The distribution appears approximately symmetric/right-skewed/left-skewed, and it contains ___ potential outliers."

Adapt the wording to the actual graph. Do not describe skewness if the evidence is insufficient.

The median-overlap rule: a useful visual clue

Students sometimes use the position of one group's median relative to the other group's box to judge how different the distributions appear.

If the median of one group lies outside the other group's box, that can suggest a noticeable difference in location.

However, if both medians lie inside the other group's boxes, the central distributions overlap.

In Tobias's example:

Riverside's median = 30, which lies inside Hillside's box (29 to 34).

Hillside's median = 31.5, which lies inside Riverside's box (26 to 34).

This indicates substantial overlap in the central portions of the distributions.

But boxplot overlap is not a formal significance test. It cannot establish whether population medians differ statistically.

For that, an appropriate inferential procedure is required.

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8. WHY MEDIAN AND IQR ARE USEFUL WHEN DATA CONTAINS OUTLIERS

Consider the Riverside data again.

The original mean is:

467 ÷ 15 = 31.13 minutes.

The original median is:

30 minutes.

Now consider what happens if the 52-minute observation is replaced by a less extreme value, such as 90.

The mean changes substantially, but the median remains 30.

The range also increases sharply.

This illustrates why the mean and range are sensitive to extreme values, while the median and IQR are generally more resistant.

There is an important distinction here.

Making an existing extreme endpoint more extreme does not necessarily change the quartiles.

However, removing an observation changes the number and positions of the observations. The quartiles may then change.

For example, removing the 52-minute observation gives:

New median = 29.5

New Q1 = 26

New Q3 = 33

New IQR = 7

The IQR has changed from 8 to 7.

Therefore, it is more accurate to say that the IQR is resistant to extreme values, not that it is guaranteed to remain unchanged whenever an outlier is removed.

When should you use median and IQR?

Use the median and IQR when:

• The distribution is strongly skewed.

• There are unusually extreme observations.

• You want a resistant measure of centre and spread.

The mean and standard deviation are often more appropriate when the distribution is reasonably symmetric and free from influential outliers.

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9. WHAT BOXPLOTS CANNOT SHOW: THE BIMODAL DISTRIBUTION PROBLEM

A boxplot summarises data using five important positions.

That makes it efficient, but it also removes information.

Consider these ten observations:

1, 1, 2, 2, 2, 10, 10, 11, 11, 11

The median is:

(2 + 10) ÷ 2 = 6

Using the median-of-halves method:

Q1 = 2

Q3 = 11

IQR = 11 − 2 = 9

The boxplot shows a box from 2 to 11, with the median at 6.

However, there is a surprising feature.

There are no observations near 6.

Instead, the data contains two distinct clusters:

• A lower cluster around 1 and 2.

• An upper cluster around 10 and 11.

The distribution is bimodal: it has two concentrations of observations.

A histogram or dot plot would make this structure much easier to see.

A boxplot alone cannot reveal every cluster, gap or peak.

Which statistical graph should you use?

Boxplot: Comparing medians, quartiles, spread and potential outliers across groups.

Histogram: Understanding the overall distribution, concentration, gaps and skewness.

Dot plot: Examining individual observations and small data sets.

Stem-and-leaf plot: Displaying numerical values while preserving much of the original data.

For a proper statistical investigation, using more than one graph can provide a much clearer picture.

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10. CREATING BOXPLOTS USING EXCEL, GOOGLE SHEETS, CALCULATORS AND PYTHON

You do not have to calculate every boxplot manually. Statistical tools can construct them, although you should understand the quartile method used.

Excel

Modern versions of Microsoft Excel include a Box and Whisker chart.

General procedure:

1. Enter numerical observations into a column.

2. Select the data.

3. Open Insert.

4. Choose the Statistical Chart options.

5. Select Box and Whisker.

Check the chart settings and quartile conventions before comparing the output with hand calculations.

Google Sheets

Google Sheets can be used to calculate the five-number summary through spreadsheet functions.

For example:

• MIN calculates the minimum.

• MEDIAN calculates the median.

• MAX calculates the maximum.

• QUARTILE functions calculate quartiles.

The results depend on the chosen quartile convention.

A boxplot can also be constructed using a suitable charting approach.

TI-83 and TI-84 graphing calculators

These calculators support statistical plots.

A general procedure is:

1. Enter observations in a list.

2. Open the statistical plotting menu.

3. Select a boxplot or modified boxplot.

4. Enable the plot.

5. Adjust the viewing window to display all observations.

The modified boxplot option identifies potential outliers separately.

Python

Python can generate boxplots using libraries such as Matplotlib.

Example:

import matplotlib.pyplot as plt

riverside = [22, 24, 25, 26, 27, 28, 29,
             30, 31, 32, 33, 34, 36, 38, 52]

hillside = [25, 27, 28, 29, 30, 30, 31,
            32, 33, 33, 34, 35, 36, 37]

plt.boxplot([riverside, hillside],
            labels=["Riverside", "Hillside"])

plt.ylabel("5K running time (minutes)")
plt.title("Comparison of Running Times")
plt.show()

Software-generated quartiles can differ from classroom calculations because different programs use different conventions.

Always check the quartile method before interpreting or reporting the graph.

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11. TEN COMMON BOXPLOT MISTAKES IN STATISTICS EXAMINATIONS

1. Finding quartiles without sorting the observations.

Always arrange numerical data in ascending order first.

2. Including the median in both halves when the sample size is odd.

Follow the quartile method specified by your course.

3. Assuming that a wider box contains more observations.

The box always represents approximately the middle 50%.

4. Confusing the IQR with the range.

IQR = Q3 − Q1.

Range = Maximum − Minimum.

5. Drawing modified-boxplot whiskers all the way to the minimum and maximum.

Check for potential outliers first.

6. Claiming that a group has a higher average when you are actually comparing medians.

Use the correct statistical terminology.

7. Declaring a distribution skewed simply because it contains an outlier.

Consider the median position and both whisker lengths.

8. Forgetting the numerical scale and measurement units.

A boxplot must have a clearly labelled axis.

9. Automatically deleting outliers.

Investigate unusual observations before deciding how to handle them.

10. Assuming that a boxplot displays the mean or sample size.

A standard boxplot does not directly show either value.

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12. BOXPLOT PRACTICE QUESTIONS WITH ANSWERS

Test your understanding of five-number summaries, quartiles, IQR, outlier detection and boxplot interpretation.

Question 1: Calculate the five-number summary

Find the five-number summary and IQR for:

4, 7, 8, 9, 11, 12, 14, 20

Use the median-of-halves method.

Also determine whether any observation is a potential outlier.

Question 2: Apply the 1.5 × IQR rule

A data set has this five-number summary:

12, 18, 25, 31, 60

Determine whether the minimum or maximum is a potential outlier.

Show the fence calculations.

Question 3: Interpret skewness

A boxplot has its median close to Q1, while the upper whisker is considerably longer than the lower whisker.

What type of skewness does this suggest?

Question 4: Understand quartiles

True or false?

The section between Q1 and the median contains more observations than the section between the median and Q3 if its box segment is wider.

Explain your answer.

Question 5: Understand limitations

Why cannot a standard boxplot directly tell you the sample size or mean?

Question 6: Compare two distributions

Two classes sit the same examination.

Class P:

Median = 68

IQR = 10

Class Q:

Median = 68

IQR = 22

Write a statistical comparison of the two classes.

ANSWERS AND EXPLANATIONS

Answer 1

There are eight observations.

Median = (9 + 11) ÷ 2 = 10

Lower half:

4, 7, 8, 9

Q1 = (7 + 8) ÷ 2 = 7.5

Upper half:

11, 12, 14, 20

Q3 = (12 + 14) ÷ 2 = 13

Five-number summary:

4, 7.5, 10, 13, 20

IQR = 13 − 7.5 = 5.5

Lower fence:

7.5 − (1.5 × 5.5) = −0.75

Upper fence:

13 + (1.5 × 5.5) = 21.25

All observations lie within the fences.

There are no potential outliers.

Answer 2

Q1 = 18

Q3 = 31

IQR = 31 − 18 = 13

1.5 × IQR = 19.5

Lower fence:

18 − 19.5 = −1.5

Upper fence:

31 + 19.5 = 50.5

The minimum of 12 is above −1.5, so it is not a potential outlier.

The maximum of 60 exceeds 50.5, so it is a potential upper outlier.

Answer 3

The distribution appears right-skewed, also called positively skewed.

The longer upper whisker indicates a more extended spread towards higher values.

Answer 4

False.

Each section represents approximately 25% of the observations.

A wider section indicates that the values are more spread out, not that it contains more observations.

Answer 5

A standard boxplot displays the five-number summary rather than every individual observation.

Different data sets can have identical five-number summaries despite having different sample sizes and means.

Answer 6

"Both classes have the same median examination score of 68. However, Class Q has a substantially larger IQR of 22 compared with 10 for Class P. This indicates that the middle 50% of Class Q's scores are more spread out, while Class P has more consistent central scores."

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13. FREQUENTLY ASKED QUESTIONS ABOUT BOXPLOTS

What is the difference between a boxplot and a box-and-whisker plot?

They are two names for the same type of statistical graph. The terms boxplot, box plot, box-and-whisker plot and box-and-whisker diagram are commonly used.

How do you calculate the five-number summary?

Arrange the data in ascending order and calculate the minimum, Q1, median, Q3 and maximum. The quartile method should follow your course or software convention.

How do you calculate the interquartile range?

Subtract the first quartile from the third quartile.

IQR = Q3 − Q1

What does the box represent in a boxplot?

The box extends from Q1 to Q3 and represents approximately the middle 50% of the observations.

What do the whiskers represent?

In a basic boxplot, whiskers generally extend to the minimum and maximum. In a modified boxplot using the 1.5 × IQR rule, they extend to the most extreme observations that are not flagged as outliers.

How do you identify outliers in a boxplot?

Calculate the lower and upper fences:

Lower fence = Q1 − 1.5 × IQR

Upper fence = Q3 + 1.5 × IQR

Values outside these boundaries are potential outliers.

Can a boxplot show negative values?

Yes. Boxplots can represent negative as well as positive numerical observations. The number line simply needs to accommodate the values.

Can a boxplot have no whiskers?

Yes. If Q1 equals the minimum, the lower whisker has zero length. If Q3 equals the maximum, the upper whisker has zero length.

Is the median always in the centre of the box?

No. The median can be closer to either quartile. Its position can provide a visual clue about the distribution's symmetry.

Does a boxplot show the mean?

No. The standard boxplot displays the quartiles, median and whiskers. Some specialised plots add a mean marker, but it is not part of the basic five-number summary.

Can you compare two boxplots?

Yes. Compare their medians, IQRs, whisker lengths, potential outliers and overall shape. For meaningful visual comparisons, use the same numerical scale.

When should you use a histogram instead of a boxplot?

Use a histogram when you want to examine the detailed distribution, including clusters, gaps, peaks and possible bimodality.

Are outliers always errors?

No. An outlier may represent a genuine observation, an unusual event or a recording error. It should be investigated rather than automatically removed.

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14. THE QUICK REVISION CARD: STOPS–FENCES–SAY

Before an examination, remember these three steps.

STOPS: FIVE-NUMBER SUMMARY

Minimum

Q1

Median

Q3

Maximum

FENCES: OUTLIER DETECTION

IQR = Q3 − Q1

Lower fence = Q1 − 1.5 × IQR

Upper fence = Q3 + 1.5 × IQR

Values outside the fences are potential outliers.

SAY: COMPARE DISTRIBUTIONS

Centre = Median

Spread = IQR

Shape = Symmetry or skewness

Outliers = Unusual observations identified in context

Use numerical evidence rather than vague descriptions.

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THE FINAL LESSON: FIVE NUMBERS CAN REVEAL MORE THAN ONE AVERAGE

Tobias initially believed Riverside and Hillside were almost identical because their average running times were close.

The boxplot revealed a more useful comparison.

Riverside had a slightly lower median, but its central running times were more variable. Hillside had a smaller IQR and no potential outliers under the 1.5 × IQR rule.

For a training group that values consistency, Hillside offers the more concentrated distribution.

That is the real purpose of descriptive statistics.

The mean gives one measure of centre. The five-number summary adds information about position and spread. The boxplot makes those differences visible.

Remember the method:

STOPS to calculate the summary. FENCES to identify potential outliers. SAY to explain what the distributions reveal.

Once you understand these three steps, boxplots become much easier to construct, interpret and compare in AP Statistics, GCSE Mathematics, A-level Statistics, IB Mathematics and introductory college statistics.