Year 11 General Univariate Data Analysis

Boxplots

20 practice questions 2 video lessons Theory + worked examples

Theory

A boxplot shows the five-number summary (min, $Q_{1}$ , median, $Q_{3}$ , max). The box spans $Q_{1}$ to $Q_{3}$ ; its width is the IQR. The line inside is the median. $25 %$ of data sits in each section. Outliers (beyond $Q_{1} - 1.5 IQR$ or $Q_{3} + 1.5 IQR$ ) are plotted as dots. Parallel boxplots compare two groups on the same axis.

A boxplot (or box-and-whisker plot) is a visual summary of the five-number summary: minimum, $Q_{1}$ , median, $Q_{3}$ , and maximum. It shows the centre, spread, and shape of a distribution at a glance.

Anatomy:

The box stretches from $Q_{1}$ to $Q_{3}$ . Its width is the IQR.
The line inside the box is the median.
The whiskers extend from the box to the minimum and maximum (or to the most extreme non-outlier values).
The whole plot covers the range = max $-$ min.

Quartile percentages (always true):

$25 %$ of values are at or below $Q_{1}$ .
$50 %$ of values lie inside the box (between $Q_{1}$ and $Q_{3}$ ).
$75 %$ of values are at or below $Q_{3}$ ; equivalently, $25 %$ are above.

Shape from a boxplot:

Symmetric: median centred in the box, whiskers similar in length.
Positively skewed: right whisker longer; median closer to $Q_{1}$ .
Negatively skewed: left whisker longer; median closer to $Q_{3}$ .

Outliers are plotted as separate dots beyond the whiskers, using the $1.5 \times IQR$ rule for the fences.

The first diagram is the anatomy of a boxplot — every component labelled. The second shows how the same five-number-summary structure looks for symmetric, positively skewed, and negatively skewed distributions.

The box runs from

Q_{1}

Q_{3}

; its width is the IQR. The line inside is the median.

Whisker length and median position inside the box tell you the shape at a glance.

The boxplot is built from the five-number summary and the IQR.

Five-number summary

$min, Q_{1}, median, Q_{3}, max$

five-number summary

IQR and range

$IQR = Q_{3} - Q_{1}, range = max - min$

Outlier fences

$lower fence = Q_{1} - 1.5 \times IQR, upper fence = Q_{3} + 1.5 \times IQR$

Values outside the fences are plotted as separate dots; the whiskers extend only to the most extreme value still inside the fences.

Quartile percentages

Region of the boxplot	% of data
Below $Q_{1}$ (left whisker)	$25 %$
Inside the box (between $Q_{1}$ and $Q_{3}$ )	$50 %$
Above $Q_{3}$ (right whisker)	$25 %$
At or below $Q_{3}$ (everything except the right whisker)	$75 %$

Reading shape. Median centred + whiskers similar = symmetric. Median near Q1 + long right whisker = positively skewed. Median near Q3 + long left whisker = negatively skewed.

Building a boxplot from raw data

Sort the data and find the five-number summary: min, $Q_{1}$ , median, $Q_{3}$ , max.
Draw a number line that covers at least min to max.
Draw the box from $Q_{1}$ to $Q_{3}$ with a vertical line at the median.
Draw whiskers from the box to the min and max — or, if there are outliers, to the most extreme non-outlier values, and plot the outliers as separate dots.

Reading off a boxplot

Read the five values left to right: min, $Q_{1}$ , median, $Q_{3}$ , max.
$IQR = Q_{3} - Q_{1}$ (the width of the box); range = max $-$ min.
Use the quartile percentages: $25 %$ below $Q_{1}$ , $50 %$ inside the box, $25 %$ above $Q_{3}$ .

Identifying outliers on a boxplot

Compute $IQR = Q_{3} - Q_{1}$ .
Compute the fences: $Q_{1} - 1.5 IQR$ and $Q_{3} + 1.5 IQR$ .
Any value outside the fences is an outlier — plot it as a separate dot.

EXAMPLE 1 — READ OFF A BOXPLOT

A boxplot has min

= 6

Q_{1} = 10

, median

= 14

Q_{3} = 20

, max

= 30

. Find the IQR and the range.

SOLUTION

Use the formulas $IQR = Q_{3} - Q_{1}$ and $range = max - min$ .

IQR	$=$	$20 - 10 = 10$
Range	$=$	$30 - 6 = 24$

Answer: IQR $= 10$ , range $= 24$ .

IQR = 10, range = 24

EXAMPLE 2 — FIVE-NUMBER SUMMARY

Find the five-number summary for these

9

scores:

3, 6, 8, 10, 12, 13, 15, 17, 20

SOLUTION

$n = 9$ (odd). The median is the middle ( $5$ th) value, which is $12$ . Lower half: $3, 6, 8, 10$ . Upper half: $13, 15, 17, 20$ .

$Q_{1}$	$=$	$\frac{6 + 8}{2} = 7$
$Q_{3}$	$=$	$\frac{15 + 17}{2} = 16$

Five-number summary: $3, 7, 12, 16, 20$ .

Answer: min $= 3$ , $Q_{1} = 7$ , median $= 12$ , $Q_{3} = 16$ , max $= 20$ .

5-num = 3, 7, 12, 16, 20

EXAMPLE 3 — OUTLIER CHECK ON A BOXPLOT

Five-number summary: min

= 12

Q_{1} = 18

, median

= 22

Q_{3} = 28

, max

= 38

. A new value of

45

appears. Is it an outlier?

SOLUTION

Compute the IQR and upper fence; compare $45$ to the fence.

IQR	$=$	$28 - 18 = 10$
Upper fence	$=$	$28 + 1.5 (10) = 43$

Since $45 > 43$ , the value lies past the upper fence.

Answer: yes — $45$ is an outlier and would be plotted as a separate dot.

45 > 43 \Rightarrow outlier

EXAMPLE 4 — PARALLEL BOXPLOT COMPARISON

Year 11 boxplot: median

= 14

Q_{1} = 8

. Year 12 boxplot: median

= 24

Q_{1} = 18

. Compare typical weekly study hours.

SOLUTION

Compare the medians (centre) and note where one group's quartiles sit relative to the other.

Median diff

=

24 - 14 = 10 hours

Year 12 students typically study $10$ hours per week more than Year 11. Note also: Year 12's $Q_{1}$ ( $18$ ) is already higher than Year 11's median ( $14$ ) — so more than $75 %$ of Year 12 students study more than the typical Year 11 student.

Answer: Year 12 has a markedly higher centre and even Year 12's lower quartile sits above Year 11's median.

median diff = 10

Common pitfalls

Boxplots hide detail. A boxplot only shows five summary numbers, not every individual value. Two datasets with very different shapes (e.g. bimodal vs uniform) can produce identical boxplots.

"Median at the centre of the box" ≠ "mean equals median". The mean is not shown on a standard boxplot. A symmetric-looking boxplot suggests mean and median are close, but doesn't directly show the mean.

Compare parallel boxplots carefully. Overlapping boxes do not always mean "no difference" — real differences exist if medians are well separated. Look at all three: median, IQR, and range.

Outliers shorten the whiskers. When outliers are present, the whisker extends only to the most extreme value inside the

1.5 IQR

fence. The outliers themselves are plotted as separate dots beyond the whisker.

IQR is the box width, not the whisker length.

IQR = Q_{3} - Q_{1}

measures the spread of the middle

50 %

. The range covers the whole plot.

Frequently asked questions

What does each part of a boxplot show?

The box runs from Q1 to Q3, so its width equals the IQR. The line inside the box is the median. The two whiskers extend to the minimum and maximum (or to the most extreme non-outlier values). The whole plot covers the range.

What percentage of data lies inside the box?

Always 50 percent. The box covers the middle half of the data — from Q1 (the 25th percentile) to Q3 (the 75th). 25 percent of values sit below Q1 in the lower whisker, and 25 percent sit above Q3 in the upper whisker.

How can I tell the shape from a boxplot?

Symmetric: median centred in the box, whiskers similar in length. Positively skewed: right whisker longer, median closer to Q1. Negatively skewed: left whisker longer, median closer to Q3.

How are outliers shown on a boxplot?

Outliers are plotted as separate dots beyond the whiskers. The whiskers themselves extend only to the most extreme value that is NOT an outlier (i.e. the most extreme value still inside the 1.5 times IQR fences).

What is the range on a boxplot?

The range is max minus min — the total span from the left whisker tip (or the smallest outlier) to the right whisker tip (or the largest outlier). The IQR is just Q3 minus Q1 — the width of the box.

Can two different datasets have the same boxplot?

Yes. A boxplot only shows five summary numbers, not every individual value. Two datasets with very different shapes or bimodal patterns can produce identical boxplots — so boxplots can hide some detail.

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Comparing Data For A Numerical Variable Across Two Or More Groups

Boxplots

📖 Theory

Five-number summary

IQR and range

Outlier fences

Quartile percentages

Building a boxplot from raw data

Reading off a boxplot

Identifying outliers on a boxplot

Common pitfalls

Frequently asked questions

🎬 Video Lessons

✏️ Practice Questions

Theory

Video Lessons

Practice Questions