01 · Summarise
Find the five-number summary for 1, 3, 5, 7, 9, 11, 13, 15 using the same quartile convention.
Hint
Average ranks two and three for Q₁, six and seven for Q₃.
Worked solution
Minimum 1; Q₁ = 4; median 8; Q₃ = 12; maximum 15.
Understand · explore · practise
Drawing box plots
Build box plots from ordered data, distinguish outlier fences from observed whiskers, label scales and avoid inventing endpoints from incomplete summaries.
Before you startQuartiles, medians, IQR and outlier fences.
01 / Define each part
A box plot locates a few summaries on one numerical scale.
Box: Q₁ to Q₃. Line inside: median. Whiskers: endpoints under the stated convention.
Here whiskers end at the smallest and largest observed non-outliers; flagged observations are plotted separately.
Some basic box plots instead use the observed minimum and maximum without separating outliers. Identify the convention before drawing or interpreting the plot.
Whiskers end at observationsExplore
Ordered data: 2, 4, 6, 8, 10, 12, 14, m. Q₁ = 5, median = 9, Q₃ = 13. Strict 1.5 IQR fences: −7 and 25.
Whiskers: 2 and 16. No outliers are flagged.
The box stays from 5 to 13. When m exceeds 25, mark it separately and end the upper whisker at the largest remaining observation, 14.
02 / Start with the five-number summary
Order the data before finding ranks.
Ordered data: 2, 4, 6, 8, 10, 12, 14, 16.Worked example
Minimum 2; maximum 16; median (8 + 10)/2 = 9
These values describe location on the axis.
Q₁ = (4 + 6)/2 = 5; Q₃ = (12 + 14)/2 = 13
Use the stated listed-data convention.
Five-number summary: 2, 5, 9, 13, 16
Keep these in increasing order.
02 · Impossible order
A proposed summary is minimum 2, Q₁ 9, median 7, Q₃ 13, maximum 20. What is inconsistent?
Hint
Quartiles and median must respect order.
Worked solution
Q₁ cannot exceed the median under this convention. Recheck the raw data and calculations before plotting.
03 / Use a labelled numerical scale
Distances along the axis must match numerical differences.
03 · Axis distances
A scale uses 2 cm for 10 units. How far apart should Q₁ = 5 and Q₃ = 20 appear?
Hint
The numerical difference is 15 units.
Worked solution
3 cm. Do not space the five summary marks equally regardless of their values.
04 · Units and labels
What should be labelled on a box-plot axis for journey times?
Hint
Identify the quantity and unit.
Worked solution
Use a title such as Journey time (minutes), with consistent numerical tick spacing. A reader must know what the numbers measure.
04 / Apply the outlier rule
The fence decides which observations get separate marks.
Watch: the whisker returns to an observed value
Pause, replay or seek freely. The notes explain the same idea and stay in view.
Change the last value in the worked list from 16 to 40.Worked example
Q₁ = 5, median = 9, Q₃ = 13 remain unchanged
The quartile ranks do not use the last value in this list.
IQR = 8; strict 1.5 IQR fences are −7 and 25
40 is above the upper fence.
Whiskers at 2 and 14; plot 40 separately
25 is a fence, not an observed endpoint.
05 · Equality at a fence
Instead use the list 2, 4, 6, 8, 10, 12, 14, 25. Where does the upper observed non-outlier whisker end?
Hint
The strict rule does not flag equality at U = 25.
Worked solution
It ends at 25, which is both an observation and exactly on the fence.
06 · Just beyond
Change that 25 to 26. Where should the upper whisker and separate point be?
Hint
26 exceeds the unchanged fence.
Worked solution
Upper whisker at 14; separate outlier point at 26. Do not put the whisker at 25.
05 / Draw in a reliable order
Keep the box, whiskers and separate points distinct.
- Label the axis and choose a scale covering all observed values, including outliers.
- Draw the box between Q₁ and Q₃.
- Add the median line inside the box.
- Extend whiskers to the endpoints required by the convention.
- Mark any separate outlier observations and identify the rule.
07 · Full range with an outlier
In the worked list ending at 40, is the full observed range 14 − 2 = 12?
Hint
The separately plotted observation still belongs to the data.
Worked solution
No. Full range = 40 − 2 = 38. The span between the non-outlier whiskers is 12, a different quantity.
08 · Orientation
Can a box plot use a vertical measurement axis?
Hint
The summary values still belong on one numerical axis.
Worked solution
Yes. The same values and conventions apply. Label the variable and preserve the scale; orientation does not change the summaries.
06 / Do not invent a whisker
A fence and maximum may not reveal the largest non-outlier.
09 · Summary only
You know Q₁ = 5, Q₃ = 13, maximum 40, and that 40 is an outlier under fences −7 and 25. Is the largest observed non-outlier necessarily 25?
Hint
You have no record showing an observation at 25.
Worked solution
No. More information is needed for an observed non-outlier whisker. State the limitation or use a specifically stated alternative plotting convention; do not invent an observed endpoint.
10 · Basic convention
A question asks for a basic box plot using the given minimum and maximum, with no outlier adjustment. Should you silently replace the maximum by an IQR fence?
Hint
Follow the requested convention.
Worked solution
No. Use the given extrema for that basic plot. If discussing outliers separately, state the additional rule and plotting convention explicitly.
07 / Label grouped estimates
Estimated quartiles do not restore the lost raw data.
11 · Grouped extremes
The lowest occupied class is [0,10) and the highest is [30,40). Are 0 and 40 necessarily the observed minimum and maximum?
Hint
Boundaries contain values; they need not be observed.
Worked solution
No. They are class boundaries. A box plot using these as outer endpoints must identify them as bounds or estimates, not exact observed extrema.
12 · Interpolated box
Quartiles were estimated by linear interpolation within grouped classes. What should accompany the resulting box plot?
Hint
The construction inherited an assumption.
Worked solution
Label the quartiles as estimates and state the within-class uniformity assumption. Distinguish any assumed outer endpoints from observed minimum and maximum values.
08 / Know what the picture omits
Four parts of a box plot are not four equal counts in every dataset.
13 · Ties
If many observations equal the median, must exactly half of the observations be strictly below it?
Hint
Strict inequalities exclude the tied values.
Worked solution
No. A median divides ordered positions, but ties can make the count strictly below it smaller than half. Avoid reading exact distinct-value counts from the diagram alone.
14 · Reconstruct the data
Can a box plot alone determine every original observation or the exact mean?
Hint
Many lists can share the same five-number summary.
Worked solution
No. It summarises selected ordered positions and endpoints. The individual values and exact mean are generally not recoverable.
09 / Plot values, explain conventions
Keep thresholds separate from observations.
Calculate ordered summaries, apply the stated outlier rule, select the required whisker convention and use a labelled scale. Include separate outlier points and acknowledge estimates or missing information.
Section 1 of 9 · Define each part