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Box plots

Read them, draw them, count the quarters

  • In 6 of 20 recent exams
  • Paper 1 or 2
  • 1 to 2 marks
How it is set in the exam

Paper 1 (no calculator) and Paper 2 (calculator), 1–2 marks a part, inside a longer statistics question. Common (6 of 20 exam versions, 3 of them on Paper 1 (no calculator); 11 logged parts).

Spot it in the exam
Exam phrases to spot (4)

  • “use the box plot to find …”

  • “how many … lie between … and …”

  • “draw a box-and-whisker diagram”

  • “write down the median”

In disguise (2)
  • a percentage of the data between two markers, rather than a number
  • one marker is not labelled, and the IQR or the range is given instead
Variants

asked: the median, the IQR, a count between two markers (the median to Q3Q_3, a quartile to an end), a missing marker, a drawing from a frequency table

Looks like it, but isn't (3)
  • "the interquartile range of a normal variable is …; find σ\sigma": Unknown mean or standard deviation
  • "is … an outlier?": Outliers
  • two box plots compared in context: Compare two distributions
Worked example

Read the diagram, count the quarters, find a missing marker

Paper 1 (no calculator)
Question

The box-and-whisker diagram summarises the lengths of 180180 fish caught in a lake. On the length axis (in cm), the left whisker starts at 1212, the box runs from aa to 3333 with a vertical line at 2727, and the right whisker ends at 4848.
(a) Write down the median length. [1]
(b) The interquartile range is 1111 cm. Find the value of aa. [2]
(c) Find the number of fish with lengths between 2727 cm and 3333 cm. [2]
(d) Find the number of fish with lengths between aa cm and 4848 cm. [2]

Try it yourself first, then compare.

Now try one like it

Write your working on paper or on screen. Paper Zero marks it step by step, like the worked example, and shows where each mark went. Free, with no account.

A handwritten page as Paper Zero marks it: ticks at the two lines that earned a mark, a cross on line 3 where +7 was copied as −7, and a short note beside that line.123456f′(x) = 4x − 124x − 12 = 0, so x = 3f(3) = 2(3)² − 12(3) − 7= 18 − 36 − 7 = −25minimum at (3, −25)M1 ✓A1 ✓A1 ✗11The +7 became −7: 18 − 36 + 7 = −11.
Steps

  1. Read the five markers:

    the ends of the whiskers are the least and greatest values, the ends of the box are Q1Q_1 and Q3Q_3, and the line inside the box is the median.

  2. A missing marker

    comes from the others: Q1=Q3−IQRQ_1 = Q_3 - \mathrm{IQR}, or greatest == least ++ range.

  3. Counts:

    each section holds 2525 % of the data. Count the sections between the two markers, then take that fraction of nn. Writing the percentage earns the method mark.

  4. To draw:

    get the five-number summary (one-variable statistics for a table on ), check the fences for outliers (Outliers), then draw to scale on a labelled axis: whiskers to the least and greatest values, a box from Q1Q_1 to Q3Q_3 and a line at the median. Any outlier is a cross.

Common mistakes

  • Quarters of the length instead of quarters of the data

    a long whisker has as many values as a short one.

  • Whiskers that stop at the quartiles,

    or that don't reach the least and greatest values: the drawing loses a mark.

  • The range for the IQR

    the IQR is the length of the box only.

More mistakes (2)

  • A diagram not to scale,

    or with no labelled axis.

  • Drawing an outlier as the end of a whisker

    check the fences first; an outlier is a cross.

Keep going

Easy to confuse with

Next typeWork backwards from the mean or median
Next: Work backwards from the mean or median
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