Stem-and-leaf diagrams, box-and-whisker diagrams, two-way tables and Venn diagrams

Last time

Before we start: last time, in Pictograms, bar charts, pie charts and percentage bar charts, you learned these points. Here they are again on one card. Keep them in mind, because today builds on them.

Diagrams that show more

Some diagrams keep every value, some summarise the spread, and some sort items into overlapping groups. This lesson covers stem-and-leaf diagrams, box-and-whisker diagrams, two-way tables and Venn diagrams, and how diagrams can mislead.

Stem-and-leaf diagrams

A stem-and-leaf diagram splits each value into a stem, such as the tens digit, and a leaf, the units digit. The leaves are written in order, and a key shows how to read the diagram, such as: 2 bar 5 means 25. A back-to-back diagram compares two sets on one stem; the leaves on the left increase outwards from the stem.

Median and quartiles

For n ordered values, the median is the value in position n plus 1, over 2. For 12, 15, 18, 21, 24, 27, 30, the median is the 4th value, 21; the lower quartile is the 2nd, 15; the upper quartile the 6th, 27. The interquartile range is the upper quartile minus the lower quartile: 27 minus 15 is 12.

Box-and-whisker diagrams

A box-and-whisker diagram shows the minimum, lower quartile, median, upper quartile and maximum against a scale. The box runs from the lower to the upper quartile, with a line at the median; the whiskers reach the extreme values. To compare two box plots, compare the medians for the average and the interquartile ranges for the spread.

Two-way tables

A two-way table counts items by two variables, such as gender and favourite sport, with row and column totals. Missing values are found from the totals: every row and every column must add up. Read the question carefully: girls who like tennis is one cell; everyone who likes tennis is a column total.

Venn diagrams

A Venn diagram shows sets as overlapping circles inside a rectangle, the universal set. Fill from the middle outwards. 40 people: 25 like A, 20 like B, 12 like both. A only is 13, B only is 8. Outside both circles: 40 minus 33 is 7. Check that all the regions add up to the total.

Misleading diagrams

A vertical axis that does not start at zero exaggerates the differences between bars. Pictures scaled in two or three dimensions make increases look far larger than they are. Unequal scales, missing labels or no key also mislead. Diagrams that are combined or compared need the same scale and units.

A two-way table

A two-way table of 50 students, with the gaps filled from the totals.

Mistakes that lose marks

Mistake one: a stem-and-leaf diagram with no key, or with its leaves out of order. Mistake two: in a Venn diagram, putting a whole set's total in the region for that set only, so the overlap is counted twice. Mistake three: comparing box plots by their ranges alone, ignoring the medians and interquartile ranges. The fix: a key, the middle first, and both average and spread.

The full-marks tip

The full-marks tip: fill a Venn diagram from the middle outwards, then check that the regions add up to the total. When comparing two distributions, make one point about the average and one about the spread, with numbers.

On one card

So, on one card: stem-and-leaf, box plots, tables and Venn diagrams. Keep it for revision.