Statistical Displays

Categorical and Numerical Data #

Categories you count, measurements you average.

Some data names a category and some data counts or measures

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Favorite pet is a category. You can count them but not average them.

Height is a measurement, so averaging it actually means something.

Bar Charts #

Taller bar, bigger count, at a glance.

A bar chart turns counts into heights you can compare at a glance

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A taller bar means a bigger count. The scale must start at zero to keep the heights fair.

Pie Charts #

How a whole is shared out.

A pie chart shows how a whole is shared out rather than how big it is

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$\frac{1}{2}$1/2 of the children walk, $\frac{1}{4}$1/4 take the bus, and $\frac{1}{4}$1/4 come by car.

Together the slices make one whole circle: $\frac{1}{2}+\frac{1}{4}+\frac{1}{4}=1$1/2 + 1/4 + 1/4 = 1, with nothing left over.

One whole is 100% and a full turn is 360°, so the $\frac{1}{4}$1/4 bus slice is 25% and 90°.

Dot Diagrams #

One dot per value, stacked where it falls.

A dot diagram puts one dot above the value it belongs to, so a stack is a count

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One dot for each child, sitting above the score that child got.

3 dots stacked above 5 means 3 children scored 5.

Count the dots, not the stacks: 8 dots, so 8 children were tested.

The scores run from 3 to 7 and pile up at 5, with nobody at 2 or 8.

Vertical Line Charts #

One line per value, as tall as its count.

A vertical line chart gives each discrete value one line, as tall as its count

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Each score gets one line, as tall as its count. Read that height off the scale.

The lines have no width because a score of 2 is one number, not a range of them.

Add the heights and nothing is counted twice: 2 + 5 + 7 + 4 + 1 makes 19 in all.

Stem and Leaf #

Keeps every digit and still shows the shape.

A stem and leaf keeps every digit while still showing the shape

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The stem is the tens, each leaf is a ones digit. Nothing is thrown away.

The longest row is where most of the data sits, like a bar on its side.

Tally and Frequency Tables #

Count in gates of five; the table keeps score.

A tally counts in gates of five and the frequency column records each row count

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Scores arrive one by one. Tally each in its row; the fifth stroke closes a gate of five.

The frequency column just counts the tally: a gate of five and two more make 7.

When the values spread too wide, group them: each row now counts a whole interval.

Two-Way Tables #

Two questions at once, and margins that agree.

A two-way table counts two questions at once and its margins total each way

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Two questions at once — girl or boy, walks or rides. Each cell counts one combination.

Total along a row: 5 + 3 makes 8 girls. Each column totals the same way, downwards.

The corner holds the grand total, and both directions agree on it: 14 children.

If a cell is hidden, use its row total: the boys row totals 6, so the hidden cell is 4.

Frequency Trees #

A total split twice, and every fork adds back.

A frequency tree splits a total into groups and then splits each group again

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100 children are split by how they travel: 60 come by bus and 40 walk.

Each group splits again by whether they were late: 12 of the 60, and 6 of the 40.

Every fork adds back to the count before it: 12 + 48 = 60, and 6 + 34 = 40.

If the 48 is covered, the fork recovers it: the bus branch is 60, and 60 − 12 = 48.

Histograms #

Bars that touch, because the ranges do.

A histogram groups measurements into intervals and has no gaps

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The bars touch because their intervals touch, and the tallest interval holds the most.

Frequency Density #

When widths differ, area shows frequency.

When intervals differ in width it is the area that shows the frequency

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By raw count the last bar ties the middle one — but it covers twice as much scale.

A wide interval catches more just by being wide, so raw height is not a fair read.

Divide each count by its width: 6 over 2 units is 3 per unit — the frequency density.

Choosing a Chart #

The question picks the picture.

The question picks the chart: comparing, sharing out, changing, or spreading

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Which club is biggest? Categories sit side by side — a bar chart compares them.

What share of the day is sleep? Parts of one whole — a pie chart shows each share.

How did the temperature move all week? Change over time — a line graph joins the days.

How are the heights spread? A histogram bands them; a stem-and-leaf keeps each value.

Misleading Graphs #

Accurate, and still leaving a false impression.

A chart can be accurate and still leave a false impression

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52 against 48 is a close result. Drawn from 0, it looks close.

Both panels show the same two numbers. Cut the scale at 40 and it looks like a landslide.