Statistical Displays

Categorical and Numerical Data #

Categories you count, measurements you average.

Some data names a category and some data counts or measures

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

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

$\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

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

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

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

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

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

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

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

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

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

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.