Reading Graphs & Estimation

A graph is an argument. Read the scale before the shape, or you will believe whatever it wants you to believe.

Three shops scored 96, 98 and 100 in a survey. Here are two bar charts of those numbers.

On the left the shops look almost identical. On the right, shop C towers over shop A.

the same three numbers, drawn twice A B C 0 axis starts at 0 A B C 94 axis starts at 94 C looks a little ahead on the left and enormously ahead on the right

Same three numbers. Same three bars. The only difference is where the axis starts.

The chart on the right begins at 94, so it shows only the tips of the bars — and a gap of 4 fills the whole picture.

Neither chart is lying about the numbers. The right one is lying about how big the difference is.

So the first thing to read on any graph is not the bars — it is the axis. Check where it starts, what the gridlines are worth, and only then look at the shape.

The same two values drawn on a full axis and on a truncated axis axis from 0 — honest 7478 0 axis from 70 — misleading 7478 70 same data — a 4-point gap looks like a doubling. Read the numbers, not the bars.

Which graph for which job

Bar: comparing separate things — favourite sports, sales by shop.

Line: change over time — temperature through the day.

Circle: how one whole splits up — a budget. Use it only when the parts really do make 100%.

Using the wrong type is itself an error: a line graph joining unrelated categories implies change that is not there.

Read the axis first

Before looking at the bars, read: what is measured, in what units, what is one step worth, and where does the axis start?

An axis starting at 90 instead of 0 makes a bar of 95 look five times taller than one of 91, when the real difference is about 4%.

This is the commonest way a truthful graph tells a misleading story.

The truncated axis

If a vertical axis does not start at zero, bar heights cannot be compared by eye. Read the numbers instead.

Other ways graphs mislead

Uneven spacing on a time axis makes growth look smooth when it is not.

Pictures scaled in two directions — doubling the height of an icon also doubles its width, so it looks four times bigger.

Missing labels, or percentages in a circle graph that do not add to 100.

Estimating to check

Round to friendly numbers and compute in your head first. 487 × 21 is about 500 × 20 = 10,000.

If your calculator says 1,023, something went wrong. That is what the estimate is for.

Use compatible numbers, not just rounded ones: for 796038, think 800040 = 200.

Estimate before, not after

Do the rough answer first, on paper. Doing it afterwards makes it very easy to talk yourself into whatever the calculator said.

Reading between the marks

If gridlines are 5 apart and a point sits halfway between 20 and 25, read it as about 22.5 — and say "about".

Never report more precision than the graph can show.

Worked examples

A misleading axis

A bar chart of test averages has a vertical axis from 70 to 80. Class A's bar reaches 74 and Class B's reaches 78. B's bar looks twice as tall. Is B twice as good?

  1. Read the actual values: 74 and 78. Numbers first, shape second.
  2. The real difference is 4 points. Not a doubling by any measure.
  3. The bars start at 70, so only the part above 70 is drawn: 4 versus 8. That is why one looks twice the other.
  4. As a percentage, 78 is about 5% more than 74. The visual impression overstates it roughly tenfold.

No — the truncated axis exaggerates a 4-point gap.

Circle graph

A circle graph of a $2,400 budget shows housing at 35% and food at 20%. How much more is spent on housing?

  1. Difference in percent: 35% - 20% = 15% Subtract the shares before converting.
  2. 0.15 × 2400 = 360 One multiplication instead of two.
  3. Check: 840 - 480 = 360 ✓ Computing both separately confirms it.

$360 more.

Estimate first

Estimate 6,13829, then say whether 211.7 is reasonable.

  1. Use compatible numbers: 6000 ÷ 30 30 divides 6000 cleanly — that is the point of "compatible".
  2. 6000 ÷ 30 = 200 A quick mental benchmark.
  3. The true value should be a bit above 200. The top was rounded down and the bottom rounded up, so both push the estimate low.
  4. 211.7 sits just above 200 ✓ Reasonable.

Yes — about 200, so 211.7 is sensible.

Words to know

Bar graph
Compares separate categories.
Line graph
Shows change over time.
Circle graph
Shows parts of one whole, as percentages summing to 100%.
Scale
What one grid step is worth — and where the axis starts.
Estimate
A deliberately rough answer used to check a precise one.

Mistakes to watch for

Comparing bar heights without checking where the axis starts. A truncated axis can make a 2% difference look like a landslide. The bars are honest only if the axis begins at zero.

Reading a value more precisely than the gridlines allow. If the lines are every 10, you can say "about 35" but not "34.7." The extra digit is invented, not read.

Estimating after calculating. The point of an estimate is to give you something to check the calculation against; doing it afterwards just copies whatever the calculation said, mistakes included.

Assuming a rising line means a rising quantity. On a graph of rate against time, a falling line can still mean the total is growing — just more slowly. Read the axis label, not the slope.

Florence Nightingale is remembered as a nurse, but she was also a formidable statistician and a fellow of the Royal Statistical Society.

Her "rose diagram" of Crimean War deaths showed that far more soldiers died of preventable disease than of wounds — and showed it in a single picture a politician could not argue with.

Sanitary reform followed. She used a graph because she had already tried tables, and tables had not worked.

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