Reading Graphs & Estimation — Print
A graph is an argument. Read the scale before the shape, or you will believe whatever it wants you to believe.
What the practice questions cover
These questions use 8 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Rate — A ratio of two different units, like miles per hour or dollars per pound.
- Percent — Parts per hundred. 37% = 37100 = 0.37.
- Area — Space inside, measured in square units.
- Bar graph — Compares separate categories.
- Line graph — Shows change over time.
- Circle graph — Shows parts of one whole, as percentages summing to 100%.
- Dimensions — Rows × columns. A 2 × 3 matrix has two rows and three columns.
- Height — The perpendicular distance from the base to the opposite corner.
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.
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.
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?
- Read the actual values: 74 and 78. Numbers first, shape second.
- The real difference is 4 points. Not a doubling by any measure.
- The bars start at 70, so only the part above 70 is drawn: 4 versus 8. That is why one looks twice the other.
- 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?
- Difference in percent: 35% - 20% = 15% Subtract the shares before converting.
- 0.15 × 2400 = 360 One multiplication instead of two.
- Check: 840 - 480 = 360 ✓ Computing both separately confirms it.
$360 more.
Estimate first
Estimate 6,13829, then say whether 211.7 is reasonable.
- Use compatible numbers: 6000 ÷ 30 30 divides 6000 cleanly — that is the point of "compatible".
- 6000 ÷ 30 = 200 A quick mental benchmark.
- 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.
- 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.
Problem 1
Which graph best shows how temperature changes over a day?
- Bar graph
- Line graph
- Circle graph
Worked answer
Line graph
A line graph shows a quantity changing over time.
Problem 2
Which graph best shows how a budget splits into categories?
- Bar graph
- Line graph
- Circle graph
Worked answer
Circle graph
A circle graph shows parts of a whole, summing to 100%.
Problem 3
A circle graph shows 25% of 800 students walk to school. How many is that?
Worked answer
200
0.25 × 800 = 200
Problem 4
Estimate 487 × 21.
- About 1,000
- About 10,000
- About 100,000
Worked answer
About 10,000
500 × 20 = 10,000
The exact value is 10,227.
Problem 5
Which graph best compares the populations of five different towns?
- Bar graph
- Line graph
- Circle graph
Worked answer
Bar graph
A bar graph compares separate categories.
Problem 6
A circle graph of a $2,400 budget shows housing at 35% and food at 20%. How much more goes on housing?
Worked answer
360
35% - 20% = 15%
0.15 × 2400 = 360
Problem 7
Estimate 6,13829 to the nearest hundred.
Worked answer
200
6000 ÷ 30 = 200
The exact value is about 211.7.
Problem 8
A bar chart has a vertical axis starting at 70. One bar reaches 74 and another 78, so the second looks twice as tall. What is the actual difference?
Worked answer
4
78 - 74 = 4
Only the part above 70 is drawn — 4 against 8 — which is why one looks double.
Problem 9
A circle graph shows sections of 30%, 25%, 20% and 15%. What percent is the remaining section?
Worked answer
10
30 + 25 + 20 + 15 = 90
100 - 90 = 10%
Problem 10
Estimate 8.9 × 31.
Worked answer
270
9 × 30 = 270
The exact value is 275.9.
Problem 11
A line graph shows temperature rising from 12° to 20° over 4 hours. What is the rate of increase per hour?
Worked answer
2
20 - 12 = 8 degrees
8 ÷ 4 = 2 degrees per hour
Problem 12
Estimate 1,997 + 4,085 to the nearest thousand.
Worked answer
6000
2000 + 4000 = 6000
The exact value is 6,082.
Problem 13
The sections of a circle graph must add up to what percent?
Worked answer
100
A circle graph shows parts of a single whole, so the sections total 100%.
If they do not, the graph is wrong.
Problem 14
A pictogram doubles both the height and the width of an icon to show a doubled value. How many times bigger does the icon LOOK?
Worked answer
4
Doubling both dimensions multiplies the area by 2 × 2 = 4
The value doubled but the picture looks four times bigger — a classic distortion.
Problem 15
Estimate 39.7 × 20.4.
Worked answer
800
40 × 20 = 800
The exact value is about 809.9.
Problem 16
A survey of 250 students found 18% prefer science. How many students is that?
Worked answer
45
0.18 × 250 = 45