Graphs that tell the truth, and chances that multiply

Word	What it means
Bar graph, line graph	Bars whose heights compare separate groups; points joined by lines to show change over time.
Circle graph, pictograph	A circle cut into sectors showing the parts of one whole; symbols standing for counts, with a key saying what one symbol is worth.
Axis, scale, interval	The axis is the number line along the edge of a graph, and the scale is the numbers on it. The interval, the gap between labelled numbers, should be the same all the way along.
Outcome	One possible result, like rolling a 4. A favourable outcome is one you are counting.
Probability, P(A)	How likely event A is, from 0 (impossible) to 1 (certain), as a fraction, decimal, percent or ratio. P(red) means the probability of red.
Independent events	Events where one happening does not change the chances of the other, like rolling a die and spinning a spinner.
Tree diagram	A diagram that branches once for each outcome of the first event, then again for the second.

Worked example. A school library lent 460 books in September, 480 in October and 500 in November. On the left, the axis starts at 450, so the bars are drawn 10, 30 and 50 units tall and November looks five times as busy as September. The real increase is 500 − 460 = 40 books, and 40 ÷ 460 ≈ 0.087: under 9%.

Other tricks to look for.

• Uneven intervals. Years 2000, 2010, 2020, 2022 and 2024 spaced evenly give two years as much room as ten, so a steady rise looks as if it slowed down.
• Pictures that grow two ways. To show that recycling doubled, a poster draws a bin twice as tall, and so twice as wide. It covers 2 × 2 = 4 times the area, so it looks like four times as much.
• Chosen time ranges. A team's last five games can show a winning streak in a losing season. Ask what came before.
• Missing labels, and 3-D effects. With no title, units or scale, a graph can mean anything. And tilting a circle graph makes the slices at the front look bigger than equal slices at the back.

Graph	Best for	Example
Circle graph	Parts of one whole, adding to 100%	How a club spends its budget
Line graph	Change over time	The temperature through a day
Bar graph	Comparing separate groups	Favourite sports in a class
Pictograph	Counts, for a quick read	Books read by each grade

Worked example: which graph? A survey asked 50 students which after-school activities they do, and they could choose more than one. Sports got 60%, music 40%, clubs 30% and a part-time job 10%.

1. Circle graph? The percents add to 60% + 40% + 30% + 10% = 140%, more than one whole, because many students were counted twice. So no.
2. Line graph? It is not a change over time, so no.
3. Separate groups to compare: a bar graph, one bar for each activity.

Reading a pictograph. If one symbol stands for 10 books, 3½ symbols is 35 books. With no key it could be 35 or 350.

Reading a line graph. The points are measurements, and the line between them is an estimate. If it was −6 °C at 8 a.m. and −2 °C at 10 a.m., it was probably about −4 °C at 9 a.m. — but nobody measured it.

Worked example. A spinner has 4 equal sections — red, blue, green and yellow — and a die has 6 faces. You spin and roll. What is the probability of red and a 6?

1. Independent? Yes: the spinner cannot affect the die.
2. Each on its own: P(red) = 1/4 and P(6) = 1/6.
3. Multiply: P(red and 6) = 1/4 × 1/6 = 1/24.
4. Check by counting: 4 colours × 6 numbers make 24 equally likely outcomes, and exactly one of them is red with a 6 ✓.

As a decimal, 1/24 is about 0.042, or 4.2%. As a ratio of favourable to possible outcomes, it is 1 : 24.

A tree diagram shows the same thing as branches. Flip a coin twice: the first flip branches into H and T, and each of those branches again, giving HH, HT, TH and TT. P(two heads) = ½ × ½ = ¼, one branch out of four.

More than one favourable outcome. What is the probability of an even number on the die and a colour that is not red? P(even) = 3/6 = ½ and P(not red) = 3/4, so P(both) = ½ × 3/4 = 3/8, which is 0.375, or 37.5%.

Putting it back. A bag has 3 red and 2 blue marbles. Draw one, put it back, and draw again: the draws are independent, so P(red, then red) = 3/5 × 3/5 = 9/25, or 36%. If the first marble is not put back, the second draw has only 4 marbles, and its chances depend on the first. Those events are not independent, and 3/5 × 3/5 would be wrong.

In real life, independence is an assumption. A player who makes 70% of her free throws makes two in a row with probability 0.7 × 0.7 = 0.49, or 49% — if one shot does not affect the next.

• Judging bars by their heights alone. Check where the axis starts.
• A circle graph for percents that add to more than 100%. Use a bar graph.
• Adding probabilities that should be multiplied. P(red and 6) is 1/4 × 1/6, not 1/4 + 1/6.
• Multiplying for events that are not independent, such as drawing without putting back.
• Reading a pictograph without its key.