Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- Plot the point (3, 5). How far across and how far up is it?
Answer
3 across and 5 up from the origin. - Find the mean of 4, 6 and 11.
Answer
(4 + 6 + 11) ÷ 3 = 7. - A graph's y axis starts at 240 instead of 0. Why might a graph do this?
Answer
All the values are between 240 and 380, so starting at 0 would waste the space.
Words used in this lesson
- bivariate data
- pairs of values, two measurements for each person or thing. For example: hours of sleep and reaction time for ten pupils.
- independent variable
- the variable that might explain the change; plotted on the x axis. For example: hours of sleep.
- dependent variable
- the variable that responds; plotted on the y axis. For example: reaction time.
- scatter graph
- a graph with one dot for each pair of values. For example: ten dots for ten pupils.
- direction
- whether the dots rise (positive) or fall (negative) from left to right. For example: negative.
Do longer sleepers react faster?
A class tests ten pupils with a ruler-drop timer and also asks how long each one slept the night before. Each pupil gives two numbers. Before you read on: do you think the pupils who slept longer reacted faster, or slower? Hold on to your guess; the dots will test it.
Bivariate data are pairs of values, two measurements on each person or thing. A scatter graph shows one dot for each pair. The independent variable (the one that might explain the change) goes on the x axis and the dependent variable (the one that responds) goes on the y axis.
When the dots rise from left to right the direction is positive; when they fall it is negative; when there is no trend there is no correlation.
Add the pairs one at a time. After three dots, say what you expect; then add the rest.
3 pairs plotted. The latest, (5, 330), is 5 hours of sleep and a reaction time of 330 ms.
Show me the steps
- One pair: (4, 360) is 4 hours of sleep and a reaction time of 360 ms, so its dot is 4 across and 360 up.
- Three pairs: (4, 360), (5, 345) and (5, 330). Two pupils slept 5 hours, so two dots share a column.
- Six pairs: the dots now run from the top left towards the bottom right.
- All ten pairs: the more sleep, the lower the reaction time, so the direction is negative.
Two slips to watch for. A dot needs both numbers of a pair, so (5, 330) is 5 across and 330 up, not two separate points. And “negative” describes the direction of the dots, not whether the news is bad: here a lower reaction time means a quicker reaction.
Worked examples
A shop records the noon temperature (°C) and the hot drinks sold: (14, 82), (18, 60), (22, 41), (27, 20). Which variable is independent, and what is the direction?
- The weather is not changed by the drinks sold, so temperature is independent: it goes on the x axis.
- Drinks sold respond to the temperature, so they are the dependent variable on the y axis.
- As x rises from 14 to 27, y falls from 82 to 20, so the direction is negative.
Choose axes for those four pairs.
- The x values run from 14 to 27, so an axis from 10 to 30 in steps of 5 holds them.
- The y values run from 20 to 82, so an axis from 0 to 100 in steps of 20 holds them.
- Label each axis with its name and unit, and give the graph a title.
Your go. The first step is given. A bus company records each bus's age in years and its repair bill in thousands of rupees: (1, 12), (3, 20), (6, 31), (9, 47). Which is independent, and what is the direction?
- Age is not changed by the repair bill, so age is the independent variable and goes on the x axis.
- Say what happens to the repair bill as the age rises, then name the direction.
Check the answer
The bill rises from 12 to 47 as the age rises from 1 to 9, so the direction is positive.
Check before you go on
1. A teacher records, for each pupil, the hours spent on homework in a week and the mark in a Friday quiz. Which variable goes on the x axis?
Your turn
Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.
Level 1 · Getting started
2. A survey records hours of sleep and reaction time for 12 pupils, but 2 pupils did not give their hours of sleep. How many dots can be plotted?
3. A school records distance from home (km) and journey time (minutes): (1, 5), (2, 9), (4, 14), (6, 22), (9, 31). How many more minutes does the 9 km pupil take than the 2 km pupil?
Level 2 · Building
4. Four pairs (x, y): (2, 90), (4, 70), (6, 55), (8, 30). On average, by how much does y fall each time x goes up by 2?
5. A scatter graph's y axis runs from 240 ms to 380 ms and is drawn 7 cm tall. How many cm tall is the gap between 300 ms and 340 ms on the page?
6. Which pair of variables would you expect to show a negative direction on a scatter graph?
Level 3 · Stretch
7. Prices (in thousands of rupees) of cars aged 1 to 7 years, in order of age: 900, 820, 760, 610, 650, 480, 390. Going from each car to the next older one, how many of the 6 moves go UP in price?
8. Every dot on a scatter graph lies exactly on the line y = 120 − 4x, for x from 5 to 15. What is the largest value of y on the graph?
Something to think about
Swap the axes so reaction time is on the x axis and hours of sleep is on the y axis. Would the dots still run downwards? Which way round would you rather show it, and why?
One way to think about it
Yes, they still run from top left to bottom right: a longer reaction time goes with less sleep, which says the same thing. The usual choice is sleep on the x axis, because we ask whether sleep explains the reaction time and not the other way round.
Exit ticket
9. A coach plots weekly training sessions (x) against the time in seconds for a 100 m sprint (y). The dots run from the top left to the bottom right. Which statement is correct?