GCSE Maths: Line graphs and time series (Foundation)
Part of the GCSE Maths statistics chapter (Foundation). Chapter overview · Higher version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Line graphs and time series
A time series is a set of measurements taken at regular times: the temperature every hour, the number of customers every month, a baby’s mass every week. We show a time series on a line graph: time goes on the horizontal axis, the measurement on the vertical axis. We plot a point for each reading and join the points with straight lines, from left to right.
What a line graph shows. Reading a line graph, look for three things. The trend is the general direction over a long time (here, sales rise from year to year). Seasonal variation is a pattern that repeats regularly (here, sales are highest in summer, Q3, every year). Unusual values do not fit the pattern and may need explaining.
Predicting. To predict the next value, continue the trend and the seasonal pattern. For example, Q3 sales were 58, 63, 68 in three years: they go up by 5 each year, so Year 4’s Q3 might be 73. This is an extrapolation (going beyond the data), so it is only an estimate and could be wrong if conditions change. Reading a value between two plotted points is interpolation: it is less risky, but it assumes the values change steadily between the readings.
Key ideas
- A time series is data measured at regular times; show it on a line graph with time on the horizontal axis.
- Trend = long-term direction. Seasonal variation = a pattern that repeats regularly.
- Compare the same season in different years to see the trend.
- Interpolation (between points) is more reliable than extrapolation (beyond the data).
- A line graph that does not start at 0, or has an uneven scale, can mislead.
Method. Find each quarter on the horizontal axis, go up to the plotted point, and read across to the vertical axis.
- (a) Find Q2 Y2 and read the point. The 6th point from the left is Q2 of Year 2.Sales = 45 hundred = 4500
- (b) Find the highest point. The highest point on the graph is the greatest sales.Highest point = 68 hundred, at Q3 of Year 3
- (c) Read the two Q3 points. Q3 Y3 and Q3 Y1.Q3 Y3 = 68, Q3 Y1 = 58
- Subtract. “How many more” means take away.68 − 58 = 10 hundred = 1000
Check. The Q3 points are 58, 63, 68, rising by 5 each year, so 68 is 10 more than 58. ✓
Watch out. Remember the units: the vertical axis is in hundreds.
Method. Compare the same quarter in different years for the trend, and compare the quarters within a year for the seasonal pattern.
- Look at the same quarter each year. This removes the seasonal effect so that only the trend is left.Q1: 24, 27, 30 (up 3 each year)
Q3: 58, 63, 68 (up 5 each year) - State the trend. The values for every quarter go up from year to year.Trend: sales are rising each year.
- Look within one year. Compare the four quarters of Year 1.Year 1: 24, 41, 58, 30: low, then rising to a peak in Q3, then falling.
- State the seasonal pattern. The same shape repeats in every year.Seasonal pattern: sales are highest in Q3 (summer) and lowest in Q1 (winter) every year.
Check. Each year looks like a bigger version of the year before. ✓
Watch out. The trend and the seasonal pattern are different: the trend compares years, the seasonal pattern compares seasons.
| Age (months) | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|---|
| Mass (kg) | 3.4 | 4.1 | 4.9 | 5.6 | 6.1 | 6.6 | 7 |
Method. For a line graph, plot a point for each time and join the points with straight lines, from left to right.
- Draw and label the axes. Time goes on the horizontal axis. The measurement goes on the vertical axis, with an equal scale.Horizontal axis: Age (months). Vertical axis: Mass (kg).
- Plot a point for each pair of values. Go along to the time, then up to the value.0: 3.399999999999999911182158029987476766109466552734375
1: 4.0999999999999996447286321199499070644378662109375
2: 4.9000000000000003552713678800500929355621337890625
3: 5.5999999999999996447286321199499070644378662109375
4: 6.0999999999999996447286321199499070644378662109375
5: 6.5999999999999996447286321199499070644378662109375
6: 7 - Join the points with straight lines. The lines show the trend. Do not draw a line before the first point or after the last one.
Check. Each point matches a value in the table. ✓
Watch out. Plot every point accurately before you join them.
Method. Find how the Q3 sales change each year, continue the pattern one more year, and comment on the risk.
- Write the Q3 sales for each year. Same quarter each year, so no seasonal effect.58, 63, 68
- Find the yearly change. Subtract each value from the next.63 − 58 = 5
68 − 63 = 5 - Continue the pattern. Sales have gone up by 5 each year, so assume they go up by 5 again.68 + 5 = 73 hundred
- Comment on reliability. This is an extrapolation: it assumes the trend continues, but weather, prices or competition could change.It is only an estimate.
Check. The sequence 58, 63, 68, 73 has a constant difference of 5. ✓
Watch out. Say that the estimate assumes the trend continues.
Watch out
The trend compares the same season in different years. Do not compare Q1 with Q3 to find the trend: that mixes the seasonal pattern with the trend. And never say a prediction is certain.Exercise 24H Line graphs and time series
Non-calculatorMethod. Find the lowest point on the graph.
- Find the lowest point. The lowest sales are at the bottom of the graph.Lowest value = 24 hundred, which is the first point.
- Name the quarter. Q1 of Year 1
Check. All other points are higher than 24. ✓
Watch out. Name the quarter and the year.
Method. Read the four Year 2 points and add them.
- Read the Year 2 values. Q1 to Q4 of Year 2.27, 45, 63, 34
- Add. 27 + 45 + 63 + 34 = 169 hundred
Check. Year 2 is bigger than Year 1 (153) and smaller than Year 3 (185). ✓
Watch out. Read all four quarters.
Method. Read the Q4 points for Year 1 and Year 3, then subtract.
- Read the two values. Q4 Y1 = 30, Q4 Y3 = 38
- Subtract. 38 − 30 = 8 hundred
Check. 30 + 8 = 38. ✓
Watch out. Remember the units are hundreds.
| Month | Jan | Feb | Mar | Apr | May | Jun |
|---|---|---|---|---|---|---|
| Visitors | 120 | 150 | 210 | 260 | 240 | 300 |
Method. For a line graph, plot a point for each time and join the points with straight lines, from left to right.
- Draw and label the axes. Time goes on the horizontal axis. The measurement goes on the vertical axis, with an equal scale.Horizontal axis: Month. Vertical axis: Number of visitors.
- Plot a point for each pair of values. Go along to the time, then up to the value.Jan: 120
Feb: 150
Mar: 210
Apr: 260
May: 240
Jun: 300 - Join the points with straight lines. The lines show the trend. Do not draw a line before the first point or after the last one.
Check. Each point matches a value in the table. ✓
Watch out. Plot every point accurately before you join them.
Method. Compare the four quarters inside one year.
- Look at Year 1. The values for the four quarters.120, 80, 40, 90
- Describe the shape. The bill falls to a low in Q3 (summer) and rises again in Q4. The same shape repeats in Year 2.Highest in winter (Q1), lowest in summer (Q3).
Check. Year 2 shows the same shape: 115, 75, 38, 85. ✓
Watch out. A seasonal pattern repeats every year.
Method. Compare the same quarter in the two years.
- Find the change in each quarter. Year 2 minus Year 1.Q1: 115 − 120 = -5
Q2: 75 − 80 = -5
Q3: 38 − 40 = -2
Q4: 85 − 90 = -5 - State the trend. Every quarter is lower in Year 2, by between £2 and £5.The bills are falling slowly.
Check. All four differences are negative. ✓
Watch out. Compare like with like: the same quarter.
Method. Compare the two lines month by month.
- Read both lines each month. Jan: A 20, B 30. Feb: A 24, B 28. Mar: A 30, B 27.In Mar: A = 30, B = 27
- Find the first month with A above B. Mar
Check. In the months before Mar the blue line (A) is below the orange line (B). ✓
Watch out. Read the correct line using the key.
Method. Read both lines in June and subtract.
- Read June. A = 40, B = 30
- Subtract. 40 − 30 = 10 thousand pounds, which is £10,000
Check. 30 + 10 = 40. ✓
Watch out. The scale is in thousands of pounds.
Method. Find how far each shop’s sales vary.
- Find the range of A. Biggest minus smallest.40 − 20 = 20
- Find the range of B. 31 − 27 = 4
- Compare. A smaller range means more steady.B (4) is smaller than A (20), so B is more steady.
Check. B’s line is much flatter. ✓
Watch out. Give a reason using numbers from the graph.
Method. Find how the Q1 sales change each year, then continue the pattern.
- Find the yearly changes. Subtract each value from the next.27 − 24 = 3
30 − 27 = 3 - Continue the pattern. The Q1 sales go up by 3 each year.30 + 3 = 33 hundred
Check. 24, 27, 30, 33 goes up by 3 each time. ✓
Watch out. Use the same quarter each year, so the seasonal pattern does not interfere.
Method. Read the numbers on the vertical axis, not just the shape of the line.
- Look at the scale. The vertical axis starts at 100, and each gridline is only 1 thousand pounds.Axis: 100 to 105
- Work out the real change. The profit went from 100 to 104 thousand.104 − 100 = 4, and 4 ÷ 100 = 4%
- Explain. Cutting off the axis makes a small rise look steep.A rise of only 4% over four years is not “very fast”.
Check. On a scale from 0 the line would look almost flat. ✓
Watch out. Check where the axis starts and read the numbers.
Method. Think about what the line between two points means.
- What does a joining line mean? A line between two points suggests values in between, and a change that happens in order.On a time graph, the line shows the value between the times.
- Apply to colours. There is nothing “between” red and blue, and the colours have no natural order.The line has no meaning.
- Say what to use instead. Categories are shown by separate bars.Use a bar chart.
Check. Line graphs are for data that is measured over time or in order. ✓
Watch out. Join points only when the horizontal axis is a continuous quantity such as time.
Method. 1 pm is halfway between 12 noon and 2 pm, so estimate halfway between the two temperatures.
- Find the halfway point of the times. 1 pm is halfway between 12 and 2.Halfway
- Find the number halfway between 14 and 18. Add and divide by 2.(14 + 18) ÷ 2 = 16
Check. 14 + 2 = 16, 18 − 2 = 16. ✓
Watch out. This assumes the temperature changes steadily.
Method. Think about what we do not know between two readings.
- What do we know? Only the two readings.14°C at 12 noon, 18°C at 2 pm
- What do we assume? We assume steady change, but the real temperature could have gone up and down in between.The 1 pm value is an interpolation, an estimate.
Check. A reading at 1 pm could be anything near 16. ✓
Watch out. Interpolation assumes a steady change between the readings.
Key words. An average is one number that stands for a whole set of data. The mean is the total divided by how many values there are. The median is the middle value when the data is in order. The mode is the value that appears most often. The range is the biggest value minus the smallest value.
Method. To find the mean, add up all the values and divide by how many values there are.
- Read the four Year 2 values from the graph. Q1, Q2, Q3 and Q4 of Year 2.27, 45, 63, 34
- Add up all the values. The mean uses the total of all the data.27 + 45 + 63 + 34 = 169
- Count how many values there are. The mean shares the total equally between all the values, so we need to know how many there are.There are 4 values.
- Divide the total by the number of values. mean = total ÷ number of values.169 ÷ 4 = 42.25
Check. Multiply back: 42.25 × 4 = 169, which is the total. ✓
Watch out. The mean is not the middle value. Add every value, and divide by how many values there are (not by the largest value).
Method. Find how much Q3 sales rise each year, then count on.
- Find the yearly rise. Q3 rises by 5 each year.63 − 58 = 5 and 68 − 63 = 5
- Work out how many more years are needed. Sales need to rise from 68 to over 100.100 − 68 = 32, and 32 ÷ 5 = 6.4, so 7 more years
- Find the year. Year 3 plus 7 years.Year 3 + 7 = Year 10, with sales 68 + 35 = 103
- Comment. Predicting 7 years beyond the data is a long extrapolation: the trend may not continue.Not reliable.
Check. In Year 9, 68 + 30 = 98, which is not over 100. In Year 10 it is 103. ✓
Watch out. Round up: 6.4 years means you need 7 whole years.
| Time | 8am | 9am | 10am | 11am | 12pm | 1pm | 2pm | 3pm |
|---|---|---|---|---|---|---|---|---|
| Temperature (°C) | 12 | 14 | 17 | 19 | 21 | 22 | 20 | 18 |
Method. For a line graph, plot a point for each time and join the points with straight lines, from left to right.
- Draw and label the axes. Time goes on the horizontal axis. The measurement goes on the vertical axis, with an equal scale.Horizontal axis: Time. Vertical axis: Temperature (°C).
- Plot a point for each pair of values. Go along to the time, then up to the value.8am: 12
9am: 14
10am: 17
11am: 19
12pm: 21
1pm: 22
2pm: 20
3pm: 18 - Join the points with straight lines. The lines show the trend. Do not draw a line before the first point or after the last one.
Check. Each point matches a value in the table. ✓
Watch out. Plot every point accurately before you join them.
Method. Think about how much evidence there is and what could change.
- Count the evidence. Only two readings (January and March) are used, which is too few to show a trend.Two points are not enough.
- Think about seasons. Umbrella sales may be seasonal: they could rise again in the wet months.The pattern may repeat.
- Think about prediction. Continuing the fall for ever is an extrapolation and could be badly wrong.Extrapolation is unreliable.
Check. Two reasons are enough. ✓
Watch out. Mention the seasonal pattern or the risk of extrapolating.