Line graphs and time series: worked examples and method

Data over time

A time series is a set of readings taken at regular times, such as the temperature each hour. A line graph shows it by joining the points in time order, with time on the horizontal axis.

Reading a value

To read a value, find the time on the horizontal axis, go up to the line, then across to the vertical scale. Check the scale first: the gaps between the grid lines may be 2, 5 or 10, not 1.

Rises and falls

A rise or fall is the difference between two readings. Subtract the earlier reading from the later one. A positive answer is a rise and a negative answer is a fall.

Trend and rate of change

The trend is the overall direction: rising, falling or staying about the same, even if the line wobbles. The steepest part of the line is where the quantity changes fastest. The mean rate of change is the total change divided by the time taken.

Drawing a line graph

Choose a scale that fits the data and use it evenly. Plot each point carefully. Join the points with straight lines in time order. Label both axes and give the graph a title.

Patterns and predictions

Some time series repeat, such as sales that are highest every winter. You can use a repeating pattern to make a sensible prediction. Be careful: a prediction beyond the data is only an estimate, and something may change.

The mean of readings

To find the mean of several readings, add them all up and divide by how many readings there are. For example, the readings 10, 12 and 17 add up to 39, and 39 ÷ 3 = 13, so the mean is 13. A mean rate of change works the same way: the total change divided by the time taken.

Reading values from a line graph

To read a value, find the time on the horizontal axis, go up to the line, then across to the vertical axis. Look at the numbers on the axis before you start. The grid lines may go up in 2s, 5s or 10s, so each small square is not always worth 1.

When a reading falls between two grid lines, work out its position. Half way between 20 and 25 is 22.5. When a question gives the readings in a table instead of a graph, you use them in exactly the same way.

Method: 1. Read the labels and the scale on both axes. 2. Find the time, go to the line, then across to the scale. 3. Write the value with its unit.

Tip: Work out what one small square is worth before you read anything off the graph.

Rises, falls and the mean rate of change

A rise or fall is a difference between two readings. If the rainfall was 55 mm in February and 35 mm in March, the change is 35 - 55 = -20, which is a fall of 20 mm. Always take the earlier reading from the later one, or take the smaller from the larger and say whether it is a rise or a fall.

The mean rate of change is the total change divided by the time. If the temperature rises from 12 degrees at 9 am to 21 degrees at 12 noon, the total change is 9 degrees over 3 hours, so the mean rise is 3 degrees each hour. Count the hours between the readings, not the number of readings.

The mean of several readings is found by adding them all up and dividing by how many readings there are. For example, the readings 10, 12 and 17 add up to 39, and 39 ÷ 3 = 13, so the mean is 13.

Method: 1. Find the two readings and subtract the earlier from the later. 2. Count the time between the readings. 3. Divide the total change by the time.

Trends and repeating patterns

The trend is the overall direction. A graph of sales might wobble from month to month but still trend upwards over the year. Describe a trend by quoting actual readings, such as rising from 20 in January to 34 in June, rather than just saying it goes up.

Some time series repeat. Sales of ice cream are high every summer, and the number of buses used is high every weekday morning. A repeating pattern helps you predict. If the quarterly sales each rose by 4 from one year to the next, the same quarter next year will probably be 4 more again.

Method: 1. Say whether the overall direction is rising, falling or steady. 2. Quote readings to back up what you say. 3. Look for a pattern that repeats and use it to predict the next value.

Tip: Always say that a prediction is an estimate, because the pattern may not carry on.