Time-Series Data
Theory
A time series records the same quantity at regular times, such as each month or each year. This Year 10A maths page shows how to read time-series graphs and tables, describe the trend and any seasonal pattern, find the biggest changes, use percentage change, fit a linear trend and use it to predict, and judge when a prediction is reliable.
A time series is a set of data collected at equal time intervals. Time is the independent variable, so it goes on the horizontal axis, and the points are joined in order to make a line graph. Data with one variable for each item (such as heights) are univariate; a time series pairs each value with a time, so it is bivariate.
The trend is the long-term direction: increasing, decreasing or steady. A seasonal pattern repeats at the same time every year, such as heating costs peaking each winter. A cycle repeats over a longer, less regular period, and irregular changes are one-off jumps or dips.
If a series rises by the same amount each time period, the trend is linear and can be modelled by \(y = a + bt\), where \(a\) is the value at \(t = 0\) and \(b\) is the change per time period. If it rises by the same percentage each period, the increases keep growing and the trend is non-linear.
Using a trend to predict inside the time span of the data is interpolation; predicting the future is extrapolation, which assumes the trend continues and is less reliable the further ahead you go.
| To find | Use |
|---|---|
| change between two times | later value \(-\) earlier value |
| biggest rise or fall | the change for each pair of neighbouring times; the largest is the steepest segment |
| rate of change | \(\dfrac{\text{change in value}}{\text{change in time}}\) |
| linear model | \(y = a + bt\) with \(t\) = time since the start |
How to describe a time series
- Read the axes: the time interval and the units of the values.
- Trend: look past the ups and downs. Is the series rising, falling or steady overall?
- Season: is there a pattern that repeats each year? Name when the peaks and dips happen and explain them.
- Changes: subtract neighbouring values to find the biggest rise or fall; use percentage change to compare the same season in different years.
- Unusual values: note any one-off jump or dip.
How to predict from a trend
- Check the form: equal increases (linear) or growing increases (for example a constant percentage).
- Linear: find \(b\) from two points, \(b = \dfrac{\text{change in value}}{\text{change in time}}\), then \(y = a + bt\). Substitute \(t\) to predict, or set \(y\) and solve for \(t\).
- Percentage growth: multiply by the growth factor once for each period.
- Comment: a prediction far beyond the data is extrapolation and may fail if conditions change.
(ii) Is there a long-term trend?
(iii) Find the percentage change in Quarter \(3\) visitors from \(2024\) to \(2025\), to \(1\) decimal place.
(i) Quarter 3 every year (\(95\), \(101\), \(108\) thousand). It is winter, the ski season, so this is a seasonal pattern.
(ii) Compare the same quarter each year. Quarter 3: \(95 \to 101 \to 108\); Quarter 1: \(12 \to 14 \to 15\). Every quarter is higher than a year before, so the trend is increasing.
| \(\text{(iii)}\ \dfrac{108 - 101}{101} \times 100\%\) | \(=\) | \(\dfrac{7}{101} \times 100\%\) |
| \(\) | \(\approx\) | \(\mathbf{6.9\%}\) |
| Year | \(2015\) | \(2016\) | \(2017\) | \(2018\) |
|---|---|---|---|---|
| Cars | \(48\) | \(51\) | \(55\) | \(54\) |
| Year | \(2019\) | \(2020\) | \(2021\) |
|---|---|---|---|
| Cars | \(60\) | \(57\) | \(62\) |
(ii) Between which two years did the number fall the most?
(iii) Describe the trend.
Subtract each year's value from the next year's.
| \(2015 \to 2016\) | \(:\) | \(+3\) |
| \(2016 \to 2017\) | \(:\) | \(+4\) |
| \(2017 \to 2018\) | \(:\) | \(-1\) |
| \(2018 \to 2019\) | \(:\) | \(+6\) |
| \(2019 \to 2020\) | \(:\) | \(-3\) |
| \(2020 \to 2021\) | \(:\) | \(+5\) |
(i) The greatest increase was 2018 to 2019 (\(6\) thousand): the steepest rising segment of the graph.
(ii) The biggest fall was 2019 to 2020 (\(3\) thousand).
(iii) Despite two small dips, the overall trend is increasing: from \(48\) thousand to \(62\) thousand in six years.
(i) Find the gradient in hours per year, and per decade.
(ii) Write a rule for \(H\), with \(t\) the number of years after \(2015\).
(iii) Predict \(H\) in \(2030\).
(iv) In which year does the trend give \(13.2\) hours? How reliable is it?
| \(\text{(i) } b\) | \(=\) | \(\dfrac{11.6 - 8.4}{2023 - 2015} = \dfrac{3.2}{8}\) |
| \(\) | \(=\) | \(\mathbf{0.4}\text{ h per year}\) |
That is \(10 \times 0.4 = \mathbf{4}\) hours per decade.
| \(\text{(ii) } H\) | \(=\) | \(\mathbf{8.4 + 0.4t}\) |
| \(\text{(iii) } t = 15:\ H\) | \(=\) | \(8.4 + 0.4(15) = \mathbf{14.4}\) |
| \(\text{(iv) } 8.4 + 0.4t\) | \(=\) | \(13.2\) |
| \(0.4t\) | \(=\) | \(4.8\) |
| \(t\) | \(=\) | \(12 \Rightarrow \mathbf{2027}\) |
Both \(2027\) and \(2030\) are after the data, so these are extrapolations. Viewing habits can change quickly, so the predictions become less reliable the further ahead they go.
(i) Is the growth linear?
(ii) Find the percentage increase each year.
(iii) If the pattern continues, predict the next value.
| \(\text{(i) increases}\) | \(:\) | \(64,\ 80,\ 100,\ 125\) |
The increases are not equal, so the growth is not linear. A straight trend line would under-predict.
| \(\text{(ii)}\ \dfrac{320}{256}\) | \(=\) | \(1.25\) |
| \(\dfrac{400}{320} = \dfrac{500}{400} = \dfrac{625}{500}\) | \(=\) | \(1.25\) |
Each value is \(1.25\) times the one before: a \(25\%\) increase each year.
| \(\text{(iii)}\ 625 \times 1.25\) | \(=\) | \(\mathbf{781.25}\text{ thousand}\) |
About \(781\,000\) subscribers, assuming the \(25\%\) growth continues.
Common pitfalls
Frequently asked questions
What is a time series?
A set of data recorded at equal time intervals, such as monthly rainfall or yearly population. Time goes on the horizontal axis and the points are joined to make a line graph.
What is the difference between a trend and a seasonal pattern?
The trend is the long-term direction of the data. A seasonal pattern is a rise and fall that repeats at the same time every year.
How do you find the biggest increase in a time series?
Subtract each value from the next one. The largest positive difference is the biggest increase, and it shows as the steepest rising part of the graph.
How do you predict from a linear trend?
Find the change per time period from two points, write y = a + bt, and substitute the time you want. To find when a value is reached, set y equal to it and solve for t.
Why are predictions far into the future unreliable?
They are extrapolations. They assume the trend carries on unchanged, but conditions such as prices, weather or habits can change.