Constructing and reading a time-series plot
A time series pairs a response with a sequence of periods. Plot every pair on the stated axes, join consecutive periods and read coordinates using the axis scale
\[\text{period }t\text{ with value }y_t\Rightarrow\left(t,y_t\right)\]
- Use time as the horizontal variable Period number is the explanatory variable and the recorded response is the vertical variable \(\displaystyle x=t,\quad y=y_t\)
- Plot each ordered pair Match each period index with the response observed at that period \(\displaystyle t=3,\quad y_t=142\Rightarrow\left(3,142\right)\)
- Join points in time order Connect consecutive observations by increasing period index, even when the table is displayed differently \(\displaystyle \left(1,120\right)\to\left(2,135\right)\to\left(3,142\right)\)
- Read the stated scale Use the correct coordinate and subtract the earlier response from the later response for a change \(\displaystyle \Delta y=162-135=27\)
Worked example
A regional ferry service records 118, 127, 123 and 139 passengers in four consecutive weeks. List the plotted pairs in time order and find the change from week 1 to week 4.
\[y_1=118,\quad y_2=127,\quad y_3=123,\quad y_4=139\]
- \(\displaystyle P_1=\left(1,118\right),\quad P_2=\left(2,127\right),\quad P_3=\left(3,123\right),\quad P_4=\left(4,139\right)\)Pair each passenger count with its consecutive week index
- \(\displaystyle 1<2<3<4\)The points are joined by increasing time index
- \(\displaystyle \Delta y_{1\to4}=139-118=21\)Subtract the week 1 response from the week 4 response
- \(\displaystyle P_1\to P_2\to P_3\to P_4\)The time-series plot joins each pair to the next pair in time order
Common mistakes
Joining in displayed order
✕\(\displaystyle \left(3,142\right)\to\left(1,120\right)\to\left(4,162\right)\to\left(2,135\right)\)✓\(\displaystyle \left(1,120\right)\to\left(2,135\right)\to\left(3,142\right)\to\left(4,162\right)\)
Connections follow increasing period index, not the order in which rows or columns are shownExchanging the axes
✕\(\displaystyle \text{horizontal: response},\quad\text{vertical: period}\)✓\(\displaystyle \text{horizontal: period},\quad\text{vertical: response}\)
Time is the explanatory variable, so period belongs on the horizontal axis and the response belongs on the vertical axisLeaving the points unconnected
✕\(\displaystyle P_1,P_2,P_3,P_4\text{ plotted separately}\)✓\(\displaystyle P_1\to P_2\to P_3\to P_4\text{ joined}\)
A time-series plot shows movement between consecutive observations by joining the pointsDifficulty levels
introChronological observations are used to select a correct plot and read one response value
standardNamed periods and table layouts are combined with a response or period-index read
challengeShuffled observations must be ordered before a change between two periods is calculated
examA compact plot-selection choice tests axes, joins and chronological order together
Questions about this topic
What makes data a time series?+
The response is observed at successive times or periods, so each value has a time index
Which axis shows time?+
Time or period number is placed on the horizontal axis because it is the explanatory variable
Why are consecutive points joined?+
The joins show how the response changes from one time period to the next in the recorded series
Can I use a calculator?+
A calculator is allowed unless a question is marked no calculator
Which year level and chapter is this?+
Year 12, VCE Units 3 & 4, General Mathematics, chapter 5: Time series
In this chapter
Construct and read a time series plotIdentify qualitative features in a time seriesCalculate moving meansMoving median smoothingSelect and compare smoothing methodsInterpret and complete seasonal indicesDeseasonalise and reseasonalise valuesCalculate seasonal indices from complete cyclesFit a time trend and forecast