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Plot, read and classify a basic time-series graph

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Time is horizontal; recorded response is vertical. Points are connected chronologically, using their actual time coordinates.246204060800DayWater (kL)

Daily water use is shown. Which quantity belongs on the horizontal axis?

Year 10, Chapter 5, 4 difficulty levels, Calculator allowedBuild a worksheet

Read a time series

Time is horizontal; the recorded response is vertical. Describe the full pattern and retain the measurement units

\[x=t,\quad y=y_t\]
  1. Read an observation Find the time coordinate, then read the response at that plotted point
  2. Read a change Subtract the earlier value from the later value. A negative result is a decrease \(\displaystyle \Delta y=y_2-y_1\)
  3. Describe the pattern Compare the full interval. Similar changes over equal time intervals suggest a linear trend; systematic changing increments suggest curvature
  4. Keep the limits clear Small fluctuations can occur around a level or trend. Joining lines do not create measured values between observations

Worked example

Library loans over four weeks are 196, 204, 211 and 207. Find the week 1 to week 3 change, then describe the overall pattern

\[196,204,211,207\]
  1. \(\displaystyle 211-196=15\)Week 3 has 15 more loans than week 1.
  2. \(\displaystyle 207-211=-4\)The last movement falls by 4 loans
  3. \(\displaystyle 196,204,211,207\)The full sequence generally rises despite that final fall. Do not classify the whole pattern from the final segment alone

Common mistakes

Using only the last movement
✕\(\displaystyle 219-223=-4\)✓\(\displaystyle 210,218,223,219\)
The first calculation describes only the last step. Read all four values to see the generally increasing pattern
Exchanging the coordinate roles
✕\(\displaystyle (42,3)\)✓\(\displaystyle (3,42)\)
If time is day 3 and the recorded response is 42, time is the first coordinate
Joining shuffled records
✕\(\displaystyle 3\to1\to4\to2\)✓\(\displaystyle 1\to2\to3\to4\)
Join by increasing time index, keeping each recorded response paired with its own time

Difficulty levels

introChronological observations with a rising or falling pattern include a direct value read
standardNamed periods are combined with a value, change or extremum read and an overall trend description
challengeShuffled records require time ordering before a change or stationary run is read
examA compact trend choice distinguishes the complete pattern from single-movement and form errors

Questions about this topic

What is a time series?+

It is a sequence of observations of a response variable recorded at successive times or periods

Why is time placed on the horizontal axis?+

Time is the explanatory variable because the response is observed as time progresses, so period number is the horizontal coordinate

How is an overall trend found?+

Read all periods and describe the long-term direction. A single final rise or fall does not decide the trend

Is a straight joining segment a measured value at every time?+

No. Values between recorded times are estimates unless separately observed. State the interpolation assumption

Can equal endpoints hide a trend change?+

Yes. The middle can rise then fall, or fall then rise. Inspect the whole series

What if observation times are unequal?+

Place them by their actual time coordinates. Do not give a three-day gap the same width as a one-day gap

Does time on the horizontal axis imply causation?+

No. Time indexes the observations; other variables or changes may explain the pattern

Can I use a calculator?+

A calculator is allowed unless a question is marked no calculator

Which year level and chapter is this?+

Year 10 Mathematics, chapter 5: Statistics

In this chapter

Classify data types5A Data collection and types: Collect data and evaluate biasBuild a frequency and percentage frequency tableBuild a bar chart from a frequency tableGroup numerical data into class intervalsConstruct a histogram from a frequency tableRead values from a histogramDescribe the shape of a distributionChoose an appropriate display for a variableRead values from a frequency table or bar chart5B Frequency tables and histograms: Critique statistical displays and claimsConstruct a dot plot from raw dataConstruct an ordered stem plot from raw data5C Dot plots and stem-and-leaf plots: Compare a numerical variable across groupsMean, mode, median and range of a data listFind the mean and the median, and choose between themFind measures of centre from a frequency displayFind the range, the quartiles and the IQRFind the fences and identify outliers5E Quartiles, interquartile range and outliers: Produce a five-number summary and construct a boxplot5F Box plots: Produce a five-number summary and construct a boxplotRead values and percentages from a boxplot5F Box plots: Compare a numerical variable across groupsPlot, read and classify a basic time-series graphConstruct and read a time series plotIdentify qualitative features in a time seriesIdentify the explanatory and response variablesConstruct a scatterplot from a data tableRead a scatterplotDescribe an association from a scatterplotAssociation is not causationFit a line of best fit by eye and make estimatesCompare categorical groups using two-way tables