Time series: moving averages, seasonal components and forecasts

Last time

Before we start: last time, in Scatter diagrams, correlation and the line of best fit by semi-averages, you learned these points. Here they are again on one card. Keep them in mind, because today builds on them.

Patterns over time

Sales, temperatures and visitor numbers often follow a pattern that repeats each year, on top of a long-term trend. This lesson covers moving averages, which reveal the trend, and seasonal components, which measure the repeating pattern and are used to forecast.

A time series

A time series is a set of values recorded at equal time intervals, such as each quarter of a year. The trend is the long-term movement; seasonal variation repeats in every cycle, such as higher sales every summer. Random variation is what is left: unpredictable ups and downs.

Moving averages

A moving average averages a fixed number of consecutive values, then moves along one step at a time. Its period should equal the length of the seasonal cycle: four for quarterly data, twelve for monthly data. Each average is plotted at the middle of the values it covers, and together they show the trend.

Centring

A four-point moving average sits midway between the second and third quarters it covers, not at a quarter. So it is centred: two consecutive moving averages are averaged, which places the result at a quarter. Centring is needed for an even period, and not for an odd one, such as a three-point average.

Quarterly data, worked

Eight quarters, the moving averages each centred value uses, the centred values and the seasonal components.

The moving averages

The first four-point average is 20 plus 34 plus 28 plus 14, over 4: 24. The next is 34 plus 28 plus 14 plus 24, over 4: 25. The next three are 26, 27 and 28. Centring 24 and 25 gives 24.5, at the third quarter of year one. The trend rises by 1 each quarter.

Seasonal components

The seasonal component is the actual value minus the centred moving average. Year one, quarter three: 28 minus 24.5 is 3.5. Quarter four: 14 minus 25.5 is minus 11.5. With several years of data, average the components for each quarter to get the average seasonal component.

Forecasting

Extend the trend line to the quarter you want, and read off the trend value. The trend reaches 27.5 at year two, quarter two, and rises by 1 a quarter, so it is 30.5 at year three, quarter one. Add the seasonal component for that quarter, minus 2.5: the forecast is 28. Forecasts further ahead are less reliable, as the trend may change.

Mistakes that lose marks

Mistake one: plotting a four-point moving average at a quarter without centring it. Mistake two: subtracting the wrong way round: trend minus actual. Mistake three: forecasting from the trend without adding the seasonal component. The fix: centre, actual minus trend, then trend plus season.

The full-marks tip

The full-marks tip: show each moving-average total, and say where each value is plotted. For a forecast, state the trend value read from your line, then add the seasonal component.

On one card

So, on one card: time series. Keep it for revision.