Time Series, Moving Averages and Quality Assurance
How do you see the real trend when data jumps up and down with the seasons? Learn to smooth a time series with moving averages, and how factories use control charts to keep quality on track.
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Data over time
A time series is data recorded at regular intervals over time - monthly sales, daily temperatures, quarterly profits - and plotted as a line graph. Two features matter: the trend (the general long-term direction - rising, falling or flat) and seasonal variation (regular ups and downs that repeat, like ice cream sales peaking each summer). The problem: the seasonal jumps can hide the trend. This module shows how a moving average fixes that, and how factories use the same statistics for quality control.
Words for a time series
Learn these four - the steps that follow put them to work.
Trend and seasonal variation
- Time series
- Trend
- Seasonal variation
- Moving average
- Data collected at regular intervals over time
- The general long-term direction of the data
- A regular up-and-down pattern that repeats each period
- An average of consecutive values, repeated to smooth the data
Read the trend
A shop's quarterly sales rise every summer and fall every winter, but each year's figures are a little higher than the last. What is the underlying trend?
- Rising - the overall level increases year on year despite the seasonal ups and downs
- Falling - because sales drop every winter
- Flat - the seasons cancel out to nothing
- There is no trend, only seasonal variation
Calculate a moving average
The first four quarterly figures are 10, 12, 14 and 16. Work out the first 4-point moving average: add the four values and divide by 4.
How the smoothing works 〰️
A 4-point moving average takes four values in a row, averages them, then slides along one step and does it again - so with quarterly data each average covers a whole year. Because every average includes one of each season, the seasonal highs and lows cancel out. Plotting the moving averages gives a much smoother line that shows the real trend. (The general version averages any n points: (x1 + ... + xn) / n. [Higher])
Why use a moving average?
What is the main reason for calculating a moving average of a time series?
- To smooth out the seasonal variation so the underlying trend is clear
- To make all the numbers bigger
- To find the most common value in the data
- To measure how spread out the data is
Quality control lines [Higher]
Factories check quality by taking samples and plotting each SAMPLE MEAN on a control chart. (Sample means are less spread out than individual items.) Two pairs of lines sit either side of the target.
Match each situation to the response [Higher]
- Sample mean within all the lines
- Sample mean outside a warning line
- Sample mean outside an action line
- The process is running normally; carry on
- Take another sample and watch closely; it may be chance
- Stop production and investigate or adjust the process
Quality assurance facts [Higher]
Select the TWO statements that are correct about quality-assurance control charts.
- Sample means are more closely distributed than individual measurements
- Action lines are drawn at plus or minus 3 standard deviations
- A point outside a warning line means production must stop at once
- A point outside an action line can safely be ignored
Smooth a time series, in order
Put the steps of using a moving average to show a trend into the right order.
- Plot the original time-series data as points over time
- Take the first group of consecutive values (for example the first four)
- Work out their average - the first moving-average value
- Slide along one step and average the next group, repeating along the series
- Plot the moving averages to draw the smooth trend line
On the factory floor
You are monitoring a control chart and interpreting a trend. Choose the best response each time.
- A sample mean falls just outside a warning line. [Higher] What is the correct response?
- The next sample mean falls outside an action line. [Higher] What should happen?
- A rising sales trend covers two years. A manager wants to predict sales ten years ahead from it. What is the wise statistical advice?
Seasonal swing, steady rise
A cafe's quarterly sales swing up in summer and down in winter, but rise a little each year. Explain how a moving average would help the owner see what is really happening, and give one caution about using the trend to predict the future.
- Explain what the seasonal variation and the trend are in this data
- Explain how a 4-point moving average would smooth the seasonal swings to reveal the trend
- Say what the owner could conclude about the direction of the business
- Give one caution about extrapolating the trend too far ahead
Say why you smoothed it
Two things win marks here. First, always say why a moving average is used: it smooths the seasonal variation so the trend shows - not to change the numbers. Second, be careful with prediction. Extending a trend a little way can be reasonable, but extrapolating far beyond the data is risky because the trend may not hold [Higher]. And for quality assurance, remember the pair: a warning line (+/-2 SD) means re-sample, an action line (+/-3 SD) means stop and investigate [Higher]. Naming the response is the mark-winner.