Moving Averages and Line of Best Fit

Smoothing out the noise

This part of the course looks like two separate topics, moving averages on one side and lines of best fit on the other. They are the same move done to different kinds of data. In both cases you have real measurements that jump about, and underneath them there is something steadier you actually want to see. A shop takes more before Christmas and less in February, every year, and that pattern hides whether the business is growing. A scatter of two measurements never sits on a neat line, and the scatter hides how strongly they are related. In both cases the answer is to smooth the noise away: a moving average for a series that wobbles by season, a line of best fit for a scatter. And once you have smoothed something away, you can look at what you removed, which is what mean seasonal variation is. Hold that and the whole of this section is one idea rather than three.

Words for trends

Five terms, with their tiers. Treat the tier labels as part of the content.

Match the tool to the noise

Work out a four-point moving average

Higher tier. A shop records its sales over eight quarters, in thousands of pounds: 34, 44, 54, 40, 42, 52, 62, 48. Work out the first 4-point moving average, using the first four values.

What the moving averages show

Work all five of them out for that series and you get 43, 45, 47, 49 and 51. Look at what has happened. The original figures went up and down by as much as twenty from one quarter to the next, and the moving averages rise by exactly two every time, without a single dip. That steady rise is the trend, and it was there all along underneath the seasonal pattern. This is why a moving average is worth the arithmetic: nobody could have told from the raw figures whether the shop was growing, because the drop from 54 to 40 looks alarming and is entirely seasonal. Two things to notice for the exam. First, you take four values at a time because the cycle here is four quarters, and averaging a whole cycle is what makes the season cancel. If the cycle were seven days you would use a 7-point moving average, which is what the specification means by choosing an appropriate number of points. Second, there are fewer moving averages than original values, because each one uses four of them, and that is expected rather than a mistake.

The season you took out

Higher tier. The trend line through those moving averages gives a trend value of 44 at the third quarter and 52 at the seventh, which are the same season a year apart. The actual figures at those quarters were 54 and 62. Work out the mean seasonal variation for that quarter, in thousands of pounds.

Two kinds of smoothing

The same move, applied to two different kinds of data. Setting them side by side shows what changes and what does not.

This time you memorise them all

If you have done the module on spread, you learned something that does not transfer here, and it is worth stopping on. There, the surprise was that the standard deviation formula is printed in the question while the plain-looking outlier rule is not. On this part of the section the split runs the other way and there is no surprise at all: the 4-point moving average, the general n-point moving average and the double mean point are all formulae you must carry in your own head, and nothing in this part is given to you. That is not difficult, because all three are means and you already know how to find a mean. But do not walk in expecting them on the page. The one thing you are not asked to produce is the regression-line equation, which is Higher only and beyond what the specification sets out; know that it exists as the more precise way of doing what a line of best fit does by eye, and leave it there.

Plot the double mean point

Higher tier. This scatter graph shows five points: (2,10), (4,16), (6,18), (8,26) and (10,30), already plotted. The x values total 30 and the y values total 100, over five points. Plot the double mean point.

Why the line goes through it

You have plotted the double mean point at (6, 20). Why must the line of best fit pass through it?

• Because it is the average position of all the points, so a line balanced among them has to run through it
• Because it is always halfway between the lowest and highest points
• Because it is one of the data points
• Because every line of best fit passes through the origin

Which claims about the trend

The moving averages for the shop were 43, 45, 47, 49 and 51. Select the TWO statements that are supported.

• The underlying trend is upward and steady across the period covered
• The fall from 54 to 40 in the raw figures does not show the business shrinking
• Sales will continue to rise by two every quarter indefinitely
• The seasonal pattern has disappeared from the business

Smoothing in a sentence

A moving average smooths a series by averaging a whole _____, so the repeating pattern cancels and the _____ shows through. Because each average uses several values, there are _____ moving averages than original figures. What was smoothed away can then be measured: the mean seasonal _____ is how far a particular season sits from the trend on average. A line of best fit does the same job for a scatter, and it must pass through the double _____ point.

A trend answer, annotated

A question gives the eight quarterly figures and asks what the moving averages show about the business. Here is an answer that would score full marks. The 4-point moving averages are 43, 45, 47, 49 and 51 thousand pounds. Each is two higher than the one before, so the underlying trend is a steady rise of about two thousand pounds a quarter across the period. The raw figures rise and fall sharply, for example from 54 down to 40, but this is seasonal rather than a decline in the business, because the same pattern repeats in the following year and the moving averages show no fall at all. The mean seasonal variation for the third quarter is plus ten thousand pounds, so that quarter runs about ten thousand above the trend. Notice four things. The moving averages are quoted with their unit. The trend is described as a rate, not just as going up. The alarming-looking fall is explained rather than ignored, which is usually where the marks are. And nothing is said about what will happen next, because a trend describes the data you have rather than predicting beyond it.

The claim that goes beyond the data

Four statements about the shop. Select the ONE that claims more than moving averages can support.

• The trend rose by about two thousand pounds a quarter over the period measured.
• The drop from 54 to 40 is seasonal rather than evidence of decline.
• Sales will reach 60 thousand a quarter within two years.
• The third quarter runs about ten thousand pounds above the trend.

Reading a series and a scatter

You are given two sets of data and asked what each shows. Take the decisions in order.

• The first is daily visitor numbers that repeat a pattern every seven days. Which moving average do you use?
• You have the moving averages and they rise steadily. What can you say?
• The second set is a scatter of two measurements. Before drawing a line, what do you work out?
• Someone asks which formulae will be printed on the paper for this section. What do you tell them?

Write up the trend

A leisure centre records visitor numbers each quarter for two years, and separately records temperature against attendance on twenty days. Explain how you would analyse each, and what you could and could not conclude.

• Explain which technique suits the quarterly figures, and how many points you would average and why
• Explain what the moving averages would tell you that the raw figures do not
• Explain what mean seasonal variation is, and what it would tell the centre
• Explain which technique suits the temperature and attendance data
• Explain what the double mean point is and why the line must pass through it
• State which of the formulae you have used are given in the exam and which must be memorised
• Give one conclusion the data could not support, and say why not