Unit 18 Section 5 : Trends

Moving averages can be used to make predictions. They do this by smoothing out monthly, seasonal or other periodic variations. For example, an ice-cream seller might expect to sell more in the summer than he does in the winter. He could use a moving average over the four seasons to find out if his sales are increasing for each 12 month period.

and so on. In each case, the oldest piece of data is replaced by the newest one. So, for the fifth moving average, the ice-cream seller would replace the winter sales figure for the first year with the winter sales figure for the second year, and so on. Because the mean of four items of data is being found every time, this is called a 4 point moving average.

Example Question

(a) Calculate the 4 point moving averages for this list of data:
            6    5    7    4    6.1    5.1    7.1    4.1

The first 4 point moving average is calculated using the mean of the first 4 items of data:
    1st moving average = (6 + 5 + 7 + 4) 4 = 5.5

For the second 4 point moving average we discard the first value (6) and include the next value (6.1) like this:
    2nd moving average = (5 + 7 + 4 + 6.1) 4 = 5.525

The next three 4 point moving averages are calculated in a similar fashion:
    3rd moving average = (7 + 4 + 6.1 + 5.1) 4 = 5.55
    4th moving average = (4 + 6.1 + 5.1 + 7.1) 4 = 5.575
    5th moving average = (6.1 + 5.1 + 7.1 + 4.1) 4 = 5.6

The 4 point moving averages for this list of data are: 5.5,   5.525,   5.55,   5.575,   5.6

(b) Estimate the next two values in the list.

Looking at the moving averages so far (showing them to 3 decimal places), we can see that they increase by 0.025 each time:
    5.500,   5.525,   5.550,   5.575,   5.600
It would be reasonable to assume that this pattern will continue, so the 6th moving average should be 5.625.

If we call the next value in the list x, this would mean:
    (5.1 + 7.1 + 4.1 + x) 4 = 5.625
     5.1 + 7.1 + 4.1 + x         = 5.625 4
                                 x         = 5.625 4 – 5.1 – 7.1 – 4.1
                                 x         = 6.2

Now we have predicted that the next value after 4.1 will be 6.2, and that the 7th moving average should be 5.650.
If we call the next value in the list y (after 6.2), this would mean:
    (7.1 + 4.1 + 6.2 + y) 4 = 5.650
     7.1 + 4.1 + 6.2 + y         = 5.650 4
                                 y         = 5.650 4 – 7.1 – 4.1 – 6.2
                                 y         = 5.2

Our predictions for the next two values in the list are 6.2 and 5.1.

Practice Question

Work out the answer to each part of the question then click Click on this button below to see the correct answer to see whether you are correct.

The table below gives the average daytime temperatures for each of the four seasons over a two-year period.

A weatherman wants to use a 4 point moving average to predict the temperature for Spring and Summer of Year 3.

(a) Start by calculating the first five moving averages:
      1st moving average:
      2nd moving average:
      3rd moving average:
      4th moving average:
      5th moving average:

The differences between the moving averages are:
      0.075,   0.1,   0.15,   0.125
When the differences are not constant, we average them to find out how much the next moving averages should increase by.

(b) What is the mean of the four differences between the moving averages?

To predict the 6th and 7th moving averages we need start with the 5th moving average and add this value on twice.

(c) What are the predictions for the 6th and 7th moving averages?

Now we can use the same method as in the first example to predict Spring and Summer temperatures for Year 3.

(d) What is the predicted temperature for Spring in Year 3

(e) What is the predicted temperature for Summer in Year 3

 

Exercises

Work out the answers to the questions below and fill in the boxes. Click on the Click this button to see if you are correct button to find out whether you have answered correctly. If you are right then will appear and you should move on to the next question. If appears then your answer is wrong. Click on to clear your original answer and have another go. If you can't work out the right answer then click on Click on this button to see the correct answer to see the answer.

Question 1
Look at the list of data below:

      4   3   5   4   3   5

(a) Calculate the 3 point moving averages for this set of data.
      1st moving average =
      2nd moving average =
      3rd moving average =
      4th moving average =

(b) What do you notice about the moving averages?
      

Question 2
Look at the list of data below:

      6   2   7   1   8   4   9   3   10

(a) Calculate the 4 point moving averages for this set of data.
      1st moving average =
      2nd moving average =
      3rd moving average =
      4th moving average =
      5th moving average =
      6th moving average =

(b) What do you notice about the moving averages?
      

(c) Predict the next two values in the list using a 4 point moving average.
      10th value =
      11th value =

Question 3
Use a 3 point moving average to predict the next two values in this list:

      4   6   5   5.5   7.5   6.5

What is the prediction for the 7th value in the list?

What is the prediction for the 8th value in the list?

Question 4
The sales of an ice-cream company are given in the table below, in thousands of ice-creams:

Use a 4 point moving average to predict the sales of ice creams (in thousands) sold in each season of 1998.
        Spring 1998: thousands
      Summer 1998: thousands
      Autumn 1998: thousands
       Winter 1998: thousands

Question 5
A school tuck shop keeps a record of the number of cans of drink it sells over a 3-week period:

Week 1Week 2Week 3
MonTuesWedThursFriMonTuesWedThursFriMonTuesWedThursFri
18229715192398162123101016

Use a 5 point moving average to estimate the sales of cans for week 4.

Do your working to 3 decimal places but as you work out each estimate round it to the nearest whole number.

     Monday:
     Tuesday:
Wednesday:
    Thursday:
        Friday:


 

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Produced by A.J. Reynolds April 2011