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

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

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

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