House price moving averages. Information sheet A Finding moving averages. Average house prices in the UK since 2000
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1 House price moving averages House prices vary a great deal. They are influenced by many factors, including the region of the UK where the houses are and the time of year. This activity introduces moving averages. These can be used to iron out seasonal fluctuations in price in order to look at the yearly trend. Information sheet A Finding moving averages The table and graph below show how average house prices in the UK have changed since 000. Q1 represents the first quarter of the year from January to March, Q the second quarter from April to June, Q3 the third quarter from July to September, and Q the fourth quarter from October to December. Average house prices in the UK since 000 Period Year Average price ( ) Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Period Year Average price ( ) Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Nuffield Free-Standing Mathematics Activity House price moving averages Student sheets Copiable page 1 of 8
2 Think about How have house prices in the UK changed since 000? Do the house prices change more in some quarters of the year than in others? What evidence does the table or graph give of this? How could fluctuations due to seasonal factors be smoothed out? Seasonally adjusted average house prices Moving averages can be used to smooth out fluctuations due to seasonal factors. In this case combining the values for quarters gives the moving averages shown below: (nearest ) 000Q where the symbol 000Q means the mean of the quarters up to and including the th quarter of 000. The calculation shows the most recent price first Q (nearest ) Q 87 3 (nearest ) Q Q (nearest ) These have been entered into the table below. Nuffield Free-Standing Mathematics Activity House price moving averages Student sheets Copiable page of 8
3 Period Year Average price ( ) Q Q Q Moving average ( ) Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Period Year Average price ( ) Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Moving average ( ) Try this A Calculate the moving averages for 00, 003 and 00. Enter the values you find into the table above. Nuffield Free-Standing Mathematics Activity House price moving averages Student sheets Copiable page 3 of 8
4 Information sheet B Alternative method for finding moving averages A moving average can also be found from the previous value as shown for 001Q below. 001Q 001Q (nearest ) So Q1 001Q (nearest ) Check that this value agrees with the value you have entered into the table above. Try this B Use the alternative method to calculate the moving averages for 005, 006 and 007. Enter the values you find into the table above. Complete the table using either of the methods for finding moving averages. Think about Both methods for finding moving averages are given in general terms below. Which method do you prefer? Why? Moving averages For data points p 1, p, the simple moving average at interval m with n data points is: m p m p m 1 p m... p m ( n1) n Alternatively, calculate successive values using m 1 m p m ( n1) n p m 1 n Nuffield Free-Standing Mathematics Activity House price moving averages Student sheets Copiable page of 8
5 Information sheet C Graph showing moving averages The 000 and 001 moving averages have been plotted on the House price graph below. Try this C Plot the other moving averages from the table onto the graph and join them with a smooth curve. Think about What effect has using moving averages had on the trend line? Can you think of a way to adjust the lag which has occurred? Nuffield Free-Standing Mathematics Activity House price moving averages Student sheets Copiable page 5 of 8
6 Information sheet D Weighted moving averages The problem of lag can be alleviated by using a weighted moving average. Here are calculations for the 000 and 001 weighted moving averages for house prices: Q (nearest ) Q (nearest ) Q (nearest ) Q (nearest ) Q (nearest ) The method for finding the linear weighted moving average is given in general terms in the bo nn 1 below. Note the formula given for finding the sum of values in the denominator. When n, n n 1 1 as used above. In this eample this is no quicker than adding, 3, and 1. However, in cases where n is large, nn 1 the formula is much quicker than adding the individual values. Weighted moving averages The linear weighted moving average is m np m n 1p m 1 n p m... n n 1 n... 1 p m ( n1) where the denominator is the triangular number with sum n n 1. Nuffield Free-Standing Mathematics Activity House price moving averages Student sheets Copiable page 6 of 8
7 The values calculated above have been entered into the table below. Average Period Year price ( ) Q Q Q Weighted moving average ( ) Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Period Year Average price ( ) Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Weighted moving average ( ) Try this D Complete the rest of the table. The graph on the following page shows the weighted moving averages for 000 and 001. Plot the other weighted moving averages from the table onto the graph, and join them with a smooth curve. You should find that the lag is less severe than on the previous graph. Nuffield Free-Standing Mathematics Activity House price moving averages Student sheets Copiable page 7 of 8
8 Reflect on your work What is a moving average? Why are they used? Describe two methods of calculating a moving average. Describe how to find a weighted moving average. Why does a weighted moving average help to overcome the problem of lag? Notes on the data The data are adapted from The Halifa House Price Inde spreadsheet at This website gives data for years before 000 which could be used for finding moving averages over longer periods. Please note that any use of the data for an individual's own or third party commercial purposes is done entirely at the risk of the person or persons making such reliance. Nuffield Free-Standing Mathematics Activity House price moving averages Student sheets Copiable page 8 of 8
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