Moving mean

A specified moving average method is used to smooth time-series data. It forms a new smoothed series in which the irregular component is reduced.

If the time series has a seasonal component, a moving mean may be used to eliminate the seasonal component.

Each value in the time series is replaced by the mean of the value and a number of neighbouring values. The number of values used to calculate a moving mean depends on the type of time-series data. For weekly data, seven values are used; for monthly data, 12 values are used; and for quarterly data, four values are used. If the number of values used is even, the moving mean must be centred by taking the two-term moving means of each pair of consecutive moving means, forming a series of centred moving means. See Example 2 for an illustration of this technique.

In terms of an additive model for time-series data, Y = T + S + C + I, where:

• T represents the trend component,
• S represents the seasonal component,
• C represents the cyclical component, and
• I represents the irregular component
• the smoothed series = T + C.

Example 1 (weekly data)

Daily sales, in thousands of dollars, for a hardware store were recorded for 21 days. There is reasonably systematic variation over each 7-day period, and so moving means of order 7 have been calculated to attempt to eliminate this seasonal component. The moving mean for the first Thursday is calculated by:

(86 + 125 + 115 + 150 + 168 + 291 + 102) ÷ 7 = 148.14

The raw data and the moving means are displayed below.

Example 2 (quarterly data)

Statistics New Zealand’s economic survey of manufacturing provided the following data on actual operating income for the manufacturing sector in New Zealand: There is reasonably systematic variation over each 4-quarter period, and so moving means of order 4 have been calculated to attempt to eliminate this seasonal component. However, these moving means do not align with the quarters; the moving means are not centred. To align the moving means with the quarters, each pair of moving means is averaged to form centred moving means.

The first moving mean (between Mar-05 and Dec-05) is calculated by:

(17,322 + 17,696 + 17,060 + 18,046) ÷ 4 = 17,531

The centred moving mean for September 2005 is calculated by:

(17,531 + 17,565.5) ÷ 2 = 17,548.25

The raw data and the centred moving means are displayed below. Note that M, J, S and D indicate quarter years ending in March, June, September, and December respectively.