**Background**

Moving averages were used for centuries prior to the current understanding of stochastic processes to stabilize estimates calculated from small numbers or to facilitate studies on the periodicity of observed events (early time-series analysis). Essentially, moving averages are methods for estimating incidence density when the time period spans several years. Although modern statistics is now helpful to estimate the variability of incidence density estimates, averaging estimates over longer time periods is useful as an intuitive method of reducing variability when there is a small number of observations in a particular time period.

There are two common methods for calculating moving averages. These methods have been used primarily in the disciplines of demography (centred moving average [CMA]) and economics (time-series analysis: prior moving average [PMA]). Both methods are frequently used in health studies but the demographic method, CMA, is recommended since this discipline more closely reflects health indicators and applications in public health. Both methods use the same data but report different reference time periods. Various other moving average methods are available but are not commonly used in public health and are not discussed here (Velleman and Hoaglin, 1981; Hseih, 1981).

**Method**

There are three practical considerations when deciding how to calculate moving averages. These issues will be discussed with a general recommendation to use the CMA method for most public health applications.

*Considerations when calculating moving averages.*

- The number of events (the numerator) is the most important consideration when calculating moving averages. Estimates for uncommon events in local populations will fluctuate and the appearance from these estimates can appear more stable if combined over several years. As the number of events for a calendar year decreases, the total number of years used to calculate moving averages should be increased.

The following example uses the mortality as a guideline for the number of years that should be averaged. These guidelines can be extrapolated to other indicators.

**Indicator****Population****Number of years of combined deaths**Crude death rate Ontario 1 to 3 years Health Units 3 to 5 years Age-specific death rates Ontario 3 years Health Units 5 years There are several special situations where an indicator estimate will fluctuate widely from year to year based on underlying factors. However, these factors are secondary to the purpose of reporting the indicator and reporting longer time periods reduces these yearly changes. For instance, a Mandatory Program and Service Guideline objective is to reduce the rate of influenza cases and deaths. A high rate of influenza deaths for a particular year may not be an indication of a poor influenza control program but rather of a particularly virulent strain of influenza for that year. Averaging several years reduces the influence of a virulent influenza strain. Similarly, total fertility rate fluctuates from underlying economic conditions. During economic recessions, the total fertility rate temporarily drops as couples delay fertility. It may be desirable to control for these economic cycles by estimating a wider time period if total fertility rate is used as an indicator of lifetime fertility.

- The type of indicator or the final use of the indicator will influence the type of moving average. The CMA is the standard method for indicators rooted in demography and should always be used in when these indicators are used. These demographic indicators include life expectancy and total fertility rate. The CMA should usually be used for other mortality indicators (crude mortality rate, direct and indirect standardization rates, age-specific mortality rates, infant and perinatal mortality rates) and fertility indicators (crude birth rate, general fertility rate) since these indicators closely resemble life expectancy and total fertility.

Frequently there is an interest in reporting the most recent data that is available. Reporting PMA will give the impression that data is more recent than a CMA; however, the same data is used for both moving averages and the PMA does not reflect more current data. For this reason, CMA can be used to provide a consistent approach for all indicators.

**Example of methods for calculating moving averages for the crude mortality rate in Ontario:**

where,

Mt = deaths

t = calendar year

Pt = population

*Central Moving Average*

M t-1 + M t + M t+1

3*x*P t

Report as the crude death rate for time t (on graphs) or time interval t-1 to t+1

*Prior Moving Average*

M t-2 + M t-1 + M t

3*x*P t-1

Report as the crude death rate for time t (on graphs) or the time interval t-2 to t.

- The final consideration is the quality the estimates used to define the at-risk population (the denominator). Traditionally, accurate estimates were available for only Census years and therefore CMAs were calculated for these mid-point years only. Statistics Canada now publishes yearly inter-Censal estimates that are widely used and reliable for most populations. Therefore, the following two methods of calculating CMA should yield the same result and most populations with stable population changes can used either method. If there are concerns regarding population estimates, CMA should be reported using method A, where t is a Census year.

*Method A*

Mt-1 + M2 + Mt+1

3*x*Pt

*Method B*

Mt-1 + Mt + Mt+1

Pt-1 + Pt + Pt+1

Addendum: Several spread sheet programs (including EXCEL) have functions to calculate PMA on graphs. SPSS has functions to derive both CMAs and PMAs on graphs and as new working variables.

**References**

1. Velleman, P. F., and D. C. Hoaglin. *Applications, basics, and computing of exploratory data analysis*. 1981; Boston: Duxbury Press.

2. Hseih, JJ. A general theory of life table construction and a precise abridge life table method. Biom J 1991:33(2);143-62.

**This page last updated: January 6, 2003**