Overview

In statistics, a moving average (rolling average or running average or moving mean or rolling mean) is a calculation to analyze data points by creating a series of averages of different selections of the full data set. Variations include: simple, cumulative, or weighted forms.

Mathematically, a moving average is a type of convolution. Thus in signal processing it is viewed as a low-pass finite impulse response filter. Because the boxcar function outlines its filter coefficients, it is called a boxcar filter. It is sometimes followed by downsampling.

Given a series of numbers and a fixed subset size, the first element of the moving average is obtained by taking the average of the initial fixed subset of the number series. Then the subset is modified by "shifting forward"; that is, excluding the first number of the series and including the next value in the series.

History

The technique of moving averages was invented by the Bank of England in 1833 to conceal the state of its bullion reserves.

Simple moving average

If the data used are not centered around the mean, a simple moving average lags behind the latest datum by half the sample width. An SMA can also be disproportionately influenced by old data dropping out or new data coming in. One characteristic of the SMA is that if the data has a periodic fluctuation, then applying an SMA of that period will eliminate that variation (the average always containing one complete cycle). But a perfectly regular cycle is rarely encountered.

For a number of applications, it is advantageous to avoid the shifting induced by using only "past" data. Hence a central moving average can be computed, using data equally spaced on either side of the point in the series where the mean is calculated. This requires using an odd number of points in the sample window.

A major drawback of the SMA is that it lets through a significant amount of the signal shorter than the window length. This can lead to unexpected artifacts, such as peaks in the smoothed result appearing where there were troughs in the data. It also leads to the result being less smooth than expected since some of the higher frequencies are not properly removed.

Cumulative average

Thus the current cumulative average for a new datum is equal to the previous cumulative average, times n, plus the latest datum, all divided by the number of points received so far, n+1. When all of the data arrive (n = N), then the cumulative average will equal the final average. It is also possible to store a running total of the data as well as the number of points and dividing the total by the number of points to get the CA each time a new datum arrives.

Weighted moving average

A weighted average is an average that has multiplying factors to give different weights to data at different positions in the sample window. Mathematically, the weighted moving average is the convolution of the data with a fixed weighting function. One application is removing pixelization from a digital graphical image. This is also known as Anti-aliasing

The graph at the right shows how the weights decrease, from highest weight for the most recent data, down to zero. It can be compared to the weights in the exponential moving average which follows.