Weighted average

A weighted average is more heavily influenced by some of the numbers it is calculated from than others.

It is calculated by:

1. multiplying each number by a weight
2. adding the results together
3. dividing the total by the sum of the weights.

Stock market indices are based on weighted averages of share prices. The weights are the market caps of the companies in the index. More complex variants on this are often used.

A useful weighted average for time series data (common in finance and economics) is and exponential one:

S_t = αx_t + (1 - α)S_{t - 1}, and
S₀ = x₀

where S is the exponentially weighted average, or smoothed value
x is the series being averaged,
subscripts denote time, with the first period being 0, and
0 < α < 1

This gives weightings to x_t, x_{t - 1}, x_{t - 2}, etc. in proportions 1, (1 - α), (1 - α)², etc.

When this is used to produce a new time series (often to draw a graph), when it is called exponential smoothing.