Averaging

Averaging is the process of computing a representative value for a set of numbers, typically to summarize data, reduce random variation, or estimate a quantity. In statistics, averages are measures of central tendency and help describe a data set’s typical value.

The arithmetic mean is the sum of values divided by their count. For a data set x1, x2, ..., xn, the mean is (x1 + x2 + ... + xn) / n. It is widely used but sensitive to outliers, which can distort the result for skewed data.

The geometric mean is the nth root of the product of the values and is suitable for rates and ratios. For positive values, the geometric mean is (x1 x2 ... xn)^(1/n). The harmonic mean is n divided by the sum of reciprocals, and it is useful for averaging rates or ratios such as speeds or densities.

A weighted average assigns different importance to values, with weights w1, w2, ..., wn. The weighted mean is (w1 x1 + w2 x2 + ... + wn xn) / (w1 + w2 + ... + wn). Weighted averages generalize arithmetic, geometric, and harmonic means.

For time series or streaming data, averages can be updated incrementally. A running or moving average uses a fixed window of recent values. An exponential moving average applies a smoothing factor to blend new observations with past averages, giving more weight to recent data.

In probability and statistics, the expected value is the average outcome of a random variable, a weighted average over its possible values according to their probabilities.

Applications include data analysis, forecasting, signal processing, image smoothing, and finance. Limitations of averaging include potential masking of distribution shape and sensitivity to outliers; robust alternatives include the median or mode.