meanaverage

Mean average, commonly referred to simply as the mean or arithmetic mean, is a measure of central tendency that summarizes a dataset with a single representative value. For a finite collection of numbers x1, x2, ..., xn, the arithmetic mean is defined as the sum of the values divided by the count: (x1 + x2 + ... + xn)/n. In probability and statistics, the population mean is denoted by μ, and the sample mean by x̄.

Related notions include the geometric mean, the harmonic mean, and weighted means. The geometric mean of positive numbers is (x1 x2 ... xn)^(1/n). The harmonic mean is n/(1/x1 + 1/x2 + ... + 1/xn). A weighted mean uses weights w1, w2, ..., wn and is defined as (w1 x1 + w2 x2 + ... + wn xn) / (w1 + w2 + ... + wn).

Calculation and interpretation: The mean takes every value into account and is sensitive to outliers; extreme values can pull the mean toward them. It is not robust to skewed distributions. The mean is appropriate for data measured on interval or ratio scales and that roughly follow a symmetric distribution. In probability theory, the mean is the expected value; the sampling distribution of the mean describes its variability, with the standard error equal to the standard deviation divided by the square root of the sample size.

Applications and considerations: The mean is widely used to summarize data in science, economics, education, and finance, and it often appears alongside the median and mode. Understanding its behavior helps in assessing data quality, distribution shape, and the impact of outliers on reported averages.