mean

There are several kinds of mean in mathematics, especially in statistics.

For a data set, the arithmetic mean, also called the expected value or average, is the central value of a discrete set of numbers: specifically, the sum of the values divided by the number of values. The arithmetic mean of a set of numbers x₁, x₂, ..., x_n is typically denoted by ${\displaystyle {\bar {x}}}$. If the data set were based on a series of observations obtained by sampling from a statistical population, the arithmetic mean is the sample mean (denoted ${\displaystyle {\bar {x}}}$) to distinguish it from the mean of the underlying distribution, the population mean (denoted ${\displaystyle \mu }$ or ${\displaystyle \mu _{x}}$).

In probability and statistics, the population mean, or expected value, is a measure of the central tendency either of a probability distribution or of the random variable characterized by that distribution. In a discrete probability distribution of a random variable X, the mean is equal to the sum over every possible value weighted by the probability of that value; that is, it is computed by taking the product of each possible value x of X and its probability p(x), and then adding all these products together, giving ${\displaystyle \mu =\sum xp(x)....}$. An analogous formula applies to the case of a continuous probability distribution. Not every probability distribution has a defined mean (see the Cauchy distribution for an example). Moreover, the mean can be infinite for some distributions.

For a finite population, the population mean of a property is equal to the arithmetic mean of the given property, while considering every member of the population. For example, the population mean height is equal to the sum of the heights of every individual—divided by the total number of individuals. The sample mean may differ from the population mean, especially for small samples. The law of large numbers states that the larger the size of the sample, the more likely it is that the sample mean will be close to the population mean.

Outside probability and statistics, a wide range of other notions of mean are often used in geometry and mathematical analysis; examples are given below.