<P> Equality holds if and only if all the elements of the given sample are equal . </P> <P> In descriptive statistics, the mean may be confused with the median, mode or mid-range, as any of these may be called an "average" (more formally, a measure of central tendency). The mean of a set of observations is the arithmetic average of the values; however, for skewed distributions, the mean is not necessarily the same as the middle value (median), or the most likely value (mode). For example, mean income is typically skewed upwards by a small number of people with very large incomes, so that the majority have an income lower than the mean . By contrast, the median income is the level at which half the population is below and half is above . The mode income is the most likely income, and favors the larger number of people with lower incomes . While the median and mode are often more intuitive measures for such skewed data, many skewed distributions are in fact best described by their mean, including the exponential and Poisson distributions . </P> <P> The mean of a probability distribution is the long - run arithmetic average value of a random variable having that distribution . In this context, it is also known as the expected value . For a discrete probability distribution, the mean is given by ∑ x P (x) (\ displaystyle \ textstyle \ sum xP (x)), where the sum is taken over all possible values of the random variable and P (x) (\ displaystyle P (x)) is the probability mass function . For a continuous distribution, the mean is ∫ − ∞ ∞ x f (x) d x (\ displaystyle \ textstyle \ int _ (- \ infty) ^ (\ infty) xf (x) \, dx), where f (x) (\ displaystyle f (x)) is the probability density function . In all cases, including those in which the distribution is neither discrete nor continuous, the mean is the Lebesgue integral of the random variable with respect to its probability measure . The mean need not exist or be finite; for some probability distributions the mean is infinite (+ ∞ or − ∞), while others have no mean . </P> <P> The generalized mean, also known as the power mean or Hölder mean, is an abstraction of the quadratic, arithmetic, geometric and harmonic means . It is defined for a set of n positive numbers x by </P>

What does the mean of probability distribution tell us