In colloquial language, an average is a single number taken as representative of a list of numbers. Different concepts of average are used in different contexts. Often "average" refers to the arithmetic mean, the sum of the numbers divided by how many numbers are being averaged. In statistics, mean, median, and mode are all known as measures of central tendency, and in colloquial usage any of these might be called an average value.

The median is the middle value that separates the higher half from the lower half of the data set. The mode is the most frequent value in the data set.

The arithmetic mean, the geometric mean and the harmonic mean are known collectively as the Pythagorean [毕达哥拉斯的] means.

Harmonic mean [调和平均数] for a non-empty collection of numbers a1, a2, ..., an, all different from 0, is defined as the reciprocal of the arithmetic mean of the reciprocals of the ai's. One example where the harmonic mean is useful is when examining the speed for a number of fixed-distance trips. For example, if the speed for going from point A to B was 60 km/h, and the speed for returning from B to A was 40 km/h, then the harmonic mean speed is given by d / (d / 60 + d / 40) = 48 km/h, where d is the distance from A to B.

A well known inequality concerning arithmetic, geometric, and harmonic means for any set of positive numbers is: AM >= GM >= HM.
The alphabetical order of the letters A, G, and H is preserved in the inequality.

A type of average used in finance is the average percentage return. It is an example of a geometric mean. When the returns are annual, it is called the Compound Annual Growth Rate (CAGR). For example, if we are considering a period of two years, and the investment return in the first year is −10% and the return in the second year is +60%, then the average percentage return or CAGR, R, can be obtained by solving the equation: (1 − 10%) × (1 + 60%) = (1 − 0.1) × (1 + 0.6) = (1 + R) × (1 + R). The value of R that makes this equation true is 0.2, or 20%. This means that the total return over the 2-year period is the same as if there had been 20% growth each year. The order of the years makes no difference – the average percentage returns of +60% and −10% is the same result as that for −10% and +60%.

The law of averages is the commonly held belief that a particular outcome or event will, over certain periods of time, occur at a frequency that is similar to its probability. Depending on context or application it can be considered a valid common-sense observation or a misunderstanding of probability. This notion can lead to the gambler's fallacy when one becomes convinced that a particular outcome must come soon simply because it has not occurred recently (e.g. believing that because three consecutive coin flips yielded heads [扔硬币有人头像的一面朝上], the next coin flip must be virtually guaranteed to be tails). Or "The football team will win next time."

六级/考研单词: seldom, arithmetic, usage, data, geometry, reciprocal, alphabet, compound, invest, equate, differentiate, valid, misunderstand, gamble, convince, lately, consecutive, flip, yield, soccer