# arithmetic mean

If ${a}_{1},{a}_{2},\mathrm{\dots},{a}_{n}$ are real numbers, their *arithmetic mean ^{}* is defined as

$$A.M.=\frac{{a}_{1}+{a}_{2}+\mathrm{\dots}+{a}_{n}}{n}.$$ |

The arithmetic mean is what is commonly called the *average* of the numbers. The value of $A.M.$ is always between the least and the greatest of the numbers (http://planetmath.org/MinimalAndMaximalNumber) ${a}_{j}$. If the numbers ${a}_{j}$ are all positive, then $A.M.>\frac{{a}_{j}}{n}$ for all $j$.

A generalization^{} of this concept is that of *weighted mean*, also known as
*weighted average*. Let ${w}_{1},\mathrm{\dots},{w}_{n}$ be numbers whose sum is not zero,
which will be known as *weights*. (Typically, these will be strictly
positive numbers, so their sum will automatically differ from zero.) Then the
weighted mean of ${a}_{1},{a}_{2},\mathrm{\dots},{a}_{n}$ is defined to be

$$W.M.=\frac{{w}_{1}{a}_{1}+{w}_{2}{a}_{2}+\mathrm{\dots}+{w}_{n}{a}_{n}}{{w}_{1}+{w}_{2}+\mathrm{\dots}+{w}_{n}}.$$ |

In the special case where all the weights are equal to each other, the weighted mean equals the arithmetic mean.

Title | arithmetic mean |

Canonical name | ArithmeticMean |

Date of creation | 2013-03-22 11:50:42 |

Last modified on | 2013-03-22 11:50:42 |

Owner | drini (3) |

Last modified by | drini (3) |

Numerical id | 14 |

Author | drini (3) |

Entry type | Definition |

Classification | msc 26D15 |

Classification | msc 11-00 |

Synonym | average |

Synonym | mean |

Related topic | GeometricMean |

Related topic | HarmonicMean |

Related topic | ArithmeticGeometricMeansInequality |

Related topic | GeneralMeansInequality |

Related topic | WeightedPowerMean |

Related topic | PowerMean |

Related topic | GeometricDistribution2 |

Related topic | RootMeanSquare3 |

Related topic | ProofOfGeneralMeansInequality |

Related topic | ProofOfArithmeticGeometricHarmonicMeansInequality |

Related topic | DerivationOfHarm |

Defines | weighted mean |

Defines | weighted average |