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Arithmetic Mean - Statistics

Published: 2016/03/04

Channel: Jain Coaching

Arithmetic Means

Published: 2012/03/22

Channel: ProfRobBob

ARITHMETIC MEAN | DIRECT | SHORT CUT | STEP DEVIATION METHODS

Published: 2016/09/13

Channel: Jain Coaching

Arithmetic Mean or Median of AP ( GMAT / GRE / CAT / Bank PO / SSC CGL)

Published: 2015/04/20

Channel: Don't Memorise

How to find the Arithmetic Mean or Average presented by TeachtopiaTV

Published: 2011/05/07

Channel: TeachtopiaTV

Arithmetic mean - Insert n arithmetic means between two given number - solved example

Published: 2013/12/04

Channel: MathsSmart

What is the Arithmetic Mean?

Published: 2014/12/20

Channel: Don't Memorise

al2 sequences and series finding the arithmetic mean

Published: 2014/02/11

Channel: maths gotserved

Find 4 arithmetic means between 6 and 36

Published: 2012/06/14

Channel: Laura Rickhoff

Average or Central Tendency: Arithmetic Mean, Median, and Mode

Published: 2010/04/19

Channel: Khan Academy

Geometric Mean vs. Arithmetic Mean

Published: 2014/02/11

Channel: FerranteMath

What is ARITHMETIC MEAN? What does ARITHMETIC MEAN mean? ARITHMETIC MEAN meaning & explanation

Published: 2017/03/26

Channel: The Audiopedia

Arithmetic Mean

Published: 2012/01/16

Channel: mathgeekstutoring

Arithmetic Means

Published: 2013/02/21

Channel: G Hill

Arithmetic Mean in Short cut method

Published: 2015/09/16

Channel: CA N Raja Natarajan

Basic Stats- Arithmetic, Geometric and Harmonic Mean

Published: 2014/01/23

Channel: FinTree

Arithmetic Mean Average Word Problems

Published: 2016/02/24

Channel: Mathodman

Economics 11th Standard CBSE:- Calculating Mean Part 1 Arithmetic Mean

Published: 2014/03/03

Channel: sangram singh

001 Statistics - Measures of Central Tendency - Arithmetic Mean

Published: 2016/08/04

Channel: studyezee

Missing Frequency in Arithmetic Mean - Statistics

Published: 2017/05/28

Channel: Jain Coaching

SAT Math - Averages (Arithmetic Mean)

Published: 2012/07/06

Channel: SAT Prep Guy

Arithmetic mean Meaning

Published: 2015/04/14

Channel: SDictionary

Properties of Arithmetic Mean - Very Important

Published: 2016/08/14

Channel: 100Centum

Arithmetic Mean

Published: 2016/03/28

Channel: PCSciHS Math Tutorials

Geometric vs. Arithmetic Average Returns

Published: 2011/06/04

Channel: Kevin Bracker

Calculating the Mean using Step deviation method

Published: 2016/03/01

Channel: AtHome Tuition

Statistics - Arithmetic Mean LECTURE 5

Published: 2017/05/30

Channel: Vyaspeeth Satyen Vyas

Arithmetic and Geometric Means

Published: 2014/01/02

Channel: talkboard.com.au

Mathematical Properties of Arithmetic Mean | Dr Asha Chawla

Published: 2017/09/03

Channel: asha chawla

Arithmetic Mean (in Hindi)

Published: 2016/08/12

Channel: 100Centum

Ch 3 The Arithmetic Mean for Grouped Data

Published: 2016/09/23

Channel: Inna Gorlova

What is Combined Arithmetic Mean?

Published: 2014/12/20

Channel: Don't Memorise

Arithmetic Mean - IIT JEE Main and Advanced Maths Video Lecture

Published: 2014/05/13

Channel: Rao IIT Academy

Arithmetic Mean Problem 1

Published: 2010/11/26

Channel: GreeneMath.com

GRE Question#19: Statistics/Average/Arithmetic Mean

Published: 2013/05/05

Channel: Quantum Grad Prep

Write a sequence that has x arithmetic means between a and b

Published: 2017/04/16

Channel: Ms Shaws Math Class

Arithmetic Mean Part 1

Published: 2016/02/13

Channel: eLearning Meridian

ARITHMETIC MEAN SHORTCUT TRICK (IN HINDI) BY GIRISH KARIYA

Published: 2017/05/30

Channel: GIRISH KARIYA

What is Arithmetic Mean | AP | GP | CA CPT | CS & CMA Foundation | Class 11 | Class 12

Published: 2015/11/22

Channel: Mera Skill

Maths Data Handling part 3 (Arithmetic Mean) CBSE Class 7 Mathematics VII

Published: 2016/09/30

Channel: ExamFear Education

Proof by Contradiction: Arithmetic Mean & Geometric Mean

Published: 2014/03/27

Channel: Eddie Woo

Arithmetic, Harmonic, and Geometric Means in Excel

Published: 2015/10/15

Channel: Todd Grande

Arithmetic Mean for Samples and Populations

Published: 2010/07/06

Channel: statslectures

FSc Math Book1, CH 6, LEC 6: Arithmetic Mean

Published: 2016/12/03

Channel: Maktab. pk

Finding the arithmetic sequence arithmetic means.wmv

Published: 2012/07/30

Channel: maths gotserved

Arithmetic Mean vs Geometric Mean - What's The Difference & How To Find It?

Published: 2015/12/28

Channel: Math & Science 2024

Statistical Analysis with Excel: Finding the arithmetic mean

Published: 2010/03/11

Channel: Hoonuit

How to Calculate an Arithmetic Mean

Published: 2015/08/06

Channel: Mad Math Monkey

Arithmetic Mean and Geometric Mean in Relation to Profit

Published: 2015/08/03

Channel: Robert A. Bonavito, CPA

Arithmetic Meaning

Published: 2015/04/18

Channel: SDictionary

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In mathematics and statistics, the **arithmetic mean** ( /ˌærɪθˈmɛtɪk ˈmiːn/, stress on third syllable of "arithmetic"), or simply the mean or **average** when the context is clear, is the sum of a collection of numbers divided by the number of numbers in the collection.^{[1]} The collection is often a set of results of an experiment, or a set of results from a survey. The term "arithmetic mean" is preferred in some contexts in mathematics and statistics because it helps distinguish it from other means, such as the geometric mean and the harmonic mean.

In addition to mathematics and statistics, the arithmetic mean is used frequently in fields such as economics, sociology, and history, and it is used in almost every academic field to some extent. For example, per capita income is the arithmetic average income of a nation's population.

While the arithmetic mean is often used to report central tendencies, it is not a robust statistic, meaning that it is greatly influenced by outliers (values that are very much larger or smaller than most of the values). Notably, for skewed distributions, such as the distribution of income for which a few people's incomes are substantially greater than most people's, the arithmetic mean may not accord with one's notion of "middle", and robust statistics, such as the median, may be a better description of central tendency.

The **arithmetic mean** (or **mean** or **average**) is the most commonly used and readily understood measure of central tendency. In statistics, the term average refers to any of the measures of central tendency. The arithmetic mean is defined as being equal to the sum of the numerical values of each and every observation divided by the total number of observations. Symbolically, if we have a data set containing the values , then the arithmetic mean is defined by the formula:

(See summation for an explanation of the summation operator).

For example, let us consider the monthly salary of 10 employees of a firm: 2500, 2700, 2400, 2300, 2550, 2650, 2750, 2450, 2600, 2400. The arithmetic mean is

If the data set is a statistical population (i.e., consists of every possible observation and not just a subset of them), then the mean of that population is called the **population mean**. If the data set is a statistical sample (a subset of the population), we call the statistic resulting from this calculation a **sample mean**.

The arithmetic mean of a variable is often denoted by a bar, for example as in (read *bar*), which is the mean of the values .^{[2]}

The arithmetic mean has several properties that make it useful, especially as a measure of central tendency. These include:

- If numbers have mean , then . Since is the distance from a given number to the mean, one way to interpret this property is as saying that the numbers to the left of the mean are balanced by the numbers to the right of the mean. The mean is the only single number for which the residuals (deviations from the estimate) sum to zero.
- If it is required to use a single number as a "typical" value for a set of known numbers , then the arithmetic mean of the numbers does this best, in the sense of minimizing the sum of squared deviations from the typical value: the sum of . (It follows that the sample mean is also the best single predictor in the sense of having the lowest root mean squared error.)
^{[2]}If the arithmetic mean of a population of numbers is desired, then the estimate of it that is unbiased is the arithmetic mean of a sample drawn from the population.

The arithmetic mean may be contrasted with the median. The median is defined such that no more than half the values are larger than, and no more than half are smaller than, the median. If elements in the sample data increase arithmetically, when placed in some order, then the median and arithmetic average are equal. For example, consider the data sample . The average is , as is the median. However, when we consider a sample that cannot be arranged so as to increase arithmetically, such as , the median and arithmetic average can differ significantly. In this case, the arithmetic average is 6.2 and the median is 4. In general, the average value can vary significantly from most values in the sample, and can be larger or smaller than most of them.

There are applications of this phenomenon in many fields. For example, since the 1980s, the median income in the United States has increased more slowly than the arithmetic average of income.^{[3]}

A weighted average, or weighted mean, is an average in which some data points count more strongly than others, in that they are given more weight in the calculation. For example, the arithmetic mean of and is , or equivalently . In contrast, a *weighted* mean in which the first number receives, for example, twice as much weight as the second (perhaps because it is assumed to appear twice as often in the general population from which these numbers were sampled) would be calculated as . Here the weights, which necessarily sum to the value one, are and , the former being twice the latter. Note that the arithmetic mean (sometimes called the "unweighted average" or "equally weighted average") can be interpreted as a special case of a weighted average in which all the weights are equal to each other (equal to in the above example, and equal to in a situation with numbers being averaged).

When a population of numbers, and any sample of data from it, could take on any of a continuous range of numbers, instead of for example just integers, then the probability of a number falling into one range of possible values could differ from the probability of falling into a different range of possible values, even if the lengths of both ranges are the same. In such a case, the set of probabilities can be described using a continuous probability distribution. The analog of a weighted average in this context, in which there are an infinitude of possibilities for the precise value of the variable, is called the *mean of the probability distribution*. The most widely encountered probability distribution is called the normal distribution; it has the property that all measures of its central tendency, including not just the mean but also the aforementioned median and the mode, are equal to each other. This property does not hold however, in the cases of a great many probability distributions, such as the lognormal distribution illustrated here.

Particular care must be taken when using cyclic data, such as phases or angles. Naïvely taking the arithmetic mean of 1° and 359° yields a result of 180°. This is incorrect for two reasons:

- Firstly, angle measurements are only defined up to an additive constant of 360° (or 2π, if measuring in radians). Thus one could as easily call these 1° and −1°, or 361° and 719°, each of which gives a different average.
- Secondly, in this situation, 0° (equivalently, 360°) is geometrically a better
*average*value: there is lower dispersion about it (the points are both 1° from it, and 179° from 180°, the putative average).

In general application, such an oversight will lead to the average value artificially moving towards the middle of the numerical range. A solution to this problem is to use the optimization formulation (viz., define the mean as the central point: the point about which one has the lowest dispersion), and redefine the difference as a modular distance (i.e., the distance on the circle: so the modular distance between 1° and 359° is 2°, not 358°).

- Average
- Fréchet mean
- Generalized mean
- Geometric mean
- Mode
- Sample mean and covariance
- Standard error of the mean
- Summary statistics

**^**Jacobs, Harold R. (1994).*Mathematics: A Human Endeavor*(Third ed.). W. H. Freeman. p. 547. ISBN 0-7167-2426-X.- ^
^{a}^{b}Medhi, Jyotiprasad (1992).*Statistical Methods: An Introductory Text*. New Age International. pp. 53–58. ISBN 9788122404197. **^**Paul Krugman, "The Rich, the Right, and the Facts: Deconstructing the Income Distribution Debate", 'The American Prospect'

- Huff, Darrell (1993).
*How to Lie with Statistics*. W. W. Norton. ISBN 978-0-393-31072-6.

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