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How to find the Arithmetic Mean or Average presented by TeachtopiaTV
How to find the Arithmetic Mean or Average presented by TeachtopiaTV
Published: 2011/05/07
Channel: TeachtopiaTV
Arithmetic Means
Arithmetic Means
Published: 2012/03/22
Channel: ProfRobBob
What is the Arithmetic Mean?
What is the Arithmetic Mean?
Published: 2014/12/20
Channel: Don't Memorise
Arithmetic Mean or Median of AP ( GMAT / GRE / CAT / Bank PO / SSC CGL)
Arithmetic Mean or Median of AP ( GMAT / GRE / CAT / Bank PO / SSC CGL)
Published: 2015/04/20
Channel: Don't Memorise
Average or Central Tendency: Arithmetic Mean, Median, and Mode
Average or Central Tendency: Arithmetic Mean, Median, and Mode
Published: 2010/04/19
Channel: Khan Academy
Find 4 arithmetic means between 6 and 36
Find 4 arithmetic means between 6 and 36
Published: 2012/06/14
Channel: Laura Rickhoff
Arithmetic mean - Insert n arithmetic means between two given number - solved example
Arithmetic mean - Insert n arithmetic means between two given number - solved example
Published: 2013/12/04
Channel: MathsSmart
al2 sequences and series finding the arithmetic mean
al2 sequences and series finding the arithmetic mean
Published: 2014/02/11
Channel: maths gotserved
001 Statistics - Measures of Central Tendency - Arithmetic Mean
001 Statistics - Measures of Central Tendency - Arithmetic Mean
Published: 2016/08/04
Channel: studyezee
Arithmetic Means
Arithmetic Means
Published: 2013/02/21
Channel: G Hill
Arithmetic Mean
Arithmetic Mean
Published: 2012/01/16
Channel: mathgeekstutoring
What is ARITHMETIC MEAN? What does ARITHMETIC MEAN mean? ARITHMETIC MEAN meaning & explanation
What is ARITHMETIC MEAN? What does ARITHMETIC MEAN mean? ARITHMETIC MEAN meaning & explanation
Published: 2017/03/26
Channel: The Audiopedia
Arithmetic Mean Average Word Problems
Arithmetic Mean Average Word Problems
Published: 2016/02/24
Channel: Mathodman
Arithmetic Mean
Arithmetic Mean
Published: 2016/03/28
Channel: PCSciHS Math Tutorials
Arithmetic Mean Formula
Arithmetic Mean Formula
Published: 2017/02/10
Channel: Ms. Milkosky ACT
Geometric Mean vs. Arithmetic Mean
Geometric Mean vs. Arithmetic Mean
Published: 2014/02/11
Channel: FerranteMath
Arithmetic Mean in  Short cut method
Arithmetic Mean in Short cut method
Published: 2015/09/16
Channel: CA N Raja Natarajan
Economics 11th Standard CBSE:- Calculating Mean Part 1 Arithmetic Mean
Economics 11th Standard CBSE:- Calculating Mean Part 1 Arithmetic Mean
Published: 2014/03/03
Channel: sangram singh
Properties of Arithmetic Mean - Very Important
Properties of Arithmetic Mean - Very Important
Published: 2016/08/14
Channel: 100Centum
Basic Stats- Arithmetic, Geometric and Harmonic Mean
Basic Stats- Arithmetic, Geometric and Harmonic Mean
Published: 2014/01/23
Channel: FinTree
Arithmetic Progressions & Arithmetic Mean | CBSE 11 Maths & competitive | Ex 9.2 intro
Arithmetic Progressions & Arithmetic Mean | CBSE 11 Maths & competitive | Ex 9.2 intro
Published: 2016/08/28
Channel: cbseclass videos
Mean using Direct Method & Shortcut Method
Mean using Direct Method & Shortcut Method
Published: 2016/07/31
Channel: DeltaStep
Arithmetic Mean - IIT JEE Main and Advanced Maths Video Lecture
Arithmetic Mean - IIT JEE Main and Advanced Maths Video Lecture
Published: 2014/05/13
Channel: Rao IIT Academy
Arithmetic Mean Problem 1
Arithmetic Mean Problem 1
Published: 2010/11/26
Channel: GreeneMath.com
Calculating the Mean using Step deviation method
Calculating the Mean using Step deviation method
Published: 2016/03/01
Channel: AtHome Tuition
Arithmetic Mean vs Geometric Mean - What
Arithmetic Mean vs Geometric Mean - What's The Difference & How To Find It?
Published: 2015/12/28
Channel: Math & Science 2024
Arithmetic Mean Continuous series Statistics
Arithmetic Mean Continuous series Statistics
Published: 2017/08/29
Channel: Facilitator Basanta
Arithmetic Mean
Arithmetic Mean
Published: 2015/06/13
Channel: Flexiguru
Finding the arithmetic sequence arithmetic means.wmv
Finding the arithmetic sequence arithmetic means.wmv
Published: 2012/07/30
Channel: maths gotserved
Arithmetic Mean for Samples and Populations
Arithmetic Mean for Samples and Populations
Published: 2010/07/06
Channel: statslectures
Arithmetic Mean (in Hindi)
Arithmetic Mean (in Hindi)
Published: 2016/08/12
Channel: 100Centum
Proof by Contradiction: Arithmetic Mean & Geometric Mean
Proof by Contradiction: Arithmetic Mean & Geometric Mean
Published: 2014/03/27
Channel: Eddie Woo
What is Arithmetic Mean | AP | GP | CA CPT | CS & CMA Foundation | Class 11 | Class 12
What is Arithmetic Mean | AP | GP | CA CPT | CS & CMA Foundation | Class 11 | Class 12
Published: 2015/11/22
Channel: Mera Skill
Types of Arithmetic Mean , Merits & Demerits | Class 11 Economics Measures of Central Tendency
Types of Arithmetic Mean , Merits & Demerits | Class 11 Economics Measures of Central Tendency
Published: 2017/05/06
Channel: Scholarslearning Classes
What is Combined Arithmetic Mean?
What is Combined Arithmetic Mean?
Published: 2014/12/20
Channel: Don't Memorise
Statistics -  Arithmetic Mean LECTURE  5
Statistics - Arithmetic Mean LECTURE 5
Published: 2017/05/30
Channel: Vyaspeeth Satyen Vyas
Statistics:  Arithmetic Mean
Statistics: Arithmetic Mean
Published: 2015/08/08
Channel: george boole
Find the arithmetic mean of first ten odd natural    numbers
Find the arithmetic mean of first ten odd natural numbers
Published: 2017/09/20
Channel: Doubtnut
Arithmetic and Geometric Means
Arithmetic and Geometric Means
Published: 2014/01/02
Channel: talkboard.com.au
Weighted Mean - GRE-GMAT-CAT-CMAT-MBA-PGDM-PGDBA
Weighted Mean - GRE-GMAT-CAT-CMAT-MBA-PGDM-PGDBA
Published: 2014/11/25
Channel: Prashant Puaar
Arithmetic Mean and Geometric Mean
Arithmetic Mean and Geometric Mean
Published: 2016/09/27
Channel: NCERT OFFICIAL
*Arithmetic Average: If the average (arithmetic mean) of 8,11,25,and p is 15, then 8+11+25+p
*Arithmetic Average: If the average (arithmetic mean) of 8,11,25,and p is 15, then 8+11+25+p
Published: 2011/03/05
Channel: TucsonMathDoc
Geometry:  Arithmetic, Geometric, Harmonic Means
Geometry: Arithmetic, Geometric, Harmonic Means
Published: 2015/09/10
Channel: Math With Mark
Understanding the Arithmetic Mean
Understanding the Arithmetic Mean
Published: 2016/09/11
Channel: John Gabriel
Relationship Between Arithmetic Mean and Geometric Mean
Relationship Between Arithmetic Mean and Geometric Mean
Published: 2016/09/27
Channel: NCERT OFFICIAL
SAT Math - Averages (Arithmetic Mean)
SAT Math - Averages (Arithmetic Mean)
Published: 2012/07/06
Channel: SAT Prep Guy
CBSE Class 11 Statistics - Arithmetic Mean
CBSE Class 11 Statistics - Arithmetic Mean
Published: 2017/07/06
Channel: Education Cafe
Inequalities - #2 Relation Between Arithmetic Mean & Geometric Mean of Two Numbers
Inequalities - #2 Relation Between Arithmetic Mean & Geometric Mean of Two Numbers
Published: 2012/08/21
Channel: homevideotutor
CA-CPT - Sum of Deviations from Arithmetic Mean - Statistics (in Hindi)
CA-CPT - Sum of Deviations from Arithmetic Mean - Statistics (in Hindi)
Published: 2016/09/19
Channel: 100Centum
Understand arithmetic mean with the given data
Understand arithmetic mean with the given data
Published: 2014/11/10
Channel: Pearson India
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WIKIPEDIA ARTICLE

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In mathematics and statistics, the arithmetic mean ( /ˌærɪθˈmɛtɪk ˈmn/, 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.

==2+2=4-1=3 quuickmaths == 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]

Definitions[edit]

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.

Contrast with median[edit]

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]

Generalizations[edit]

Weighted average[edit]

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).

Continuous probability distributions[edit]

Comparison of mean, median and mode of two log-normal distributions with different skewness.

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.

Angles[edit]

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°).

See also[edit]

References[edit]

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

Further reading[edit]

External links[edit]

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