Arithmetic mean

In mathematics and statistics, the arithmetic mean (or simply the mean) of a list of numbers is the sum of all the members of the list divided by the number of items in the list. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Statistics is a mathematical science pertaining to the collection analysis interpretation or explanation and presentation of Data. In Statistics, mean has two related meanings the Arithmetic mean (and is distinguished from the Geometric mean or Harmonic mean If the list is a statistical population, then the mean of that population is called a population mean. In Statistics, a statistical population is a set of entities concerning which Statistical inferences are to be drawn often based on a Random sample If the list is a statistical sample, we call the resulting statistic a sample mean. Sampling is that part of Statistical practice concerned with the selection of individual observations intended to yield some knowledge about a population of concern A statistic (singular is the result of applying a function (statistical Algorithm) to a set of data.

The mean is the most commonly-used type of average and is often referred to simply as the average. In Mathematics, an average, or central tendency of a Data set refers to a measure of the "middle" or " expected " value of The term "mean" or "arithmetic mean" is preferred in mathematics and statistics to distinguish it from other averages such as the median and the mode. In Mathematics, an average, or central tendency of a Data set refers to a measure of the "middle" or " expected " value of In Probability theory and Statistics, a median is described as the number separating the higher half of a sample a population or a Probability distribution

1 Introduction
- 1.1 Examples
2 Problems with some uses of the mean
3 See also
4 Further reading
5 External links

Introduction

If we denote a set of data by X = (x₁, x₂, . . . , x_n), then the sample mean is typically denoted with a horizontal bar over the variable ( $\bar{x} \,$ , enunciated "x bar").

The symbol μ (Greek: mu) is used to denote the arithmetic mean of an entire population. Or, for a random number that has a defined mean, μ is the probabilistic mean or expected value of the random number. Random number may refer to A number generated for or part of a set exhibiting Statistical randomness. If the set X is a collection of random numbers with probabilistic mean of μ, then for any individual sample, x_i, from that collection, μ = E{x_i} is the expected value of that sample.

In practice, the difference between μ and $\bar{x} \,$ is that μ is typically unobservable because one observes only a sample rather than the whole population, and if the sample is drawn randomly, then one may treat $\bar{x} \,$ , but not μ, as a random variable, attributing a probability distribution to it (the sampling distribution of the mean). A random variable is a rigorously defined mathematical entity used mainly to describe Chance and Probability in a mathematical way In Probability theory and Statistics, a probability distribution identifies either the probability of each value of an unidentified Random variable In Statistics, a sampling distribution is the Probability distribution, under repeated sampling of the population, of a given Statistic

Both are computed in the same way:

$\bar{x} = \frac{1}{n}\sum_{i=1}^n x_i = \frac{1}{n} (x_1+\cdots+x_n).$

If X is a random variable, then the expected value of X can be seen as the long-term arithmetic mean that occurs on repeated measurements of X. A random variable is a rigorously defined mathematical entity used mainly to describe Chance and Probability in a mathematical way This is the content of the law of large numbers. The law of large numbers (LLN is a theorem in Probability that describes the long-term stability of the mean of a Random variable. As a result, the sample mean is used to estimate unknown expected values.

Note that several other "means" have been defined, including the generalized mean, the generalized f-mean, the harmonic mean, the arithmetic-geometric mean, and various weighted means. A generalized mean, also known as power mean or Hölder mean, is an abstraction of the Pythagorean means including arithmetic, geometric In Mathematics and Statistics, the quasi-arithmetic mean or generalised f -mean is one generalisation of the more familiar Means such In Mathematics, the harmonic mean (formerly sometimes called the subcontrary mean) is one of several kinds of Average. In Mathematics, the arithmetic-geometric mean (AGM of two positive Real numbers x and y is defined as follows First compute the Arithmetic The weighted mean is similar to an Arithmetic mean (the most common type of Average) where instead of each of the data points contributing equally to the final average

Examples

If you have 3 numbers then add them and divide them by 3: $\frac{x_1 + x_2 + x_3}{3}.$
If you have 4 numbers add them and divide by 4: $\frac{x_1 + x_2 + x_3 + x_4}{4}.$

Problems with some uses of the mean

While the mean is often used to report central tendency, it may not be appropriate for describing skewed distributions, because it is easily misinterpreted. In Mathematics, an average, or central tendency of a Data set refers to a measure of the "middle" or " expected " value of In Probability theory and Statistics, skewness is a measure of the asymmetry of the Probability distribution of a real -valued The arithmetic mean is greatly influenced by outliers. In Statistics, an outlier is an observation that is numerically distant from the rest of the data. These distortions can occur when the mean is different from the median. When this happens the median may be a better description of central tendency. In Probability theory and Statistics, a median is described as the number separating the higher half of a sample a population or a Probability distribution

A classic example is average income. The arithmetic mean may be misinterpreted to imply that most people's incomes are higher than is in fact the case. When presented with an "average" one may be led to believe that most people's incomes are near this number. This "average" (arithmetic mean) income is higher than most people's incomes, because high income outliers skew the result higher (in contrast, the median income "resists" such skew). However, this "average" says nothing about the number of people near the median income (nor does it say anything about the modal income that most people are near). Nevertheless, because one might carelessly relate "average" and "most people" one might incorrectly assume that most people's incomes would be higher (nearer this inflated "average") than they are. For instance, reporting the "average" net worth in Medina, Washington as the arithmetic mean of all annual net worths would yield a surprisingly high number because of Bill Gates. For the film entitled Net Worth see Net Worth (film. In business net worth (sometimes called net assets) is the total Assets Medina is a city located in the Eastside, a region of King County Washington. If you would like to experiment with Wikipedia please copy Consider the scores (1, 2, 2, 2, 3, 9). The arithmetic mean is 3. 17, but five out of six scores are below this.

In certain situations, the arithmetic mean is the wrong measure of central tendency altogether. For example, if a stock fell 10 % in the first year, and rose 30 % in the second year, then it would be incorrect to report its "average" increase per year over this two year period as the arithmetic mean (−10 % + 30 %)/2 = 10 %; the correct average in this case is the geometric mean which yields an average increase per year of only 8. The geometric mean in Mathematics, is a type of Mean or Average, which indicates the central tendency or typical value of a set of numbers 2 %. The reason for this is that each of those percents have different starting points. If the stock starts at $30 and falls 10 %, it is now at $27. If the stock then rises 30 %, it is now $35. 1. The arithmetic mean of those rises is 10 %, but since the stock rose by $5. 1 in 2 years, an average of 8. In Mathematics, an average, or central tendency of a Data set refers to a measure of the "middle" or " expected " value of 2 % would result in the final $35. 1 figure [$30(1-10 %)(1+30 %) = $30(1+8. 2 %)(1+8. 2 %) = $35. 1]. If one used the arithmetic mean 10 % in the same way, one would not get the actual increase [$30(1+10 %)(1+10 %) = $36. 3].

Particular care must be taken when using cyclic data such as phases or angles. Taking the arithmetic mean of 1 degree and 359 degrees yields a result of 180 degrees, whereas 1 and 359 are both adjacent to 360 degrees which may be a more correct average value. 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, and redefine the difference as a modular distance. (See directional statistics. Directional statistics is the subdiscipline of Statistics that deals with directions ( Unit vectors in R n) axes (lines through )

External links

Dictionary

-noun

(mathematics) The measure of central tendency of a set of values computed by dividing the sum of the values by their number; commonly called the mean or the average.

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Contents

Introduction

Examples

Problems with some uses of the mean

See also

Further reading

External links

Dictionary

arithmetic mean

-noun