mean

Dictionary: mean³ (mēn) pronunciation

Something having a position, quality, or condition midway between extremes; a medium.
Mathematics.
1. A number that typifies a set of numbers, such as a geometric mean or an arithmetic mean.
2. The average value of a set of numbers.
Logic. The middle term in a syllogism.
means (used with a sing. or pl. verb) A method, a course of action, or an instrument by which an act can be accomplished or an end achieved.
means (used with a pl. verb)
1. Money, property, or other wealth: You ought to live within your means.
2. Great wealth: a woman of means.

adj.

Occupying a middle or intermediate position between two extremes.
Intermediate in size, extent, quality, time, or degree; medium.

idioms:

by all means

Without fail; certainly.

by any means

In any way possible; to any extent: not by any means an easy opponent.

by means of

With the use of; owing to: They succeeded by means of patience and sacrifice.

by no means

In no sense; certainly not: This remark by no means should be taken lightly.

[Middle English mene, middle, from Old French meien, from Latin mediānus, from medius.]

USAGE NOTE In the sense of "financial resources" means takes a plural verb: His means are more than adequate. In the sense of "a way to an end," means may be treated as either a singular or plural. It is singular when referring to a particular strategy or method: The best means of securing the cooperation of the builders is to appeal to their self-interest. It is plural when it refers to a group of strategies or methods: The most effective means for dealing with the drug problem have generally been those suggested by the affected communities. • Means is most often followed by of: a means of noise reduction. But for, to, and toward are also used: a means for transmitting sound; a means to an end; a means toward achieving equality.

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Investment Dictionary: Mean

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The simple mathematical average of a set of two or more numbers. The mean for a given set of numbers can be computed in more than one way, including the arithmetic mean method, which uses the sum of the numbers in the series, and the geometric mean method. However, all of the primary methods for computing a simple average of a normal number series produce the same approximate result most of the time.

Investopedia Says:
If stock XYZ closed at $50, $51 and $54 over the past three days, the arithmetic mean would be the sum of those numbers divided by three, which is $51.67.

In contrast, the geometric mean would be computed as third root of the numbers' product, or the third root of 137,700, which approximately equals $51.64. While the two numbers are not exactly equal, most people consider arithmetic and geometric means to be equivalent for everyday purposes.

Related Links:
Take a closer look at the linearly weighted moving average and the exponentially smoothed moving average. Basics of Weighted Moving Averages

Marketing Dictionary: mean

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Arithmetic average calculated by summing a set of values and dividing by the number of values in the set. The mean is frequently confused with the median or mode. The mean of 4 + 8 + 6 + 12 equals 30/4, or 71/2. Mean is a good representation of quantitative data, such as the mean number of items purchased per catalog order or the mean dollar value of each order. Qualitative data, such as the item purchased most often from a catalog, is better suited to a mode calculation.

Accounting Dictionary: Mean

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Measure of central tendency; also called average. Mean and Standard Deviation are the two most widely used statistical measures that summarize the characteristics of the data. Suppose a new car dealer sells 630 cars during a 30-day period. Then the mean (average) daily sales is obtained by dividing the total number of cars by the number of days as follows:

Mean daily sales per day = 630/30 = 21 per day

Symbolically,

x- = Sx_i/n

where x- = the mean, x_i = the values in the data, S (read as sigma) is the summation sign, and n = the number of observations in the data.

Dental Dictionary: mean

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n
x

A measure of central tendency that is the calculated arithmetic average of a series of scores.

Philosophy Dictionary: mean

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(mathematical) That which occupies a middle position. In mathematics an arithmetical mean of n quantities is their sum, divided by the number n. The geometrical mean of n quantities is the nth root of their product. The harmonic mean is the reciprocal of the arithmetical mean of their reciprocals. In statistics the mean value of a distribution of a random variable x is a weighted mean of its values, where the weight of a value f(x) is the probability of the value. The mode of a distribution is the most common value, and the median is the value such that the probabilities of x being less than or greater than this value are each 0·5 (or as near 0·5 as the distribution permits).

Sports Science and Medicine: mean

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Statistical value computed from the sum of a set of numbers divided by the number of terms. See also descriptive statistic.

Columbia Encyclopedia: mean

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mean, in statistics, a type of average. The arithmetic mean of a group of numbers is found by dividing their sum by the number of members in the group; e.g., the sum of the seven numbers 4, 5, 6, 9, 13, 14, and 19 is 70 so their mean is 70 divided by 7, or 10. Less often used is the geometric mean (for two quantities, the square root of their product; for n quantities, the nth root of their product).

Science Dictionary: mean

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In statistics, an average of a group of numbers or data points. With a group of numbers, the mean is obtained by adding them and dividing by the number of numbers in the group. Thus the mean of five, seven, and twelve is eight (twenty-four divided by three). (Compare median and mode.)

Economics Dictionary: mean

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An average in statistics. (See under “Physical Sciences and Mathematics.”)

Veterinary Dictionary: mean

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An average; a numerical value intermediate between two extremes. Called also arithmetic mean.

m. arterial pressure — average pressure in artery for one heartbeat.
m. cell constants — see erythrocyte indices.
m. corpuscular hemoglobin (MCH) — see mch.
m. corpuscular hemoglobin concentration (MCHC) — see mchc.
m. corpuscular volume (MCV) — see mcv.
m. deviation — the average value of a set of absolute deviations from the mean of a set of observations.
m. electrical axis (MEA) — in electrocardiography, a calculation based on the relative amplitude of Q, R and S waves in the three bipolar limb leads. It is an aid to recognizing right ventricular enlargement and various intraventricular conduction defects.
geometric m. — the antilog of the mean of the logarithm of the calculated values, the same as the n^th root of the product of the values. It is often a more useful mean for growth curves.
harmonic m. — the reciprocal of the arithmetic mean of values converted to their reciprocals (used in dealing with skewed data).
rolling m. — see moving average.

Wikipedia: Mean

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This article is about the statistical concept. For other uses, see Mean (disambiguation).

It has been suggested that this article or section be merged with Average. (Discuss)

In statistics, mean has two related meanings:

the arithmetic mean (and is distinguished from the geometric mean or harmonic mean).
the expected value of a random variable, which is also called the population mean.

It is sometimes stated that the 'mean' means average. This is incorrect if "mean" is taken in the specific sense of "arithmetic mean" as there are different types of averages: the mean, median, and mode. Other simple statistical analyses use measures of spread, such as range, interquartile range, or standard deviation. For a real-valued random variable X, the mean is the expectation of X. Note that not every probability distribution has a defined mean (or variance); see the Cauchy distribution for an example.

For a data set, the mean is the sum of the observations divided by the number of observations. The mean of a set of numbers x₁, x₂, ..., x_n is typically denoted by $\bar{x}$ , pronounced "x bar". The mean is often quoted along with the standard deviation: the mean describes the central location of the data, and the standard deviation describes the spread.

An alternative measure of dispersion is the mean deviation, equivalent to the average absolute deviation from the mean. It is less sensitive to outliers, but less mathematically tractable.

As well as statistics, means are often used in geometry and analysis; a wide range of means have been developed for these purposes, which are not much used in statistics. These are listed below.

Examples of means

Arithmetic mean

Main article: Arithmetic mean

The arithmetic mean is the "standard" average, often simply called the "mean".

$\bar{x} = \frac{1}{n}\cdot \sum_{i=1}^n{x_i}$

The mean may often be confused with the median, mode or range. The mean is the arithmetic average of a set of values, or distribution; however, for skewed distributions, the mean is not necessarily the same as the middle value (median), or the most likely (mode). For example, mean income is skewed upwards by a small number of people with very large incomes, so that the majority have an income lower than the mean. By contrast, the median income is the level at which half the population is below and half is above. The mode income is the most likely income, and favors the larger number of people with lower incomes. The median or mode are often more intuitive measures of such data.

Nevertheless, many skewed distributions are best described by their mean - such as the Exponential and Poisson distributions.

For example, the arithmetic mean of six values: 34, 27, 45, 55, 22, 34 is:

$\frac{34+27+45+55+22+34}{6} = \frac{217}{6} \approx 36.167.$

Assumed Mean

Main article: Assumed Mean Formula

The assumed mean is used when calculating the mean where using the arithmetic mean is tiring.

Geometric mean

Main article: Geometric mean

The geometric mean is an average that is useful for sets of positive numbers that are interpreted according to their product and not their sum (as is the case with the arithmetic mean) e.g. rates of growth.

$\bar{x} = \left ( \prod_{i=1}^n{x_i} \right ) ^{1/n}$

For example, the geometric mean of six values: 34, 27, 45, 55, 22, 34 is:

$(34 \cdot 27 \cdot 45 \cdot 55 \cdot 22 \cdot 34)^{1/6} = 1,699,493,400^{1/6} \approx 34.545.$

Harmonic mean

Main article: Harmonic mean

The harmonic mean is an average which is useful for sets of numbers which are defined in relation to some unit, for example speed (distance per unit of time).

$\bar{x} = n \cdot \left ( \sum_{i=1}^n \frac{1}{x_i} \right ) ^{-1}$

For example, the harmonic mean of the six values: 34, 27, 45, 55, 22, and 34 is

$\frac{6}{\frac{1}{34}+\frac{1}{27}+\frac{1}{45} + \frac{1}{55} + \frac{1}{22}+\frac{1}{34}} = \frac{60588}{1835} \approx 33.0179836.$

Generalized means

Power mean

The generalized mean, also known as the power mean or Hölder mean, is an abstraction of the quadratic, arithmetic, geometric and harmonic means. It is defined for a set of n positive numbers x_i by

$\bar{x}(m) = \left ( \frac{1}{n}\cdot\sum_{i=1}^n{x_i^m} \right ) ^{1/m}$

By choosing the appropriate value for the parameter m we get

$m\rightarrow\infty$	maximum
$m = 2$	quadratic mean,
$m = 1$	arithmetic mean,
$m\rightarrow0$	geometric mean,
$m = - 1$	harmonic mean,
$m\rightarrow-\infty$	minimum.

f-mean

This can be generalized further as the generalized f-mean

$\bar{x} = f^{-1}\left({\frac{1}{n}\cdot\sum_{i=1}^n{f(x_i)}}\right)$

and again a suitable choice of an invertible $f$ will give

$f(x) = \frac{1}{x}$	harmonic mean,
$f (x) = x m$	power mean,
$f (x) = ln x$	geometric mean.

Weighted arithmetic mean

The weighted arithmetic mean is used, if one wants to combine average values from samples of the same population with different sample sizes:

$\bar{x} = \frac{\sum_{i=1}^n{w_i \cdot x_i}}{\sum_{i=1}^n {w_i}}.$

The weights $w i$ represent the bounds of the partial sample. In other applications they represent a measure for the reliability of the influence upon the mean by respective values.

Truncated mean

Sometimes a set of numbers might contain outliers, i.e. a datum which is much lower or much higher than the others. Often, outliers are erroneous data caused by artifacts. In this case one can use a truncated mean. It involves discarding given parts of the data at the top or the bottom end, typically an equal amount at each end, and then taking the arithmetic mean of the remaining data. The number of values removed is indicated as a percentage of total number of values.

Interquartile mean

The interquartile mean is a specific example of a truncated mean. It is simply the arithmetic mean after removing the lowest and the highest quarter of values.

$\bar{x} = {2 \over n} \sum_{i=(n/4)+1}^{3n/4}{x_i}$

assuming the values have been ordered.

Mean of a function

In calculus, and especially multivariable calculus, the mean of a function is loosely defined as the average value of the function over its domain. In one variable, the mean of a function f(x) over the interval (a,b) is defined by

$\bar{f}=\frac{1}{b-a}\int_a^bf(x)\,dx.$

(See also mean value theorem.) In several variables, the mean over a relatively compact domain U in a Euclidean space is defined by

$\bar{f}=\frac{1}{\hbox{Vol}(U)}\int_U f.$

This generalizes the arithmetic mean. On the other hand, it is also possible to generalize the geometric mean to functions by defining the geometric mean of f to be

$\exp\left(\frac{1}{\hbox{Vol}(U)}\int_U \log f\right).$

More generally, in measure theory and probability theory either sort of mean plays an important role. In this context, Jensen's inequality places sharp estimates on the relationship between these two different notions of the mean of a function.

Mean of angles

Most of the usual means fail on circular quantities, like angles, daytimes, fractional parts of real numbers. For those quantities you need a mean of circular quantities.

Other means

Properties

All means share some properties and additional properties are shared by the most common means. Some of these properties are collected here.

Weighted mean

A weighted mean $M$ is a function which maps tuples of positive numbers to a positive number

$M : (0,\infty)^n \to (0,\infty)$

such that the following properties hold:

"Fixed point": M(1,1,...,1) = 1
Homogeneity: M(λ x₁, ..., λ x_n) = λ M(x₁, ..., x_n) for all λ and x_i. In vector notation: M(λ x) = λ Mx for all n-vectors x.
Monotony: If x_i ≤ y_i for each i, then Mx ≤ My

It follows

Boundedness: min x ≤ Mx ≤ max x
Continuity: $\lim_{x\to y} M x = M y$
There are means which are not differentiable. For instance, the maximum number of a tuple is considered a mean (as an extreme case of the power mean, or as a special case of a median), but is not differentiable.
All means listed above, with the exception of most of the Generalized f-means, satisfy the presented properties.
- If $f$ is bijective, then the generalized f-mean satisfies the fixed point property.
- If $f$ is strictly monotonic, then the generalized f-mean satisfy also the monotony property.
- In general a generalized f-mean will miss homogeneity.

The above properties imply techniques to construct more complex means:

If C, M₁, ..., M_m are weighted means and p is a positive real number, then A and B defined by

$A x = C(M_1 x, \dots, M_m x) ,$

$B x = \sqrt[p]{C(x_1^p, \dots, x_n^p)} ,$

are also weighted means.

Unweighted mean

Intuitively spoken, an unweighted mean is a weighted mean with equal weights. Since our definition of weighted mean above does not expose particular weights, equal weights must be asserted by a different way. A different view on homogeneous weighting is, that the inputs can be swapped without altering the result.

Thus we define M to be an unweighted mean if it is a weighted mean and for each permutation π of inputs, the result is the same.

Symmetry: Mx = M(πx) for all n-tuples π and permutations π on n-tuples.

Analogously to the weighted means, if C is a weighted mean and M₁, ..., M_m are unweighted means and p is a positive real number, then A and B defined by

$A x = C(M_1 x, \dots, M_m x) ,$

$B x = \sqrt[p]{M_1(x_1^p, \dots, x_n^p)} ,$

are also unweighted means.

Convert unweighted mean to weighted mean

An unweighted mean can be turned into a weighted mean by repeating elements. This connection can also be used to state that a mean is the weighted version of an unweighted mean. Say you have the unweighted mean M and weight the numbers by natural numbers $a_1,\dots,a_n$ . (If the numbers are rational, then multiply them with the least common denominator.) Then the corresponding weighted mean $A$ is obtained by

$A(x_1,\dots,x_n) = M(\underbrace{x_1,\dots,x_1}_{a_1},x_2,\dots,x_{n-1},\underbrace{x_n,\dots,x_n}_{a_n}).$

Means of tuples of different sizes

If a mean $M$ is defined for tuples of several sizes, then one also expects that the mean of a tuple is bounded by the means of partitions. More precisely

Given an arbitrary tuple $x$ , which is partitioned into $y_1, \dots, y_k$ , then it holds $M x \in \mathrm{convexhull}(M y_1, \dots, M y_k)$ . (See Convex hull)

Population and sample means

The mean of a population has an expected value of μ, known as the population mean. The sample mean makes a good estimator of the population mean, as its expected value is the same as the population mean. The sample mean of a population is a random variable, not a constant, and consequently it will have its own distribution. For a random sample of n observations from a normally distributed population, the sample mean distribution is

$\bar{x} \thicksim N\left\{\mu, \frac{\sigma^2}{n}\right\}.$

Often, since the population variance is an unknown parameter, it is estimated by the mean sum of squares, which changes the distribution of the sample mean from a normal distribution to a Student's t distribution with n − 1 degrees of freedom.

References

Hardy, G.H.; Littlewood, J.E.; Pólya, G. (1988), Inequalities (2nd ed.), Cambridge University Press, ISBN 978-0521358804

External links

Statistics

Descriptive statistics

Continuous data

Location	Mean (Arithmetic, Geometric, Harmonic) · Median · Mode

Dispersion	Range · Standard deviation · Coefficient of variation · Percentile

Moments	Variance · Semivariance · Skewness · Kurtosis

Categorical data

Frequency · Contingency table

Inferential statistics
and
hypothesis testing

Inference	Confidence interval (Frequentist inference) · Credible interval (Bayesian inference) · Significance · Meta-analysis

Design of experiments	Population · Sampling · Stratified sampling · Replication · Blocking · Sensitivity and specificity

Sample size estimation	Statistical power · Effect size · Standard error

General estimation	Bayesian estimator · Maximum likelihood · Method of moments · Minimum distance · Maximum spacing

Specific tests	Z-test (normal) · Student's t-test · F-test · Chi-square test · Pearson's chi-square test · Wald test · Mann–Whitney U · Wilcoxon signed-rank test

Survival analysis	Survival function · Kaplan–Meier · Logrank test · Failure rate · Proportional hazards models

Correlation
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Correlation	Pearson product-moment correlation · Rank correlation (Spearman's rho, Kendall's tau) · Confounding variable

Linear regression	Linear model · General linear model · Generalized linear model · Analysis of variance · Analysis of covariance

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