“mean”的概念、定义、翻译、参考文献-科学参考

mean

Physics

A representative or expected value for a set of numbers.
1. 1 The arithmetic mean or average (usually just called the mean) of n values x₁, x₂,…, x_n is given by:
  
  $(x_{1} + x_{2} + x_{3} + \dots + x_{n}) / n$
  
  If x₁, x₂,…, x_k occur with frequencies (weights) w₁, w₂,…, w_k then the weighted mean is:
  
  $(w_{1} x_{1} + w_{2} x_{2} + \dots + w_{k} x_{k}) / (w_{1} + w_{2} + \dots + w_{k})$
2. 2 The harmonic mean H is given by:
  
  $n / [(1 / x_{1}) + (1 / x_{2}) + \dots + (1 / x_{n})]$
3. 3 The geometric mean G is given by:
  
  ${(x_{1}, x_{2}, \dots, x_{n})}^{1 / n}$

Mathematics

The mean of the numbers a₁, a₂,…, a_n is equal to

$\frac{a_{1} + a_{2} + \dots + a_{n}}{n} .$

This is the number most commonly used as the average. It may be called the arithmetic mean to distinguish it from other means such as those described below. When each number a_i is to have weight w_i > 0, the weighted mean is equal to

$\frac{w_{1} a_{1} + w_{2} a_{2} + \dots + w_{n} a_{n}}{w_{1} + w_{2} + \dots + w_{n}} .$

The geometric mean of the positive numbers a₁, a₂,…, a_n is

$\sqrt[n]{a_{1} a_{2} \dots a_{n} .}$

It is a theorem of elementary algebra that, for any positive numbers a₁, a₂,…, a_n, the arithmetic mean is greater than or equal to the geometric mean; that is,

$\frac{1}{n} (a_{1} + a_{2} + \dots + a_{n}) \geq \sqrt[n]{a_{1} a_{2} \dots a_{n}} .$

The harmonic mean of positive numbers a₁, a₂,…, a_n is the number h such that 1/h is the arithmetic mean of 1/a₁, 1/a₂,…, 1/a_n. Thus

$h = \frac{n}{(1 / a_{1}) + \dots + (1 / a_{n})} .$

For any set of positive numbers, a₁, a₂,…, a_n the harmonic mean is less than or equal to the geometric mean, and therefore also the arithmetic mean, and equality takes place for all three measures if and only if all the terms in the set are equal.

In statistics, the mean of a sample of observations x₁, x₂,…, x_n is called the sample mean and is denoted by $\bar{x}$ so that

$\bar{x} = \frac{x_{1} + x_{2} + \dots + x_{n}}{n} .$

The mean of a random variable X is equal to the expected value E(X). This may be called the population mean and denoted by µ. A sample mean $\bar{x}$ may be used to estimate µ.

See also arithmetic-geometric mean iteration.

Statistics

Familiarly known as the average, the word ‘mean’ is used as a shorthand for either the population mean or the sample mean, depending on context. See also expected value.

Chemical Engineering

A representative value of a set of values. The arithmetic mean or average value of a number of values is obtained by summing the values in the set and dividing by the number of values, n. The geometric mean is obtained by the 1/n^th power of the product of the values. The harmonic mean is obtained from:
$H = \frac{n}{\frac{1}{x_{1}} + \frac{1}{x_{2}} + … \frac{1}{x_{n}}}$

Computer

See measures of location.

Electronics and Electrical Engineering

In signal statistics, the average value computed for each signal data point by taking it and a fixed number of its immediate neighbours and calculating their mean.

Biology

The average value of a set of n numbers, i.e. the sum of the numbers divided by n.

Geology and Earth Sciences

In statistics, the average as calculated by the sum of each data point divided by the total number of data points. See also variance.

Philosophy

In many ethical systems the right path is presented as one that strikes a happy medium. It strays neither one way nor the other, but represents moderation, harmony, balance, and the avoidance of pitfalls on either side. Aristotle’s doctrine of the mean represents all virtues as striking a balance between vices of excess and vices of defect. The man who fears everything is a coward, but the man who fears nothing is rash; the man who indulges every pleasure is self-indulgent, but the man who indulges none is a boor. A similar idea was already present in the Philebus of Plato and is derived from Pythagoras. The doctrine is prominent in Confucianism: in the Analects, Confucius writes of the harmonious life as one avoiding excesses and deficiencies, and in which wisdom is gleaned from both old and young, low and high. K’ung Chi, the grandson of Confucius, wrote a work entitled the Chung Yung, or mean of equilibrium and harmony. The anonymous work The Doctrine of the Mean was a basic text for civil service examinations in China from 1313 until 1905. In the Buddhist System of the Middle Way the principle repudiates both exaggerated asceticism and easy-going hedonism. The term ‘golden mean’ is from the Latin poet Horace, whose aurea mediocritas is described in Odes 2. 10. 5.
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).

Economics

In a sample, a measure of central tendency of a set of data points. Most commonly referred to as the mean is the arithmetic mean, that is, the unweighted average of a set of numbers. Other frequently used quantities are the geometric mean, the arithmetic-geometric mean, the harmonic mean, the generalized or power mean, and the root-mean-square. For a random variable, the mean is the same as the expected value.

单词	mean
释义	mean Physics A representative or expected value for a set of numbers. 1 The arithmetic mean or average (usually just called the mean) of n values x₁, x₂,…, x_n is given by: $(x_{1} + x_{2} + x_{3} + \dots + x_{n}) / n$ If x₁, x₂,…, x_k occur with frequencies (weights) w₁, w₂,…, w_k then the weighted mean is: $(w_{1} x_{1} + w_{2} x_{2} + \dots + w_{k} x_{k}) / (w_{1} + w_{2} + \dots + w_{k})$ 2 The harmonic mean H is given by: $n / [(1 / x_{1}) + (1 / x_{2}) + \dots + (1 / x_{n})]$ 3 The geometric mean G is given by: ${(x_{1}, x_{2}, \dots, x_{n})}^{1 / n}$ Mathematics The mean of the numbers a₁, a₂,…, a_n is equal to $\frac{a_{1} + a_{2} + \dots + a_{n}}{n} .$ This is the number most commonly used as the average. It may be called the arithmetic mean to distinguish it from other means such as those described below. When each number a_i is to have weight w_i > 0, the weighted mean is equal to $\frac{w_{1} a_{1} + w_{2} a_{2} + \dots + w_{n} a_{n}}{w_{1} + w_{2} + \dots + w_{n}} .$ The geometric mean of the positive numbers a₁, a₂,…, a_n is $\sqrt[n]{a_{1} a_{2} \dots a_{n} .}$ It is a theorem of elementary algebra that, for any positive numbers a₁, a₂,…, a_n, the arithmetic mean is greater than or equal to the geometric mean; that is, $\frac{1}{n} (a_{1} + a_{2} + \dots + a_{n}) \geq \sqrt[n]{a_{1} a_{2} \dots a_{n}} .$ The harmonic mean of positive numbers a₁, a₂,…, a_n is the number h such that 1/h is the arithmetic mean of 1/a₁, 1/a₂,…, 1/a_n. Thus $h = \frac{n}{(1 / a_{1}) + \dots + (1 / a_{n})} .$ For any set of positive numbers, a₁, a₂,…, a_n the harmonic mean is less than or equal to the geometric mean, and therefore also the arithmetic mean, and equality takes place for all three measures if and only if all the terms in the set are equal. In statistics, the mean of a sample of observations x₁, x₂,…, x_n is called the sample mean and is denoted by $\bar{x}$ so that $\bar{x} = \frac{x_{1} + x_{2} + \dots + x_{n}}{n} .$ The mean of a random variable X is equal to the expected value E(X). This may be called the population mean and denoted by µ. A sample mean $\bar{x}$ may be used to estimate µ. See also arithmetic-geometric mean iteration. Statistics Familiarly known as the average, the word ‘mean’ is used as a shorthand for either the population mean or the sample mean, depending on context. See also expected value. Chemical Engineering A representative value of a set of values. The arithmetic mean or average value of a number of values is obtained by summing the values in the set and dividing by the number of values, n. The geometric mean is obtained by the 1/n^th power of the product of the values. The harmonic mean is obtained from: $H = \frac{n}{\frac{1}{x_{1}} + \frac{1}{x_{2}} + … \frac{1}{x_{n}}}$ Computer See measures of location. Electronics and Electrical Engineering In signal statistics, the average value computed for each signal data point by taking it and a fixed number of its immediate neighbours and calculating their mean. Biology The average value of a set of n numbers, i.e. the sum of the numbers divided by n. Geology and Earth Sciences In statistics, the average as calculated by the sum of each data point divided by the total number of data points. See also variance. Philosophy In many ethical systems the right path is presented as one that strikes a happy medium. It strays neither one way nor the other, but represents moderation, harmony, balance, and the avoidance of pitfalls on either side. Aristotle’s doctrine of the mean represents all virtues as striking a balance between vices of excess and vices of defect. The man who fears everything is a coward, but the man who fears nothing is rash; the man who indulges every pleasure is self-indulgent, but the man who indulges none is a boor. A similar idea was already present in the Philebus of Plato and is derived from Pythagoras. The doctrine is prominent in Confucianism: in the Analects, Confucius writes of the harmonious life as one avoiding excesses and deficiencies, and in which wisdom is gleaned from both old and young, low and high. K’ung Chi, the grandson of Confucius, wrote a work entitled the Chung Yung, or mean of equilibrium and harmony. The anonymous work The Doctrine of the Mean was a basic text for civil service examinations in China from 1313 until 1905. In the Buddhist System of the Middle Way the principle repudiates both exaggerated asceticism and easy-going hedonism. The term ‘golden mean’ is from the Latin poet Horace, whose aurea mediocritas is described in Odes 2. 10. 5. 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). Economics In a sample, a measure of central tendency of a set of data points. Most commonly referred to as the mean is the arithmetic mean, that is, the unweighted average of a set of numbers. Other frequently used quantities are the geometric mean, the arithmetic-geometric mean, the harmonic mean, the generalized or power mean, and the root-mean-square. For a random variable, the mean is the same as the expected value.
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