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Average This term has other meanings, see average meaning. On average (in mathematics and statistics) sets of numbers - the sum of all numbers divided by their number. It is one of the most common measures of central tendency.
It was proposed (along with the geometric mean and harmonic mean) by the Pythagoreans.
Special cases of the arithmetic mean are the mean (of the general population) and the mean of the sample (of the samples).
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Indicate the data set X = ( X 1, X 2,…, X n ), then the sample mean is usually denoted by a horizontal bar above the variable (x ¯ (\ displaystyle (\ bar (x))), pronounced " X with a dash ").
The Greek letter μ is used to denote the arithmetic mean of the whole population. For a random variable for which an average value is defined, μ is the average probability or the mathematical expectation of a random variable. If the set X is a collection of random numbers with an average of the probability μ, then for any sample X i in this collection μ = E ( X i ) is the expectation of this sample.
In practice, the difference between μ and x ¯ (\ displaystyle (\ bar (x))) is that μ is a typical variable because you can see the sample rather than the entire population. Therefore, if the sample is represented randomly (in terms of probability theory), then x ¯ (\ displaystyle (\ bar (x))) (but not μ) can be treated as a random variable with a probability distribution on the sample ( probability distribution of the mean).
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Both quantities are calculated in the same way:
X ¯ = 1 n �� i = 1 nxi = 1 n (x 1 + ⋯ + xn). (\ displaystyle (\ bar (x)) = (\ frac (1) (n)) \ sum _ (i = 1) ^ (n) x_ (i) = (\ frac (1) (n)) (x_ (1) + \ cdots + x_ (n)).)
If an X is a random variable, then the mathematical expectation X can be considered as the arithmetic mean of the values in repeated measurements of the quantity X. This is a manifestation of the law of large numbers. Therefore, the sample mean is used to estimate the unknown mathematical expectations.
In elementary algebra, it is shown that the mean n + 1 numbers above the mean n numbers if and only if the new number is greater than the old average, unless and only if the new number is less than the mean and does not change if and only if the new number equals the average. The more n , the smaller the difference between new and old environments.
Note that there are several other "means" available, including power law averages, Kolmogorov averages, harmonic averages, arithmetic-geometric averages, and various weighted averages (e.g., arithmetic weighted averages, geometric weighted averages, harmonic weighted averages).
