Mean

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Dina statistik, mean (rata-rata) mibanda dua harti:

average, leuwih cocog lamun disebut aritmetik mean, dibandingkeun jeung geometrik mean atawa harmonik mean. Average biasa oge disebut sample mean (rata-rata sampel).
nilai ekspektasi tina variabel acak, biasa disebut oge population mean (rata-rata populasi).

Sampel mean biasa dipake keur estimator ti central tendency saperti populasi mean. Sanajan kitu, estimator sejen oge dipake. Contona, median estimator nu leuwih robust keur central tendency tinimbang sampel mean.

Keur nilai-real variabel acak X, mean nyaeta nilai ekspektasi X. Lamun ekspektsi euweuh, variabel random teu ngabogaan mean.

Keur runtuyan data, mean ngan sakadar jumlah sakabeh observasi dibagi ku lobana observasi. Keur ngajelaskeun komunal tina susuna data, geus ilahar dipake simpangan baku, nu ngajelaskeun sabaraha beda tina observasi. Simpangan baku ngarupakeun akar kuadrat tina average atawa deviasi kuadrat tina mean.

Mean ngarupakeun nilai unik ngeunaan jumlah kuadrat deviasi nu minimum. Lamun ngitung jumlah kuadrat deviasi tina ukuran central tendency sejen, bakal leuwih gede tinimbang keur mean. Ieu nerangkeun kunaon simpangan baku sarta mean ilahar dipake babarengan dina laporan statistik.

Alternatip keur ngukur dispersi nyaeta simpangan mean, sarua jeung average simpangan mutlak tina mean. Ieu kurang sensitip keur outlier, tapi kurang nurut waktu kombinasi susunan data.

Nilai mean tina fungsi, $f (x)$ , dina interval, $a < x < b$ , bisa diitung (ngagunakeun proses limit dina definisi susunan data) saperti:

$E(f(X))=\frac{\int_a^b f(x)\,dx}{b-a}.$

Catetan, teu sakabeh probability distribution ngabogaan mean atawa varian - keur conto tempo sebaran Cauchy .

Di handap ngarupakeun kasimpulan tina sababaraha metoa keur ngitung mean tina susunan wilangan n.Tempo table of mathematical symbols keur nerangkeun simbol nu dipake.

[édit] Aritmetik Mean

The arithmetic mean is the "standard" average, often simply called the "mean". It is used for many purposes but also often abused by incorrectly using it to describe skewed distributions, with highly misleading results. The classic example is average income - using the arithmetic mean makes it appear to be much higher than is in fact the case. Consider the scores {1, 2, 2, 2, 3, 9}. The arithmetic mean is 3.16, but five out of six scores are below this!

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

[édit] Geometrik Mean

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

$\bar{x} = \sqrt[n]{\prod_{i=1}^n{x_i}}$

[édit] Harmonik 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).

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

[édit] Generalized Mean

The generalized mean is an abstraction of the Arithmetic, Geometric and Harmonic Means.

$\bar{x}(m) = \sqrt[m]{\frac{1}{n}\sum_{i=1}^n{x_i^m}}$

By choosing the appropriate value for the parameter m we can get the arithmetic mean (m = 1), the geometric mean (m -> 0) or the harmonic mean (m = -1)

This could be generalised further as

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

and again a suitable choice of an invertible f(x) will give the arithmetic mean with f(x)=x, the geometric mean with f(x)=log(x), and the harmonic mean with f(x)=1/x.

[édit] Weighted Mean

The weighted 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.

[édit] Truncated mean

Sometimes a set of numbers (the data) might be contaminated by inaccurate outliers, i.e. values which are much too low or much too high. 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.

[édit] 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.

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