Mean

Artikel ieu keur dikeureuyeuh, ditarjamahkeun tina basa Inggris. Bantuanna didagoan pikeun narjamahkeun.

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

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

Sampel méan biasa dipaké keur estimator ti central tendency saperti populasi méan. Sanajan kitu, éstimator séjén ogé dipaké. Contona, median éstimator nu leuwih robust keur central tendency tinimbang sampel méan.

Keur nilai-réal variabel acak X, méan nyaéta nilai ekspektasi X. Lamun ekspektsi euweuh, variabel random teu mibanda méan.

Keur runtuyan data, méan ngan sakadar jumlah sakabéh observasi dibagi ku lobana observasi. Keur ngajelaskeun komunal tina susuna data, geus ilahar dipaké simpangan baku, nu ngajelaskeun sabaraha béda tina observasi. Simpangan baku mangrupa akar kuadrat tina average atawa deviasi kuadrat tina méan.

Méan mangrupa nilai unik ngeunaan jumlah kuadrat deviasi nu minimum. Lamun ngitung jumlah kuadrat deviasi tina ukuran central tendency séjén, bakal leuwih gedé tinimbang keur méan. Ieu nerangkeun kunaon simpangan baku sarta méan ilahar dipaké babarengan dina laporan statistik.

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

Nilai méan tina fungsi, ${\displaystyle f(x)}$, dina interval, ${\displaystyle a<x<b}$, bisa diitung (ngagunakeun prosés limit dina definisi susunan data) saperti:

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

Catetan, teu sakabéh probability distribution mibanda méan atawa varian - keur conto tempo sebaran Cauchy .

Di handap mangrupa kasimpulan tina sababaraha metoa keur ngitung méan tina susunan wilangan n.Tempo table of mathematical symbols keur nerangkeun simbol nu dipaké.

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 misléading results. The classic example is average income - using the arithmetic méan makes it appéar to be much higher than is in fact the case. Consider the scores {1, 2, 2, 2, 3, 9}. The arithmetic méan is 3.16, but five out of six scores are below this!

${\displaystyle {\bar {x}}={1 \over n}\sum _{i=1}^{n}{x_{i}}}$

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 méan). For example rates of growth.

${\displaystyle {\bar {x}}={\sqrt[{n}]{\prod _{i=1}^{n}{x_{i}}}}}$

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).

${\displaystyle {\bar {x}}={\frac {n}{\sum _{i=1}^{n}{\frac {1}{x_{i}}}}}}$

The generalized mean is an abstraction of the Arithmetic, Géometric and Harmonic Méans.

${\displaystyle {\bar {x}}(m)={\sqrt[{m}]{{\frac {1}{n}}\sum _{i=1}^{n}{x_{i}^{m}}}}}$

By choosing the appropriate value for the paraméter m we can get the arithmetic méan (m = 1), the géometric méan (m -> 0) or the harmonic méan (m = -1)

This could be generalised further as

${\displaystyle {\bar {x}}=f^{-1}\left({{\frac {1}{n}}\sum _{i=1}^{n}{f(x_{i})}}\right)}$

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

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

${\displaystyle {\bar {x}}={\frac {\sum _{i=1}^{n}{w_{i}\cdot x_{i}}}{\sum _{i=1}^{n}{w_{i}}}}}$

The weights ${\displaystyle w_{i}}$ represent the bounds of the partial sample. In other applications they represent a méasure for the reliability of the influence upon the méan by respective values.

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 éach end, and then taking the arithmetic méan of the remaining data. The number of values removed is indicated as a percentage of total number of values.

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

${\displaystyle {\bar {x}}={2 \over n}\sum _{i=(n/4)+1}^{3n/4}{x_{i}}}$

assuming the values have been ordered.

• Arithmetic-geometric mean
• Geometric-harmonic mean
• Arithmetic-harmonic mean
• Central tendency
• Statistik deskriptif
• Kurtosis
• Median
• Mode
• Summary statistics

• Comparison between artihmetic and geometric mean