open-index/open-wikipedia-markdown huggingface dataset:
Pilihan probabiliti

Loba ahli statistik nyoko hiji panempo **pilihan** dina debat keur ngadukung interpretasi frekuensi tina probabiliti sarta ngadukung probabiliti personal. Hal nu penting tina ieu pilihan nyaéta, "lamun leumpang siga entog...". kabéhanna hayang mastikeun yén unggal kaahengan bakal nurut kana tiori aksioma probabiliti keur alesan interpretasi. Kadangkala maranéhanna bakal maké métodeu  nu pakait jeung métodeu  frekuensi, kadangkala ogé maké métodeu  Bayes.

Sababaraha ahli filosofi probabiliti ngadukung probabiliti pilihan salaku prinsip tinimbang tina panempo pragmatis. Alesanna yén beda interpretasi dina probabiliti mangrupa hal nu beda tina hiji hal jeung hal séjénna, dipaké dina kontek nu beda. Conto, interpretasi pobabiliti maké métodeu  frekuensi bisa ditarima keur percobaan nu berulang, interpretasi logika probabiliti dipaké dina kasus tunggal nu mibanda informasi awal nu husus, sarta interpretasi probabiliti personal dipaké keur kaayaan nu teu pasti. Hal ieu teu bisa disebutkeun yén nu nganalisa teu konsisten.
Stein's lemma

'''Stein's lemma'*, ngaran keur ngahargaan ka Charles Stein, may be characterized as a théorem of probability theory that is of interest primarily because of its application to statistical inference—in particular, its application to James-Stein estimation and empirical Bayes methods.

## Statement of the lemma
Suppose *X* is a normally distributed random variable with nilai ekspektasi &mu; and varian &sigma;2.  Further suppose *g* is a function for which the two expectations E( *g*(*X*) (*X* &minus; &mu;) ) and E( *g* &prime;(*X'') ) both exist (the existence of the expectation of any random variable is equivalent to the finiténess of the expectation of its absolute value).  Then

:E(g(X)(X-\mu))=\sigma^2 E(g'(X)).

In order to prove this lemma, recall that the probability density function for the normal distribution with expectation 0 and variance 1 is

:\varphi(x)={1 \over \sqrt{2\pi}}e^{-x^2/2}

and that for a normal distribution with expectation &mu; and varian &sigma;2 is

:{1\over\sigma}\varphi\left({x-\mu \over \sigma}\right).

Then use integration by parts.
Rumus prédiksi Spearman-Brown

**Rumus prédiksi Spéarman-Brown** (ogé dipikanyaho salaku *rumus ramalan Spearman-Brown*) nyaéta rumus hubungan reliabiliti psikometri kana panjangna uji:

{\rho}^*_{xx'}=\frac{N{\rho}_{xx'}}{1+(N-1){\rho}_{xx'}}

nu mana {\rho}^*_{xx'} nyaéta prédiksi reliabiliti; N nyaéta jumlah kombinasi "test" (tempo di handap); sarta {\rho}_{xx'} nyaéta reliabiliti "tes" ayeuna.  Rumus prédiksi réability mangrupa uji anyar nu nyimpen ulangan N ayeuna (atawa sarua jeung nambahkeun bentuk N paralél tina tes ayeuna ka tes ayeuna).  Maka N=2 akibatna duka kali panjang tes item nu ditambahkeun ku cara maké sipat nu sarua dina tes ayeuna.  Nilai N kurang ti hiji mungkin bisa dipaké keur prediksi pangaruh dina tes nu pondok.

Rumus bisa ogé disusun keur prediksi jumlah ulangan nu diperlukeun keur nangtukeun tingkat réalibiliti:

N=\frac{{\rho}_{xx'}(1-{\rho}^*_{xx'})}
{{\rho}^*_{xx'}(1-{\rho}_{xx'})}

Rumus ieu ilahar dipaké ku ahli psikometri keur prediksi réabiliti sanggeus ngarobah panjang tes. Hubungan ieu salah sahiji hal nu penting keur ngarobah-sabagian sarta métodeu  nu pakait jeung estimasi réabiliti.

Rumus ieu ogé mantuan ngarti kana hubungan nu teu linier antara tes réabiliti jeung panjang tes.

Lamun panjang/pondok tes teu paralel kana tes ayeuna, maka hasil prediksi moal akurat. Contona, lamun luhurna tes reliabiliti dipanjangan ku cara nambahan item nu goreng mangka reliabiliti bisa leuwih handap tinimbang nu diprediksi tina rumus ieu.

Tiori respon item merlukeun *informasi item* keur ngahasilkeun prediksi nu leuwih hadé dina ngaramal parobahan kualitas ukuran ku cara nambahan atawa ngurangan individu item.
Spésifisitas

Dina tés binér, saperti dina tés diagnosa médis keur panyakit di hiji wewengkon, **specificity** nyaéta proporsi négatip bener tina sakabéh sampel négatip nu dités, nyaéta
:{\rm specificity}=\frac{\rm number\ of\ true\ negatives}{{\rm number\ of\ true\ negatives}+{\rm number\ of\ false\ positives}}.

## Tempo Ogé
* klasifikasi binér
* receiver operating characteristic
* sensitivity (tests)
* statistical significance
* Type I error
Estimasi statistik

Aya sababaraha cara keur ngadiskusikeun **estimasi statistik**...

## Sasaran 
#Titik estimasi
#Interval estimasi
#multivariate estimation??
#Regression/Model building

## Pendekatan/Filosofi
#frequentist, or *classical*
#Bayesian estimation
#fiducial??

## Metoda 
(*gabungan tina sakabeh nu di luhur*)

#Maximum likelihood estimation
#Method of moments
#Least squares
#Influence functions?
#Interval kapercayaan (atawa wewengkon)
#Highest posterior density interval/region
#Fiducial??
#*etc...*
Sampel acak basajan

Dina statistik, **sampel acak basajan** tina hiji populasi nyaéta sampel nu dicokot sacara acak, unggal anggota dina éta populasi mibanda kamungkinan nu sarua pikeun dicokot.  Dina populasi nu saeutik, cara nyokot sampelna digawékeun ku cara "dicokot teu disimpen deui", nyaéta ku cara sacara tarapti milih unggal conto tina populasi.

Sacara konsép, nyokot sampel acak basajan mangrupa téhnik nyokot sampel probibiliti nu pangbasajanna, sanajan kitu langka dipaké sabab loba masalah dina makéna. Nyokot sampel acak basajan lain mangrupa métodeu  nu épéktip. métodeu  ieu merlukeun rohangan sampel nu kacida gedéna sarta hasilna merlukeun waktu ngitung nu lila sarta biaya nu mahal. Lamun panalungtik mibanda informasi nu cukup ngeunaan populasi, métodeu  ieu cukup épisién.

Kauntungannana nyaéta bébas tina klasifikasi kasalahan sarta merlukeun kanyaho nu teu loba ngeunaan populasi. Dua kaayaan nu hadé nu mana populasi ampir sarua sarta saeutikna informasi ngeunaan populasi. Lamun henteu dina kaayaan tadi, mangka leuwih hadé ngagunakeun *stratified sampling*.

## Tempo Ogé
* Statistik 
* Risét pamasaran 
* Risét pamasaran kuantitatif
* Sampling
* Cluster sampling 
* Multistage sampling 
* Nonprobability sampling 
* Systematic sampling 
* Stratified sampling''
Multistage sampling

**Multistage sampling** nyaéta bentuk kompleks tina cluster sampling. Ngagunakeun sakabel unsur sampel dina sakabéh klaster nu dipilih teu dimeunangkeun atawa teu penting. Dina kaayaan aturan ieu, nyokot sampel dina sababaraha tingkatan klaster jadi penting. Gantina maké kabéh unsur nu dipiboga ku klaster nu dipilih, panalungtik sacara acak milih unsur tina unggal klaster. Wangun klaster dina léngkah kahiji. Tangtukeun unsur mana nu dina jero klaster éta bakal dipaké, dina léngkah kadua. tehnik ieu remen dipaké lamun runtuyan lengkep sakabéh anggota dina populasi euweuh.

**Nyokot sampel Probabilitas-sabanding-kana-ukuran** mangrupa salah sahiji tipe nyokot sampel klaster multiléngkah. Dina métodeu  ieu, pamilihan probabilitas unsur anu ditangtukeun dina klaster variasina tibalik jeung ukuran dina klaster.

Sanajan nyokot sampel sacara klaster jeung nyokot sampling ku cara ngabédakeun lapisan sampel meh ampir sarua, tapi sacara substansi duanana béda. Dina nyokot sampel ku cara ngabédakeun lapisan, sampel acak dicokot tina sakabéh lapisan, nu mana dina nyokot sampel sacara klaster ngan tina klaster nu ditalungtik, boh dina saléngkah atawa multiléngkah

*Tempo ogé : statistika, marketing research, quantitative marketing research, sampling, cluster sampling, simple random sampling, nonprobability sampling, systematic sampling, stratified sampling*
Sebaran sampling

Dina statistik, **sebaran sampling** nyaéta sebaran probabiliti, dina kaayaan sampling dina populasi diulang, tina statistik nu dibérékeun (kuantitas numerik diitung tina nilai data dina sample).

Bentuk (atawa rumus keur) sebaran sampling gumantung kana sebaran populasi, statistik nu ditempo sarta ukuran sampel nu dipaké.

(Catetan yén rumus nu leuwih présisi nyebutkeun yén sebaran statistik bisa dipaké keur *sakabeh sampel nu mungkin* tina ukuran nu dibérékeun, henteu ngan sakadar "dina kaayaan sampling diulang".)

Contona, anggap populasi nu kacida gedéna nuturkeun normal (ilaharna disebut kurva bel). Asumsi urang mindo nyokot sampe dina sampel nu dibérékeun tina populasi sarta ngitung sample mean (arithmetic mean nilai data) keur unggal sampel. Sampel nu béda bakal mangaruhan kana bédana sampel méans. Sebaran méan ieu nyaéta "sebaran sampling tina sampel mean" (keur ukuran sampel nu dibérékeun). Sebaran ieu bakal jadi normal saprak populasina normal. (Dumasar kana teorema central limit, lamun populasi teu normal tapi "bentukna cukup hade",
sebaran sampling  tina sampel méan bakal masih kénéh cukup normal sanajan ukuran sampel kacida gedéna).

Simpangan baku tina sebaran sampling ngacu ka standar kasalahan (statistis) sarta di-estimasi maké rumus:

:\frac{\sigma}{\sqrt{N}}

nu mana \sigma mangrupa simpangan baku tina sebaran populasi sarta N nyaéta ukuran sampe (jumlah barang) dina sampel.

Hal nu penting dina rumus ieu nyaéta kudu ngalikeun opat kali ukuran sampel (4X) ngarah ukuran kasalahanna jadi (1/2).  Waktu rarangkay elmu statistik nu mana biaya mangrupa faktor, ieu mangrupa faktor nu kudu dipikaharti ngeunaan pentingna biaya.

Alternatipna, anggap median sampel tina populasi. Ieu mibanda sebaran sampling nu béda sacara umum teu normal (tapi masih kénéh pakait dina aturan ka-normal-an).

### Tempo oge
central limit theorem, standard error (statistics)
Latin square

**Latin square** mangrupa tabel *n* × *n* nu dieusi ku *n* lambang nu béda dina sababaraha jalan mangka éta lambang pasti ngan aya sakali dina unggal baris jeung kolom.

\begin{bmatrix}
 1 & 2 & 3 \\
 2 & 3 & 1 \\
 3 & 1 & 2 \\
\end{bmatrix}

\begin{bmatrix}
 a & b & c \\
 b & c & a \\
 c & a & b \\
\end{bmatrix}

Latin squares salaku multiplication table dina quasigroup, ilahar dipaké dina desain percobaan.

Ngaran Latin square asalna ti Leonard Euler nu ngagunakeun simbol ku hurup Latin.

## Tempo ogé 
* Graeco-Latin square
G-test

Dina statistik, '*tes-*G'''* nyaéta tes rasio likelihood atawa maximum likelihood nu loba dipaké dina statistical significance sarta saméméhna disarankeun maké tes chi-kuadrat.

Ilaharna maké tes chi-kuadrat keur *goodness of fit* tina hiji sebaran sarta keur kabébasan dina *contingency table* nu dina kanyataanna dideukeutan ku *log-likelihood ratio* keur **tes G** ieu. Perkiraan ieu diwanohkeun ku Karl Péarson alatan perluna waktu lila keur ngitung *log-likelihood''. Ku mecenghulna kalkulator sarta komputer pribadi éléktronik, hal ieu henteu jadi masalah deui. **Tes-G** mucunghul tur loba dipaké, saprak direkomendasikan di édisi 1994 dina sababaraha buku teks statistika ku Sokal sarta Rohlf.

Rumus umum keur tes statistik chi-kuadrat Péarson's nyaéta

 \chi^2 = \sum {(O - E)^2 \over E}

nu mana O nyaéta frékuénsi observasi dina sel, E nyaéta ekspektasi frékuénsi dina null hipotesa, sarta jumlah dina sakabéh sel. Rumus pakait nu ilahar dipaké keur *G* nyaéta 

 G = -2\sum {O\times \ln(O/E) }

nu mana ln ngalambangkeun logaritma alami (log dina *e*) sarta jumlah dina sakabéh sel.

Dina null hipotesa yén hasil frekuesni observasi tina nyokot sampel sacara acak tina hiji sebaran ditangtukeun ku ekspektasi frékuénsi nu dipikanyaho, sebaran *G* meh sarua jeung chi-kuadrat, nu mana jumlah wilangan tingkat kabebasan saperti dina tes chi-kuadrat.

Keur sampel nu meh sarua, tes-*G* jeung tes chi kuadrat bakal ngahasilkeun kasimpulan nu sarua. Sanajan kitu, pamarekankana téori sebaran chi-kuadrat keur tes-*G* leuwih hadé tina tes chi-kuadrat Péarson dina pasualan di mana keur sakabéh sel |*O* &minus; *E*| > *E*, sarta dina sababaraha pasualan tes*G* geus ilahar dipaké.

Keur sampel nu saeutik tes binomial keur *goodness of fit*, sarta tes kapastian Fisher keur *contingency tables*, leuwih hadé maké tes chi-kuadrat atawa tes-*G*.

## Rujukan
*Sokal, R. R., & Rohlf, F. J. (1994). *Biometry: the principles and practice of statistics in biological research.*, 3rd edition. New York: Freeman. ISBN 0-7167-2411-1.
Latin hypercube sampling

métodeu  statistik **Latin hypercube sampling** diwangun sarta dikumpulkeun ku Ronald L. Iman, J. C. Helton, and J. E. Campbell, saparakanca keur nga-generate sebaran koleksi nu asup akal tina nilai paramater dina sebaran multidimensi.

Paperna *An approach to sensitivity analysis of computer models, Part I. Introduction, input variable selection and preliminary variable assessment.* dina Journal of Quality Technology taun 1981.
Téoréma Lehmann-Scheffé

Dina statistik, **téoréma Lehmann-Scheffé** netepkeun yén unggal éstimator anu lengkep, cukup, sarta teu bias mangrupa éstimator teu bias nu *unik* tina nilai ekspektasi-na.

Cara nu ilahar dipaké keur néangan nilai éstimator nyaéta ngagunakeuntéoréma Rao-Blackwell.
Parameter lokasi

Dina statistik, lamun kulawarga densiti sebaran diparaméterisasi ku paraméter nilai -skalar atawa -vektor μ mangrupa bentuk

:*f*&mu;(*x*) = *f*(*x* &minus; &mu;)

mangka μ disebut **paraméter lokasi**, sabab ieu nilai nangtukeun "lokasi" tina sebaran probabiliti.

## Tempo oge
* numerical parameter
* statistical dispersion
* scale parameter
Kurva Lorenz

**Kurva Lorenz** diwangun ku Max O. Lorenz salaku grapik nu ngagambarkeun kateusajajaran pangala. Kurva bisa dipaké ogé keur ngukur kateusajajaran aset atawa sebaran séjénna.

Dina diskusi panghasilan pribadi, urang remen nyebutkeun yen, "duapuluh persen sahandapeun sakabeh rumahtangga mibanda sapuluh persen total panghasilan". Kurva Lorenz dumasar kana ieu *pernyataan*; unggal titik dina kurva ngagambarkeun unggal *pernyataan*.

Kurva Lorenz mangrupa grafik nu nembongkeun, di handap x% rumahtangga, persentase y% total panghasilan no dipibanda. Persentase rumah tangga di plot dina sumbu x, persentasi panghasilan dina sumbu y.

Sebaran panghasilan persis sarua di masarakat bakal jadi hiji di mana unggal jalma mibanda panghasilan nu sarua. Dina kasus ieu, handapeun N% masarakat bakal salawasna mibanda N% panghasilan. Mangka sebaran nu persis sarua digambarkeun ku garis lempeng y=x; bisa disebut yén ieu garis mangrupa garis persis sarua.

Sebaran nu persis teusarua, sacara jelas, nyaéta saurang boga kabeg penghasilan sarta nu séjénna teu boga nanaon. Dina kasus eta, kurva bakal dina y=0 keur sakabrh x<100, sarta y=100 waktu x=100. Garis dina kurva ieu disebut garis kateusaruaan sampurna.

Catetan yén dumasar kana panghasilan (atawa naon baé nu bakal diukur) teu bisa negatip, mangka teu mungkin keur kurva Lorenz naek di saluhureun garis sarua sampurna, atawa nurun sahandapeun garis teusarua sampurna.  Kurva kudu naek sarta konkav.

Kurva Lorenz dipeke keur ngitung koefisien Gini, nu mana daerah antara garis sampurna jeung kurva, salaku persentasi daerah antara garis sarua sampurna jeung garis teu sarua sampurna.

Tipikal kurva Lorenz bisa ditempo di handap ieu:

## Baca ogé
* Ékonomi kamakmuran
* Méktika kateusajajaran pangala
* Koefisien Gini
* Kamakmuran sosial (élmu pulitik)
* Analisis ROC
Loss function

Dina statistik, téori kaputusan sarta ékonomi, **loss function** nyaéta fungsi nu metakeun hiji kajadian (sacara teknis hiji unsur dina rohangan sampel) kana wilangan riil keur ngagambarkeun biaya ekonomi atawa hanjelu pakait jeung hiji kajadian.

*Loss functions* sacara tipikal digambarkeun dina watesan "keuangan" dumasar kana ukuran biaya nu mungkin, contona kalahiran jeung "kematian" dina widang public health.

*Loss functions* are complementary to utility functions which represent benefit and satisfaction. Typically, for utility *U*, loss is equal to *k*-*U*, where *k* is some arbitrary constant.

## Expected loss
*Loss function* satisfies the definition of a random variable so we can establish a cumulative distribution function and an nilai ekspektasi. However, more commonly, the *loss function* is expressed as a function of some other random variable. For example, the time that a light bulb operates before failure is a random variable and we can specify the loss, arising from having to cope in the dark and/or replace the bulb, as a function of failure time. For a continuous random variable, *X* with probability density function *f*(*x*) and *loss function*, &lambda;(*x*), the *expected loss* is:

:\Lambda = \int_{-\infty}^\infty \lambda (x) f(x) dx.

Minimum *expected loss* is widely used as a criterion for choosing between prospects. It is closely related to the criterion of maximum *expected utility*.

## Loss functions in Bayesian statistics
One of the consequences of Bayesian inference is that in addition to experimental data, the loss function  does not in itself wholly determine a decision.  What is important is the relationship between the loss function and the prior probability.  So it is possible to have two different loss functions which léad to the same decision when the prior probability distributions associated with éach compensate for the details of éach loss function.

Combining the three elements of the prior probability, the data, and the loss function then allows decisions to be based on maximising the subjective expected utility, a concept introduced by Leonard J. Savage.  He also argued that using non-Bayesian methods such as minimax, the loss function should be based on the idéa of regret, i.e. the loss associated with a decision should be the difference between the consequences of the best decision that could have been taken had the underlying circumstances been known and the decision that was in fact taken before they were known.
Hukum Zipf-Mandelbrot

**Hukum Zipf-Mandelbrot** (oge dipikanyaho minangka hukum Pareto-Zipf) 
nyaéta power-law distribution dina ranked data, dipaké sanggeus Harvard linguistic professor George Kingsley Zipf (1902-1950) 
nu ngusulkeun dina teks regularity, sarta mathematician Benoit Mandelbrot (lahir November 20, 1924), no nga-generalisir.

Distribution kecap dirangking ku frequency-na dina random
corpus text mangrupa power-law distribution umum, dipikanyaho minangka Zipf's law.

Lamun hiji plot rangking frequency kecap dikandung dina 
corpus nu gedé tina data teks ka jumlah kajadian atawa frequencies sabenerna, meunangkeun power-law distribution, mibanda exponent raket ka hiji (tapi tempo Gelbukh jeung Sidoro 2001).

## Tumbu kaluar
* Z. K. Silagadze: Citations and the Zipf-Mandelbrot's law
* NIST: Zipf's law
* W. Li's References on Zipf's law
Yates' correction for continuity

**Koreksi Yates keur kontinyu**, atawa *tes chi-kuadrat Yates*, ngatur rumus keur uji kuadrat-chi Pearson nu ngurangan 0.5 tina unggal nilai dina tabel kontingensi 2 X 2. Rumus ieu dipaké lamun dina hiji sel dina tabel ngabogaan nilai ekspektasi kurang ti 5.

Sumber séjén nyebutkeun yén koreksi ieu kudu dipaké lamun ekspektasi frekuensi kurang ti 10.
Studentized residual

In statistics, a **Studentized residual**, named in honor of William Sealey Gosset, who wrote under the pseudonym ***Student***, is a residual adjusted by dividing it by an estimate of its standard deviation.  Studentization of residuals is an important technique in the detection of outliers.

## Errors versus residuals
It is very important to understand the difference between errors and residuals in statistics.  Consider simple linear regression modél

:Y_i=\alpha_0+\alpha_1 x_i+\varepsilon_i,

where the **errors** &epsilon;*i*, *i* = 1, ..., *n*, are independent and all have the same variance &sigma;2.  The **residuals** are not the true, and unobservable, errors, but rather are *estimates*, based on the observable data, of the errors.  When the method of léast squares is used to estimate &alpha;0 and &alpha;1, then the residuals, unlike the errors, cannot be independent since they satisfy the two constraints

:\sum_{i=1}^n \hat{\varepsilon}_i=0

and

:\sum_{i=1}^n \hat{\varepsilon}_i x_i=0.

(Here \varepsilon_i is the *i*th error, and \hat{\varepsilon}_i is the *i*th residual.)  Moréover, the residuals, unlike the errors, do not all have the same variance: the variance incréases as the corresponding *x*-value gets farther from the average *x*-value.  ***The fact that the variances of the residuals differ, even though the variances of the true errors are all equal to éach other, is the principal réason for the need for Studentization.***

## How to Studentize
For this simple modél, the "design matrix" is

:X=\left[\begin{matrix}1 & x_1 \\  \vdots & \vdots \\ 1 & x_n \end{matrix}\right]

and the "hat matrix" *H* is the matrix of the orthogonal projection onto the column space of the design matrix:

:H=X(X^T X)^{-1}X^T.

The "leverage" *h**ii* is the *i*th diagonal entry in the hat matrix.  The variance of the *i*th residual is

:\mbox{var}(\hat{\varepsilon}_i)=\sigma^2(1-h_{ii}).

The corresponding **Studentized residual** is then

:{\hat{\varepsilon}_i\over \hat{\sigma} \sqrt{1-h_{ii}\  }}

where \hat{\sigma} is an appropriate estimate of &sigma;.

## Internal and external Studentization
The estimate of &sigma;2 may be

:\hat{\sigma}^2={1 \over n-2}\sum_{j=1}^n \hat{\varepsilon}_j^2.

But it is desirable to exclude the *i*th observation from the process of estimating the variance when one is considering whether to consider the *i*th case to be an outlier.  Consequently one may use the estimate

:\hat{\sigma}_{(i)}^2={1 \over n-3}\sum_{j=1}^n \hat{\varepsilon}_j^2,

based on all but the *i*th case.  If the latter estimate is used, *excluding* the *i*th case, then the residual is said to be ***externally Studentized***, if the former is used, *including* the *i*th case, then it is ***internally Studentized***.

If the errors are independent and  normally distributed with nilai ekspektasi 0 and varian &sigma;2, then the probability distribution of the *i*th externally Studentized residual is a sebaran-t student nu mana *n* &minus; 3 tingkat kabebasan.
Model linier

Dina statistik **modél linier** bisa ditembongkeun ku nyebutkeun

* Y = X \beta + \epsilon

di mana Y mangrupa nx1 vektor kolom variabel random, X mangrupa matrik kuantitas nxp "dipikanyaho" (contona, bisa di-observasi sarta non-random), vektor baris pakait jeung statistical unit, &beta; mangrupa  px1 vektor paraméter (teu ka-observasi), sarta &epsilon; mangrupa  nx1 vektor "error", nu teu pakait ka variabel random nu mibanda nilai ekspektasi 0 sarta varian  &sigma;2.  Salawasna komponen vektor kasalahan nu dicokot bakal independent sarta kasebar normal.  Anggap nilai X sarta Y ka observasi, statistikawan kudu nga-estimate &beta; sarta &sigma;2.  Sacara tipikal paraméter &beta; di-estimasi maké métodeu  least squares.

Tinimbang nyokot varian &epsilon; jadi &sigma;2I, nu mana I mangrupa nxn matrik identitas, anggap varian mangrupa &sigma;2M,
nu mana M mangrupa matrik séjén nu dipikanyaho salian ti matrik identitas, mangka estimate &beta; maké métodeu  "generalized least squares", nu mana, ku ngaminimalkeun kuadrat residu, minimalkeun bentuk béda kuadrat dina residu—bentuk kuadrat dibérékeun ku matrik M−1.
Lamun sakabéh diagonal dina matrik M sarua jeung 0, mangka estimasi normal &beta; ku métodeu  "weighted least squares", nu mibanda beurat sarua jeung diagonal asupan.

Linear regression ordiner mangrupa topik nu raket pakait.

## Generalisasi 
### Generalisasi model linier 
Generalized linear models, tinimbang 
* E(Y)=X&beta;,
leuwih ilahar
*f(E(Y))=X&beta;,
di mana f mangrupa "fungsi pakait".  Contona "model regresi Poisson", ku nuliskeun 
*Yi mibanda sebaran Poisson nu mibanda nilai ekspektasi e&gamma;+&delta;xi.
Fungsi pakait mangrupa fungsi logaritma natural.
Observasi nu dipiboga xi sarta Yi keur
i=1,...,n, bisa nga-estimasi &gamma; and &delta; ku métodeu  maximum likelihood.

### Model linier general 
General linear model (atawa multivariate regression model) mangrupa modél linier nu mibanda sababara ukuran dina unggal obyek.
Unggal obyek diwakilan dina bentuk vektor.
Mean kuadrat kasalahan

Dina statistik, **méan kuadrat kasalahan** tina estimator *T* dina paraméter nu teu ka-observasi &theta; nyaéta

:\operatorname{MSE}(T)=\operatorname{E}((T-\theta)^2),

dina hal ieu, mangrupa nilai ekspektasi kuadrat "kasalahan".  "Kasalahan" nyaéta jumlah nu mana éstimator béda jeung jumlah nu keur di-estimasi.  Méan kuadrat kasalahan nyukupan identitas

:\operatorname{MSE}(T)=\operatorname{var}(T)+(\operatorname{bias}(T))^2

di mana

:\operatorname{bias}(T)=\operatorname{E}(T)-\theta,

dina hal ieu, the bias nyaéta lobana nu mana nilai ekspektasi tina éstimator béda keur jumlah nu teu ka-observasi nu keur di-estimasi.

Conto kongkritna.  Anggap

:X_1,\dots,X_n\sim\operatorname{N}(\mu,\sigma^2),

dina hal ieu, ukran sampel random *n* tina populasi sebaran normal.  Dua éstimators  &sigma;2 kadangkala dipaké (atawa nu séjénna):

:\frac{1}{n}\sum_{i=1}^n\left(X_i-\overline{X}\,\right)^2\ {\rm and}\ \frac{1}
{n-1}\sum_{i=1}^n\left(X_i-\overline{X}\,\right)^2 

nu mana

:\overline{X}=(X_1+\cdots+X_n)/n

mangrupa "sampel mean".   Kahiji tina éstimator ieu nyaéta éstimator maximum likelihood, sarta bias, dina hal ieu, bias teu sarua jeung nol, tapi mibanda varian nu leuwih leutik tinimbang nu kadua, anu teu bias.  Varian leutik tina akibat séjén keur bias, mangka méan kuadrat kasalahan tina bias éstimator leuwih leutik tinimbang éstimator unbiased.
Tés nilai tengah

Dina statistika, **tes nilai tengah** atawa **tes median** mangrupa kasus husus dina uji kuadrat-chi Pearson. Tes null hypothesis nu median tina population tina dua samples digambarkeun identik. Data unggal sampe dibagi kana dua grup, hiji grup ngandung nilai data nu leuwih luhur tina median, sarta grup séjénna ngandung nilai data dina median atawa sahandapeunnana. Tes chi-kuadrat Péarson dipaké keur nangtukeun frékuénsi observasi dina unggal grup nu béda tina ekspektasi frékuénsi nu asalna tina kombinasi dua grup sebaran.

Statistical power dina tes ieu kadangkala ditingkatkeun ku maké nilai séjén tinimbang median nu dihartikeun tina grup - hal ieu, ku maké nilai nu ngabagi grup kana nilai grup nu ampir sarua tinimbang median-na.

Tes kacida kapakena waktu sebaran data béda tina normal, contona waktu raw score dina klasifikasi teu teratur kana *rentang pendekatan* nu dijieun. Ku kitu, ieu biasa dipaké salaku léngkah awal dina exploratory analysis.
Statistik matematis

**Statistik matematis** ngagunakeuntiori probabiliti sarta cabang séjén matematik pikeun ngulik statistik tina jihat matématis murni.

Jadi, pikeun conto, hiji conto acak dituliskeun sabagé runtuyan variabel acak,
:X_1, X_2, \ldots, X_n\,\!  
nu masing-masing nuturkeun hiji sebaran probabiliti nu tangtu (ilaharna dianggap independent and identically distributed|kasebar bebas sarta identik).
Sample mean digambarkeun salaku fungsi tina variabel,
:\bar{X} = {1\over n} \sum_{i=1}^n X_i\,\!  
sarta mangrupa variabel acak tina hiji sebaran.

Ku jalan ngagunakeun ieu cara, **statistik matematik** mangrupa salah sahiji usaha keur ngabuktikeun téori-téori nu loba dipaké dina statistik terapan.
Sebaran binomial négatip

Dina téori probabilitas, **sebaran binomial négatip** nyaéta anggota tina kulawarga sebaran probabilitas diskrit.
Léma Neyman-Pearson

Dina statistik, **léma Neyman-Péarson** nangtukeun yén waktu ngagawekeun tes hipotesa antara dua titik hipotesa *H*0:&nbsp;*&theta;*=*&theta;*0 sarta *H*1:&nbsp;*&theta;*=*&theta;*1, mangka likelihood-ratio test nu nolak *H*0 dina ngadukung *H*1 waktu

:\Lambda(x)=\frac{ L( \theta _{0} \mid x)}{ L (\theta _{1} \mid x)} \leq k \mbox{ where } Pr(\Lambda(X)\leq k|H_0)=\alpha

leuwih ilahar salaku tes power ukuran *&alpha;*.
Statistik non-parametrik

**Métode kaputusan statistik** **Non-parametrik** (atawa sebaran-bébas) mangrupa prosedur matematik keur tes hipotesa statistik, teu siga dina statistik parametrik, taya asumsi ngeunaan sebaran frekuensi tina variabel nu ditaksir. Pamakéan nu ilahar ieu métodeu  nyaéta dina tes chi-kuadrat. Pamakéan séjén nu maké métodeu  non-parametrik kaasup Mann-Whitney U, Kruskal-Wallis one-way analysis of variance, median test, Spearman's &rho;, sarta estimasi sebaran maké sebaran binomial.

kabéhanna *mungkin* leuwih mibanda statistical power tinimbang tes parametrik waktu asumsi dina kaayaan tes parametrik teu nyugemakeun.

Tempo ogé statistik parametrik.
Memorylessness

Dina tiori probabiliti, **memorylessness** nyaéta sipat penting sebaran probabiliti: sebaran eksponensial sarta sebaran geometrik.

## Discrete memorylessness
Suppose *X* is a discrete random variable whose values lie in the set { 0, 1, 2, ... } or in the set { 1, 2, 3, ... }.
The probability distribution of *X* is **memoryless** precisely if for any *x*, *y* in { 0, 1, 2, ... } or in { 1, 2, 3, ... }, (as the case may be), we have

:P(X>x+y \mid X>x)=P(X>y).

It can réadily be shown that the *only* probability distributions that enjoy this discrete memorylessness are geometric distributions.  These are the distributions of the number of independent Bernoulli trials needed to get one "success", with a fixed probability *p* of "success" on éach trial.

## Example and motivation for the name *memorylessness*
For example, suppose a die is thrown as many times as it takes to get a "1", so that the probability of "success" on éach trial is 1/6, and the random variable *X* is the number of times the die must be thrown.  Then *X* has a géometric distribution, and the conditional probability that the die must be thrown at léast four more times to get a "1", given that it has alréady been thrown 10 times without a "1" appéaring, is no different from the original probability that the die would be thrown at léast four times.  In effect, the random process does not "remember" how many failures have occurred so far.

### A frequent misunderstanding
Memorylessness is often misunderstood by students taking courses on probability: the fact that P(*X* > 16 | *X* > 12) = P(*X* > 4) does *not* méan that the events *X* > 16 and *X* > 12 are independent; i.e., it does *not* méan that P(*X* > 16 | *X* > 12) = P(*X* > 16).  To summarize: "memorlessness" of the probability distribution of the number of trials *X* until the first success méans

:\mathrm{(Right)}\ P(X>16 \mid X>12)=P(X>4).

It does *not* méan

:\mathrm{(Wrong)}\ P(X>16 \mid X>12)=P(X>16).

(That would be independence.  These two events are *not* independent.)

## Continuous memorylessness
Suppose that rather than considering the discrete number of trials until the first "success", we consider continuous waiting time *T* until the arrival of the first phone call at a switchboard.  To say that the probability distribtuion of *T* is **memoryless** méans that for any positive real numbers *s* and *t*, we have

:P(T>t+s \mid T>t)=P(T>s).

The only difference between this and the discrete version is that instéad of requiring *s* and *t* to be positive (or, in some cases, nonnegative) integers, thus achieving discreteness, we allow them to be réal numbers that are not necessarily integers.

It can be shown that the only probability distributions that enjoy this continuous memorylessness are the sebaran eksponensial.
Rankit

Dina statistik, **rankits** titik data tina susunan data nu ngandung jejer skalar sederhana nyaéta nilai ekspektasi tina order statistik dina standar  sebaran normal corresponding to data points in a manner determined by the order in which the data points appéar.

This is perhaps most réadily understood by méans of an example.  If an i.i.d. sample of six items is taken from a kasebar normal population with nilai ekspektasi 0 and varian 1 and then sorted into incréasing order, the expected values of the resulting order statistics are:

:-1.2816,\ \  -0.64335,\ \  -0.20189,\ \  0.20189,\ \  0.64335,\ \  1.2816

Suppose the numbers in a data set are

: 65,  75,  16,  22,  43,  40.

The corresponding ranks are

: 5,  6,  1,  2,  4,  3,

i.e., the number appéaring first is the 5th-smallest, the number appéaring second is 6th-smallest, the number appéaring third is smallest, the number appéaring fourth is 2nd-smallest, etc.  One réarranges the expected normal order statistics accordingly, getting the **rankits** of this data set:

:\begin{matrix}
\mbox{data}\ \mbox{point} & & \mbox{rankit} \\  \\
65 & & 0.64335 \\
75 & & 1.2816 \\
16 & & -1.2816 \\
22 & & -0.64335 \\
43 & & 0.20189 \\
40 & & -0.20189\end{matrix}

A graph ploting the rankits on the horizontal axis and the data points on the vertical axis is a **rankit plot** or **normal probability plot**.  Such a plot is necessarily nondecréasing.  In large samples from a normally distributed population, such a plot will approximate a straight line.  Substantial deviations from straightness are considered evidence against normality of the distribution.

The word ***rankit*** was introduced by the statistician Chester Bliss (not to be confused with the politician Chester Bliss Bowles).
Semivarian

Dina statistik spasial, **semivarian** dijelaskeun ku 
:\gamma(h)=\sum_{i=1}^{n(h)}\frac{(z(x+h)-z(x))^2}{n(h)}
nu mana *z* mangrupa nilai data tina sabagian lokasi, *h* nyaéta jarak antara nilai data, sarta *n*(*h*) dianggap angka pasangan nilai data dibérékeun, spasi jarak tina bagian *h*.

Pot semivarian ka jarak antara data nilai dipikanyaho salaku semivariogram, atawa keur gampang nyebutna variogram.

Topik pakait: géostatistik

Rujukan:
# Shine, J.A., Wakefield, G.I.: A comparison of supervised imagery classification using analyst-chosen and géostatistically-chosen training sets, 1999, http://www.geovista.psu.edu/sites/geocomp99/Gc99/044/gc_044.htm
Rentang tolérans

Ogé katelah wates toléran (Ing. *tolerance limits*).

## Pamakéan rentang toléran
Rentang toléran patali jeung rékayasa wates toléran. Sacara husus, rékayasa wates toléran nangtukeun niléy maksimum jeung minimum pikeun hiji produk sangkan bisa lumaku. Rékayasa wates toléran bisa ditangtukeun maké *Geometric Dimensioning and Tolerancing*. Rentang tolérans mangrupa dasar keur chart kontrol dina sababaraha rupa.
Statistik interval tolérans (atawa watés) dihasilekun tina mroses data.
Lobana ieu interval gumantung kana prosésna.
Rentang toléran dihartikeun salaku kamampuan prosés keur nangtukeun nilai minimum jeung maksimum.
Nilai wates dina ieu wewengkon mibanda bagéan nu sarimbag jeung sakabéh populasi sarta mibanda probabiliti atawa kapercayaan husus.

## Rujukan kaluar
Tempo The NIST/SEMATECH e-Handbook of Statistical Methods, 2004 July 17.

Dina bagian 
Section 7.2.6.3 - 
Tolerance intervals for a normal distribution

Tempo "Quality Control Handbook", Juran.
Métode Taguchi

**métodeu  Taguchi** nyaéta métodeu  statistik nu diwangun ku Genichi Taguchi keur ningkatkeun kualitas produksi pabrik. *Metoda Taguchi* mangrupa hal nu kontroversi keur statistikawan urang kulon.

Prinsip Taguchi nu mangaruhan kana statistik nyaéta

#Taguchi loss-function;
#Pilosopi *off-line quality control*; sarta
#Hal anyar dina desain percobaan.

## Pustaka
*León, R. V., A. C. Shoemaker & R. N. Kacker. 1987. Performance méasures independent of adjustment: an explanation and extension of Taguchi's signal-to-noise ratios (with discussion), *Technometrics* 29: 253-285
* Moen, R. D., T. W. Nolan & L. P. Provost. 1991. *Improving Quality Through Planned Experimentation*. ISBN 0070426732
*Nair, V. N. (éd.). 1992. Taguchi's paraméter design: a panel discussion. *Technometrics* 34: 127-161
Outlier

Dina statistik, **outlier** nyaéta hiji observasi nu jauh tina data nu keur ditempo.

Harti "jauh tina" di hal ieu nyaéta:
:kurang ti *Q1* &minus; 1.5 &times; *IQR* **atawa** leuwih ti *Q3* + 1.5 &times; *IQR*
nu mana *Q1* sarta *Q3* kuartil kahiji jeung katilu, sarta *IQR* interquartile range (sarua jeung *Q3* &minus; *Q1*).

Nilai ieu ngahartikeun yén hartina aya dina "jero pager", saluareun éta observasi bisa dingaranan**mild outlier**.

**Outliers ekstrim** nyaéta observasi ayana *saluareun pager*:
:kurang ti *Q1* &minus; 3 &times; *IQR* **atawa** leuwih ti *Q3* + 3 &times; *IQR*

Dina kasus data kasebar normal, maké harti di luhur, ngan 1 tina 150 observasi bakal aya *mild outlier* sarta ngan 1 tina 425,000 aya *outlier ekstrim*.

*Outliers* ilaharna merlukeun perhatian saprak nembongkeun masalah dina sampling atawa kumpulan data atawa transkrip.  Alternatipna, outlier bisa jadi hasil, contona, respon nu teu umum kana *perlakuan* nu ditempo, nu jadi perhatian *peneliti*.

### Tempo ogé 
* box plot
*Studentized residual
Masalah Urn

An **urn problem** is an idéalized thought experiment in which some objects of réal interest (such as atoms, péople, cars, etc.) are represented as colored balls in an urn or other container.
One pretends to draw (remove) one or more balls from the urn;
the goal is to determine the probability of drawing one color or another, 
or some other properties.

## Basic urn model 
In this basic urn modél in probability theory, the urn contains *x* white and *y* black balls; one ball is drawn randomly from the urn and its color observed; it is then placed back in the urn, and the selection process is repéated.

Possible questions that can be answered in this modél are:
* can I infer the proportion of white and black balls from n observations ? With what degree of confidence ?
* knowing *x* and *y*, what is the probability of drawing a specific sequence (e.g. one white followed by one black)?
* if I only observe n white balls, how sure can I be that there is no black balls?

## Other models 
Many other variations exist:
* the urn could have numbéréd balls instéad of colored ones
* balls may not be returned to the urns once drawn.

## Conto masalah urn
* Turunan sebaran binomial
* Turunan sebaran hipergeometrik
* Statistik fisika: turunan tanaga sarta sebaran kecepatan

## Historical remarks
Urn problems have been a part of the theory of probability since at léast the publication of the *Ars conjectandi* by Jakob Bernoulli (1713).
Bernoulli's inspiration may have been lotteries, elections, or games of chance which involved drawing balls from a container.
It has been asserted 
http://mathforum.org/epigone/historia_matematica/sningzahzhil/3DEFCC9A.73AA528D@earthlink.net 
that

:*Elections in medieval and renaissance Venice, including that of the doge, often included the choice of electors by lot, using balls of different colors drawn from an urn.*

Bernoulli himself, in *Ars conjectandi*, considered the problem of determining, from a number of pebbles drawn from an urn, the proportions of different colors.
This problem was known as the *inverse probability* problem, and was a topic of reséarch in the eighteenth century,
attracting the attention of Abraham de Moivre and Thomas Bayes.

## See also
* Coin-tossing problems
Ahmad H. Zewail

**Ahmad Hassan Zewail** (Basa Arab: أحمد زويل) (wedal 26 Pébruari 1946) kimiawan urang Mesir, dilélér Hadiah Nobel widang Kimia taun 1999 pikeun karyana ngeunaan fémtokimia. Lahir di Damanhur (60&nbsp;km kidul-wétaneun Iskandariah) sarta digedékeun di Disuq, anjeunna nampa gelar kahijina ti Universitas Iskandariah méméh pindah ka US pikeun ngaréngsékeun PhD-na di University of Pennsylvania. 

## Atikan
Sanggeus sababaraha karya posdoktorat di UC Berkeley, anjeunna dileler a faculty appointment di Caltech taun 1976, sarta anjeunna terus ngabakti di dinya dugi ka pupusna. Dina taun 1990, anjeunna diangkat janten jalma munggaran anu nyepeng jabatan "Linus Pauling Chair in Chemical Physics", hiji jabatan akademik anu kacida dipikaajrihna.

Karya penting Zewail nyaéta mitambeyan fémtokimia - ulikan réaksi kimiawi dina jangka waktu fémtodetik. ngagunakeun téhnik laser nu gancang (céntangan laser pondok kacida), anu matak ngamungkinkeun para élmuwan ningali kumaha atom-atom pindah sarta kumaha beungkeut kimia pegat atawa ngahiji sacara *real-time*.

## Pangajen
Taun 1999, Zewail jadi urang Mesir katilu nu nampa Hadiah Nobel, nuturkeun Anwar Sadat (1978 widang perdamaian) jeung Naguib Mahfouz (1988 widang Sastra). Pangleler internasional séjénna di antarana Hadiah Wolf (1993) jeung Robert A. Welch Award (1997). Taun 1999 anjeunna nampa panghargaan nagara pangluhurna, Grand Collar of the Nile.

Taun 1999, anjeunna nampa pangajén nagara pangluhurna ti Mesir, nyaéta Grand Collar of the Nile anu dilélér langsung ku Présidén Mesir harita.

## Rujukan
Rasio ganjil

**Rasio ganjil** nyaéta ukuran statistik, bagéan penting dina statistika Bayes sarta régrési logistik.

Rasio ganjil dihartikeun salaku rasio hiji kajadian ahéng dina hiji grup kana kaganjilan nu kajadian dina grup séjén, atawa estimasi data dumasar kana rasio.  Grup ieu bisa awewe jeung lalaki, grup percobaan sarta grup kontrol, atawa klasifikasi nu ngabingungkeun.  Maka lamun kamungkinan kajadian unggal grup *p* jeung *q* mangka rasio-ganjil nyaéta

:{ p/(1-p) \over q/(1-q)}=\frac{\;p(1-q)\;}{\;(1-p)q\;}.

Logaritma rasio ganjil mangrupa béda logit tina probabilitas.

Régrési logistik loba dipaké dina widang kadokteran jeung panalungtikan élmu sosial, hartina *rasio-ganjil* ilahar dipaké geus ngagambarkeun sababaraha bentuk uji klinis, saperti uji kasus-kontrol, sarta dina panalungtikan survéy.
Order statistik

Dina statistik, **order statistik** ka-*n* tina sampel saruna jeung nilai sampel ka-*n* pangleutikna.

Contona, lamun nilai sampe nyaéta:

:6, 9, 3, 8

mangka **order statistik kadua** nyaéta:

:*x*(2) = 6.

**Order statistik kahiji** (atawa *order statistik pangleutikna*) salawasna *minimum* tina sampel.  Keur ukuran sampel *n*, '*order statistik ka-*n'''* (atawa *order statistik panggedéna*) nyaéta *maximum*.

Dina notasi konvensional, ditulis:

:*x*1 = 6
:*x*2 = 9
:*x*3 = 3
:*x*4 = 8

**teu maké kurung()** dina *subscripts*, dumasar kana urutan data nu dikumpulkeun, sarta:

:*x*(1) = 3
:*x*(2) = 6
:*x*(3) = 8
:*x*(4) = 9

**make kurung ()** dina *subscripts*, dumasar kana order statistik, dina hal ieu, data disusun dumasar kana naekna order.

*Beda jeung quantiles....''

## Tempo oge
* Rankit
* Box plot
* Fisher-Tippett distribution
Overfitting

Dina statistik, **overfitting** nyaéta modél statistik keur nyocogkeun niley nu mibanda sababaraha paraméter.  modél *absurd* sarta falsu bakal cocog sacara sampurna lamun ieu modél mibanda karudetan nu cukup dibandingkeun jeung jumlah data nu aya. **Overfitting** ilaharna diwangun ku susunan Occam's razor.

Widang anyar nu keur ilahar dipaké dina konsép **overfitting** nyaéta *machine learning*. Ilaharna algorithm diajar nyaéta ngajarkeun ku ngagunakeun sababaraha susunan latihan, upamana ngagunakeun sababaraha kaayaan nu hasil ahirna geus dipikawanoh. Nu diajar nganggap yén ngaramalkeun yén hasil nu bakal kahontal bakal nembongkeun hasil nu bener lamun dipaké kana conto séjénna, ku sabab kitu kaayaan umum ieu teu katembong salila diajarkeun (dumasar kana *inductive bias*). Sanajan kitu, umumna dina kasus nu aya salila diajar ngabutuhkeun waktu nu lila atawa conto latihan nu kawilang langka, nu diajar bakal ngadeukeutkeun manéh kana sababaraha hal husus variabel acak salila diajar, nu taya hubungan baraya jeung fungsi target nu bakal kahontal. Dina prosés **overfitting** ieu, hasil nu ditembongkeun dina nu diajar masih alus sedengkeun hasil dina data nu teu katempo bakal leuwih goreng.

Boh dina statistik jeung machine learning, dina usaha keur meunang hasil nu cocog, perlu ditambahkeun téhnik séjénna(upamana validasi silang, *early stopping*), nu bisa nembongkeun yén léngkah diajar saterusna lain dihasilkeun tina ka-ilaharan nu panghadéna. Dina *treatment learning*, niley minimum nu ngadukung pangalusna nu dipaké keur nangtukeun niley nu cocog dina **overfitting**.
Percentile rank

**Rangking persentil** tina skor nyaéta perséntase skor panghandapna tina sebaran frekuensina. Conto, skor uji leuwih ti 90% tina skor nu dicokot tina uju maka bakal disebutkeun dina persentil ka-90.

Rangking persentil ilaharna dipaké keur ngajelaskeun interpretasi skor dina uji standar. Lamun sebaran ieu kasebar normal, rangking persentil bisa dicokot tina skor standar.
Statistik parametrik

**métodeu  kaputusan statistik parametrik** mangrupa prosedur matematik keur tes hipotesa statistik nu nganggap yén sebaran variabel bakal bisa ditaksir ku karakter nu ditempo. Analisa varian nganggap yén dina kaayaan sebaran normal sarta varian tina sebaran nu dibandingkeun dianggap sarua. Koefisien korelasi momen-produk Pearson dianggap normal.

Kadangkala tenknik parametrik mangrupa hal nu kuat – hal ieu, ku nahankeun tetempoan kana power keur ngalacak béda atawa sarua waktu ieu asumsi kalanggar – sababaraha sebaran ngalanggar asumsi mangka ditandaan salaku statistik non-parametrik, alternatip nu leuwih ilahar dipaké keur ngalacak béda atawa sarua.
Interpolasi Pareto

**Interpolasi Pareto** mangrupa métodeu  interpolasi nonlinier keur manggihkeun median tina susunan data.  Ilahar dipaké dina widang ekonomi keur ngagambarkeun analisa panghasilan.  Anggapan nu dipaké yén data nuturkeun kurva nu disebut sebaran Pareto.

Median dirumuskeun ku

:{\rm median}=\kappa\,2^{1/\theta},

nu mana paraméter &kappa; jeung &theta; dirumuskeun ku:

:
K =
\left( 
\frac{P_b - P_a}
{ \frac{1}{a^{\theta}} - \frac{1}{b^{\theta}}} 
\right) ^{ \frac{1} {\theta}}

sarta

:
\theta \; = \;
\frac{\log(1-P_a) - \log(1-P_b)}
{\log(b) - \log(a)}

nu mana

:*a* = wates handap kategori nu ngandung median

:*b* = wates luhur kategori nu ngandung median

:*P**a* = proporsi sebaran nu katutup di handapeun wates handap

:*P**b* = proporsi sebaran nu katutup di handapeun wates handap
Pearson product-moment correlation coefficient

Dina matematik, sarta dina sabagéan statistik, the **Pearson product-moment correlation coefficient** (*r*) is a méasure of how well a linear equation describes the relation between two variables *X* and *Y* méasured on the same object or organism. It is defined as the sum of the products of the skor standard of the two méasures divided by the degrees of freedom:

: r = \frac {\sum z_x z_y}{N - 1}

The result obtained is equivalent to dividing the covariance between the two variables by the product of their standard deviations. In general the quantity of a correlation coefficient is the square root of the coefficient of determination (*r2*), which is the ratio of explained variation to total variation:

: r^2 = {\sum (Y' - \overline Y)^2 \over \sum (Y - \overline Y)^2}

where:

:***Y*** = a score on a random variable *Y*
:***Y***' = corresponding predicted value of *Y*, given the correlation of *X* and *Y* and the value of *X*
:\overline Y = mean of *Y*

The correlation coefficient adds a sign to show the direction of the relationship. The formula for the Péarson coefficient conforms to this definition, and applies when the relationship is linéar.

The coefficient ranges from -1 to 1. A value of 1 shows that a linéar equation describes the relationship perfectly and positively, with all data points lying on the same line and with *Y* incréasing with *X*. A score of -1 shows that all data points lie on a single line but that *Y* incréases as *X* decréases. A value of 0 shows that a linéar modél is inappropriate &#8211; that there is no linéar relationship between the variables.

The Péarson coefficient is a statistic which estimates the correlation of the two given random variables.

The linéar equation that best describes the relationship between *X* and *Y* can be found by linear regression. If *X* and *Y* are both normally distributed, this can be used to "predict" the value of one méasurement from knowledge of the other. That is, for éach value of *X* the equation calculates a value which is the best estimate of the values of *Y* corresponding the specific value of *X*. We denote this predicted variable by *Y*.

Any value of *Y* can therefore be defined as the sum of *Y* and the difference between *Y* and *Y*:

:Y = Y^\prime + (Y - Y^\prime)

The varian of *Y* is equal to the sum of the variance of the two components of *Y*:

:s_y^2 = S_{y^\prime}^2 + s^2_{y.x}

Since the coefficient of determination implies that *sy.x2* = *sy2*(1 &minus; *r2*) we can derive the identity

:r^2 = {s_{y^\prime}^2 \over s_y^2}

The square of *r* is conventionally used as a méasure of the strength of the association between *X* and *Y*. For example, if the coefficient is .90, then 81% of the variance of *Y* is said to be explained by the changes in *X* and the linéar relation between *X* and *Y*.

*r* is a statistik parametrik. It assumes that the variables being assessed are normally distributed. If this assumption is violated, a non-parametric alternative such as Spearman's &rho; *may* be more successful in detecting a linéar relationship.