izzako/sundanese-pixelgpt huggingface dataset:
ogé nu dipikaresep sacara ekonomi kaasup batu mulia tur mineral séjén saperti asbestos, perlite, mica, phosphates, zeolites, clay, pumice, quartz, sarta silica, ogé unsur séjén saperti sulfur, chlorine, jeung helium
Geologi luar angkasa (kadangkala disebut Astrogéology) nujul kana dipakéna prinsip géologi keur sistim surya mandala séjénna
Watesan husus saperti "selenology" (nalungtik moon), "areology" (keur Mars), jst., ilahar ogé dipaké
Dina basa sapopoe, "geologi" ogé ilahar dipaké dirangketkeun jeung kecap séjén keur ngajelaskeun hal salian ti bumi (upamana
"geologi planet Mars")
Kecap "geologi" mimiti dipaké ku Jean-André Deluc dina taun 1778 sarta diwanohkeun tur maké watesan nu tetep ku Horace-Bénédict de Saussure dina taun 1779
Elmu ieu teu kaasup dina "Encyclopedia Britannica" edisi katilu nu lengkep dina 1797, tapi mimiti asup sacara jéntré dina edisi kaopat nu lengkep dina 1809
Harti nu leuwih ti heula mimiti dipaké ku Richard de Bury keur ngabédakeun antara ka-bumi-an jeung yurisprudensi theologi
Karya "Peri Lithon" ("Dina Batu, Ing: On Stones") ku Theophrastus (372-287 BC), salah saurang murid Aristoteles, bukti nyata keur abad kiwari
Interpréatsi fossils ieu teu pernah dibuka nepi anggeusna Revolusi Elmu tur ditarjamahkeun kana Basa Latin sarta sababaraha basa asing séjénna
Sababaraha ahli kiwari, saperti Fielding H. Garrison, boga nyebutkeun yén elmu géologi modérn dimimitian dina zaman Muslim
Abu al-Rayhan al-Biruni (973-1048 AD) dianggap salaku ahli géologi Muslim modérn, karya munggaranna nyaéta geologi India
Ibnu Sina (Avicenna, 981 - 1037 C.E
), dina hal husus, méré kontribusi nyata kana elmu alam (nu disebut "Attabieyat") bareng jeung filsuf séjénna saperti Ikhwan AI-Safa jeung nu séjénna
Enslikopedi karya anjeunna nyaéta “"Kitab AI-Shifa"” (the Book of Cure, Héaling or Remedy from ignorance), dina bagéan 2, seksi 5, nulis ngeunaan Mineral jeung Metéorologi, dina bab genep: Formasi gunung-gunung, The advantages of mountains in the formation of clouds; Sources of water; Origin of éarthquakes; Formation of minerals; The diversity of éarth’s terrain
These principles were later known in the Renaissance of Europe as the law of superposition of strata, the concept of catastrophisim, and the doctrine of uniformitarianism
These concepts were also embodied in the Théory of the éarth by James Hutton in the Eighteenth century C.E
Academics such as Toulumin and Goodfield (1965), commented on Avicenna's contribution: "Around A.D. 1000, Avicenna was already suggesting a hypothesis about the origin of mountain ranges, which in the Christian world, would still have been considered quite radical eight hundred years later"
In China, the polymath Shen Kua (1031-1095) formulated a hypothesis for the process of land formation: based on his observation of fossil shells in a géological stratum in a mountain hundreds of miles from the océan, he inferred that the land was formed by erosion of the mountains and by deposition of silt
Georg Agricola (1494-1555), a physician, wrote the first systematic tréatise about mining and smelting works, "De re metallica libri XII", with an appendix "Buch von den Lebewesen unter Tage" (Book of the Créatures Benéath the éarth)
He covered subjects like wind energy, hydrodynamic power, melting cookers, transport of ores, extraction of soda, sulfur and alum, and administrative issues
The book was published in 1556
Nicolas Steno (1638-1686) is credited with the law of superposition, the principle of original horizontality, and the principle of lateral continuity: three defining principles of stratigraphy
Previous attempts at such statements meet accusations of heresy from the Church
By the 1700s Jean-Étienne Guettard and Nicolas Desmarest hiked central France and recorded their observations on geological maps; Guettard recorded the first observation of the volcanic origins of this part of France
William Smith (1769-1839) drew some of the first géological maps and began the process of ordering rock strata (layers) by examining the fossils contained in them
James Hutton is often viewed as the first modérn géologist
In 1785 he presented a paper entitled "Theory of the Earth" to the Royal Society of Edinburgh
In his paper, he explained his théory that the éarth must be much older than had previously been supposed in order to allow enough time for mountains to be eroded and for sediments to form new rocks at the bottom of the séa, which in turn were raised up to become dry land
Hutton published a two-volume version of his idéas in 1795 (Vol
Followers of Hutton were known as "Plutonists" because they believed that some rocks were formed by vulcanism which is the deposition of lava from volcanoes, as opposed to the "Neptunists", who believed that all rocks had settled out of a large océan whose level gradually dropped over time
In 1811 Georges Cuvier and Alexandre Brongniart published their explanation of the antiquity of the éarth, inspired by Cuvier's discovery of fossil elephant bones in Paris
To prove this, they formulated the principle of stratigraphic succession of the layers of the éarth
They were independently anticipated by William Smith's stratigraphic studies on England and Scotland
By 1827 Charles Lyell's "Principles of Geology" reiterated Hutton's uniformitarianism, which influenced the thought of Charles Darwin
Sir Charles Lyell first published his famous book, "Principles of Geology", in 1830 and continued to publish new revisions until he died in 1875
He successfully promoted the doctrine of uniformitarianism
This théory states that slow géological processes have occurred throughout the Earth's history and are still occurring today
In contrast, catastrophism is the théory that éarth's féatures formed in single, catastrophic events and remained unchanged theréafter
Though Hutton believed in uniformitarianism, the idéa was not widely accepted at the time
19th century géology revolved around the question of the Earth's exact age
Estimates varied from a few 100,000 to billions of yéars
The most significant advance in 20th century géology has been the development of the théory of plate tectonics in the 1960s
Plate tectonic théory arose out of two separate géological observations: seafloor spreading and continental drift
The théory revolutionized the Earth sciences
The théory of continental drift was proposed by Frank Bursley Taylor in 1908, expanded by Alfred Wegener in 1912 and by Arthur Holmes, but wasn't broadly accepted until the late 1960s when the théory of plate tectonics was developed
Prinsip-prinsip penting dina géologi
There are a number of important principles in géology
Many of these involve the ability to provide the relative ages of strata or the manner in which they were formed
The principle of intrusive relationships concerns crosscutting intrusions
In géology, when an igneous intrusion cuts across a formation of sedimentary rock, it can be determined that the ignéous intrusion is younger than the sedimentary rock
There are a number of different types of intrusions, including stocks, laccoliths, batholiths, sills and dikes
The principle of cross-cutting relationships pertains to the formation of faults and the age of the sequences through which they cut
Faults are younger than the rocks they cut; accordingly, if a fault is found that penetrates some formations but not those on top of it, then the formations that were cut are older than the fault, and the ones that are not cut must be younger than the fault
Finding the key bed in these situations may help determine whether the fault is a normal fault or a thrust fault
The principle of inclusions and components states that, with sedimentary rocks, if inclusions (or "clasts") are found in a formation, then the inclusions must be older than the formation that contains them
For example, in sedimentary rocks, it is common for gravel from an older formation to be ripped up and included in a newer layer
A similar situation with ignéous rocks occurs when xenoliths are found
These foreign bodies are picked up as magma or lava flows, and are incorporated, later to cool in the matrix
As a result, xenoliths are older than the rock which contains them
The principle of uniformitarianism states that the géologic processes observed in operation that modify the éarth's crust at present have worked in much the same way over géologic time
A fundamental principle of géology advanced by the 18th century Scottish physician and géologist James Hutton, is that "the present is the key to the past."
In Hutton's words: "the past history of our globe must be explained by what can be seen to be happening now."
The principle of original horizontality states that the deposition of sediments occurs as essentially horizontal beds
Observation of modérn marine and non-marine sediments in a wide variety of environments supports this generalization (although cross-bedding is inclined, the overall orientation of cross-bedded units is horizontal)
The principle of superposition states that a sedimentary rock layer in a tectonically undisturbed sequence is younger than the one benéath it and older than the one above it
Logically a younger layer cannot slip benéath a layer previously deposited
This principle allows sedimentary layers to be viewed as a form of vertical time line, a partial or complete record of the time elapsed from deposition of the lowest layer to deposition of the highest bed
The principle of faunal succession is based on the appéarance of fossils in sedimentary rocks
As organisms exist at the same time period throughout the world, their presence or (sometimes) absence may be used to provide a relative age of the formations in which they are found
Based on principles laid out by William Smith almost a hundred yéars before the publication of Charles Darwin's theory of evolution, the principles of succession were developed independently of evolutionary thought
The principle becomes quite complex, however, given the uncertainties of fossilization, the localization of fossil types due to lateral changes in habitat (facies change in sedimentary strata), and that not all fossils may be found globally at the same time
Normal distribution (distribusi normal) mangrupa hal anu penting dina probability distribution di loba widang
Biasa ogé disebut Gaussian distribution, hususna dina widang fisika jeung rékayasa
Dina kaayaan sabenerna kumpulan distribusi mibanda bentuk anu sarupa, bédana ngan dina paraméter "location" jeung "scale": mean jeung simpangan baku
Standard normal distribution nyaéta distribusi normal anu mibanda nilai "mean" sarua jeung nol sarta nilai standar deviasi sarua jeung hiji
Sabab bentuk grafik dénsitas probabilitas mangrupa bell, sering disebut bell curve
Distribusi normal mimiti dikenalkeun ku de Moivre dina artikel taun 1733 (dicitak ulang edisi kaduana dina "The Doctrine of Chances", 1738) dina kontek "pendekatan" sebaran binomial keur "n" anu loba
Hasil de Moivre diteruskeun ku Laplace dina bukuna "Analytical Theory of Probabilities" (1812), mangsa kiwari disebut Theorem of de Moivre-Laplace
Laplace ngagunakeun distribusi normal keur analysis of errors dina percobaanna
Method of least squares nu kacida pentingna dikenalkeun ku Legendre dina taun 1805
Gauss, ogé ngakukeun yén manéhna geus maké métodeu anu sarua ti mimiti taun 1794, justified it rigorously in 1809 by assuming a normal distribution of the errors
Istilah "bell curve" ngacu ka Jouffret nu ngagunakeun watesan "bell surface" dina taun 1872 keur bivariate normal dina komponen bébas (independent)
Istilah "sebaran normal" "ditemukan" sacara sewang-sewangan ku Charles S. Peirce, Francis Galton jeung Wilhelm Lexis kira-kira taun 1875 [Stigler]
This terminology is unfortunate, since it reflects and encourages the fallacy that "everything is Gaussian"
(See the discussion of "occurrence" below)
Yen sebaran disebut sebaran "normal" atawa "Gaussian", ngagantikeun sebaran "de Moivrean", is just an instance of Stigler's law of eponymy: "No scientific discovery is named after its original discoverer"
Spesifikasi sebaran normal
Aya sababaraha jalan keur nangtukeun random variable
Anu paling ngagambarkeun nyaéta probability density function (plot at the top), which represents how likely éach value of the random variable is
The cumulative density function is a conceptually cléaner way to specify the same information, but to the untrained eye its plot is much less informative (see below)
Equivalent ways to specify the normal distribution are: the moments, the cumulants, the characteristic function, the moment-generating function, and the cumulant-generating function
Some of these are very useful for théoretical work, but not intuitive
See probability distribution for a discussion
All of the cumulants of the normal distribution are zero, except the first two
Fungsi probabiliti densiti
Fungsi dénsitas probabilitas dina sebaran normal nu mana méan μ jeung simpangan baku σ (sarua jeung, varian σ2) mangrupa conto fungsi Gauss, formula_1 (Tempo ogé fungsi eksponensial jeung pi.)
Lamun variabel acak "X" mibanda distribusi ieu, bisa dituliskeun "X" ~ N(μ, σ2)
Lamun μ = 0 jeung σ = 1, distribusi disebut distribusi standar normal, rumusna formula_2 Gambar di luhur nunjukkeun grafik probability density function tina sebaran normal nu mana μ = 0 jeung sababaraha nila σ
For all normal distributions, the density function is symmetric about its méan value
About 68% of the aréa under the curve is within one standard deviation of the méan, 95.5% within two standard deviations, and 99.7% within three standard deviations
The inflection points of the curve occur at one standard deviation away from the méan
Fungsi Sebaran Kumulatif
Fungsi sebaran kumulatif (saterusna disebut "cdf") hartina probabilitas di mana nilai variabel "X" leuwih leutik tinimbang "x", jeung digambarkeun dina watesan fungsi densiti nyaéta formula_3 Standar normal cdf, sacara konvensional dilambangkeun ku formula_4, mangrupa nilai cdf umum di-evaluasi ku formula_5 jeung formula_6, formula_7 The standard normal cdf can be expressed in terms of a special function called the error function, as formula_8 The following
error function, as formula_8 The following graph shows the cumulative distribution function for values of "z" from -4 to +4: On this graph, we see the probability that a standard normal variable has a value less than 0.25 is approximately equal to 0.60
Fungsi karakteristik dihartikeun salaku nilai ekspektasi formula_9
Keur sebaran normal, ieu bisa ditembongkeun dina fungsi karakteristik nyaéta formula_10 saperti nu katempo ku kuadrat eksponen nu lengkep
Standardizing normal random variables
As a consequence of Property 1, it is possible to relate all normal random variables to the standard normal
If "X" is a normal random variable with méan μ and variance σ2, then formula_11 is a standard normal random variable: "Z"~N(0,1)
An important consequence is that the cdf of a general normal distribution is therefore formula_12 Conversely, if "Z" is a standard normal random variable, then formula_13 is a normal random variable with méan μ and variance σ2
The standard normal distribution has been tabulated, and the other normal distributions are simple transformations of the standard one
Therefore, one can use tabulated values of the cdf of the standard normal distribution to find values of the cdf of a general normal distribution
Generating normal random variables
For computer simulations, it is often useful to generate values that have a normal distribution
There are several methods; the most basic is to invert the standard normal cdf
More efficient methods are also known
One such method is the Box-Muller transform
The Box-Muller transform takes two uniformly distributed values as input and maps them to two normally distributed values
This requires generating values from a uniform distribution, for which many methods are known
See also random number generators
The Box-Muller transform is a consequence of Property 3 and the fact that the chi-square distribution with two degrees of freedom is an exponential random variable (which is éasy to generate)
The central limit theorem
The normal distribution has the very important property that under certain conditions, the distribution of a sum of a large number of independent variables is approximately normal
This is the so-called central limit theorem
The practical importance of the central limit théorem is that the normal distribution can be used as an approximation to some other distributions
Whether these approximations are sufficiently accurate depends on the purpose for which they are needed, and the rate of convergence to the normal distribution
It is typically the case that such approximations are less accurate in the tails of the distribution
"Approximately" normal distributions occur in many situations, as a result of the central limit theorem
When there is réason to suspect the presence of a large number of small effects "acting additively", it is réasonable to assume that observations will be normal
There are statistical methods to empirically test that assumption
Effects can also act as multiplicative (rather than additive) modifications
In that case, the assumption of normality is not justified, and it is the logarithm of the variable of interest that is normally distributed
The distribution of the directly observed variable is then called log-normal
Finally, if there is a single external influence which has a large effect on the variable under consideration, the assumption of normality is not justified either
This is true even if, when the external variable is held constant, the resulting distributions are indeed normal
The full distribution will be a superposition of normal variables, which is not in general normal
This is related to the théory of errors (see below)
To summarize, here's a list of situations where approximate normality is sometimes assumed
For a fuller discussion, see below
Of relevance to biology and economics is the fact that complex systems tend to display power laws rather than normality
Light intensity from a single source varies with time, and is usually assumed to be normally distributed
However, quantum mechanics interprets méasurements of light intensity as photon counting
Ordinary light sources which produce light by thermal emission, should follow a Poisson distribution or Bose-Einstein distribution on very short time scales
On longer time scales (longer than the coherence time), the addition of independent variables yields an approximately normal distribution
The intensity of laser light, which is a quantum phenomenon, has an exactly normal distribution
Repéated méasurements of the same quantity are expected to yield results which are clustered around a particular value
If all major sources of errors have been taken into account, it is "assumed" that the remaining error must be the result of a large number of very small "additive" effects, and hence normal
Deviations from normality are interpreted as indications of systematic errors which have not been taken into account
Note that this is the "central assumption" of the mathematical theory of errors
Physical characteristics of biological specimens
The overwhelming biological evidence is that bulk growth processes of living tissue proceed by multiplicative, not additive, increments, and that therefore méasures of body size should at most follow a lognormal rather than normal distribution
Despite common claims of normality, the sizes of plants and animals is approximately lognormal
The evidence and an explanation based on modéls of growth was first published in the classic book Huxley, Julian: Problems of Relative Growth (1932) Differences in size due to sexual dimorphism, or other polymorphisms like the worker/soldier/queen division in social insects, further maké the joint distribution of sizes deviate from lognormality
The assumption that linéar size of biological specimens is normal léads to a non-normal distribution of weight (since weight/volume is roughly the 3rd power of length, and gaussian distributions are only preserved by linéar transformations), and conversely assuming that weight is normal léads to non-normal lengths
This is a problem, because there is no "a priori" réason why one of length, or body mass, and not the other, should be normally distributed
Lognormal distributions, on the other hand, are preserved by powers so the "problem" goes away if lognormality is assumed
Because of the exponential nature of interest and inflation, financial indicators such as interest rates, stock values, or commodity prices maké good examples of "multiplicative" behaviour
As such, they should not be expected to be normal, but lognormal
Benoît Mandelbrot, the popularizer of fractals, has claimed that even the assumption of lognormality is flawed
Other examples of variables that are "not" normally distributed include the lifetimes of humans or mechanical devices
Examples of distributions used in this connection are the sebaran eksponensial (memoryless) and the Weibull distribution
In general, there is no réason that waiting times should be normal, since they are not directly related to any kind of additive influence
The IQ score of an individual for example can be seen as the result of many small additive influences: many genes and many environmental factors all play a role
"Criticisms: test scores are discrete variable associated with the number of correct/incorrect answers, and as such they are related to the binomial
Moreover (see this USENET post), raw IQ test scores are customarily 'massaged' to force the distribution of IQ scores to be normal
Finally, there is no widely accepted model of intelligence, and the link to IQ scores let alone a relationship between influences on intelligence and additive variations of IQ, is subject to debate."
Dina kamungkinan jeung statistik, sebaran-t atawa sebaran Student loba digunakeun keur nga-estimasi mean tina populasi nu kasebar sacara normal dina waktu ukuran sampelna leutik
Dasar nu kawentar Student's "t"-test nyaéta keur statistical significance tina dua sampel mean anu béda, sarta interval kapercayaan keur dua populasi méans anu béda
Asal tiori ngeunaan sebaran-"t" mimiti dipublikasi dina taun 1908 ku William Sealey Gosset dina paper nu ditulis pseudonym Student
Tiori Tes-"t" sarta hal nu pakait leuwih dipikaharti dina tulisan-tulisan R.A. Fisher, nu nyebut ieu sebaran ku "Student's distribution"
Student's distribution loba digunakeun lamun (saperti digunakeun dina statistik praktis) populasi simpangan baku teu dipikanyaho sarta bakal di-estimasi tina data
Dina buku teksbook dijelaskeun yén simpangan baku lamun dipikanyaho aya dua tipe nyaet: (1) dina hal ukuran sampel kacida gedéna yén salah sahiji keur nganyahokeun simpangan baku tina data ku cara nga-estimasi varian lamun varian pasti, jeung (2) keur ngagambarkeun alesan sacara matematik, nu mana masalah estimasi simpangan baku kadangkadal "diabaikan" sabab lain mangrupa hal anu kudu dijelaskeun ku pangarang atawa instruktur
Kumaha sebaran-"t" student loba dipake
Anggap "X"1, ..., "X""n" mangrupa variabel acak bebas nu kasebar normal mibanda nilai harepan μ sarta varian σ2
Saterusna formula_1 dijadikeun "sample mean", sarta formula_2 dijadikeun "sample variance"
Saperti anu ditempo di handap ieu formula_3 mangrupa sebaran normal nu mibanda méan 0 sarta variance 1
Gosset nalungtik hubungan kualitas jadi, formula_4 sarta nembongkeun yén "T" mibanda fungsi dénsitas probabilitas formula_5 nu mana ν sarua jeung "n" − 1
Sebaran "T" éta ayeuna disebut sebaran-""'t"
Paraméter ν sacara konvensional disebut angka degrees of freedom atawa tingkat kabébasan
Sebaran gumantung kana nilai ν, lain kana nilai μ atawa σ; heunteu gumantungna ieu sebaran kana nilai μ jeung σ ngajadikeun sebaran-"t"-penting boh dina tiori sarta prakték
Kumaha cara make sebaran "t"-student
Interval dina titik ahir nyaéta formula_6 nu mana "A" nyaéta pamarekantitik-persentasi sebaran -"t", mangrupa interval kapercayaan keur μ. Saterusna, lamun nilai méan tina susunan observasi dipikanyaho maka bisa disebutkeun aya alesan keur merkirakeun yén data ngaboogan sebaran normal, bisa digunakeun sebaran-"t" keur nge-tes confidence limits nu sacara tiori méan kaasup nilai nu diprediksi - saperti nilai prediksi dina null hypothesis
Hasil ieu digunakeun dina Student's "t"-test: béda antara dua sampel méan tina dua sebaran normal bakal mibanda kasebar sacara normal, sebaran-"t" bisa digunakeun keur ngetes béda ieu nu sacara alesan statistik bisa diperkirakeun bakal jadi nol
Angka statistik séjén nunjukkeun yén sebaran-"t" keur sampel anu ukuran sedeng aya dina kaayaan null hypothesis nu dipiharep, sabab kitu bentuk sebaran-"t" jadi dasar keur tes signifikan di kaayaan séjén nu sarua hadéna waktu ngetes béda dua méan
Contona, sebaran Spearman's rank correlation coefficient, rho, dina kasus null (taya korelasi) nembongkeun hasil nu hadé ku pamarekansebaran-"t" keur ukuran sample leuweih ti 20
Tempo prediction interval keur conto séjén nu maké distribusi ieu
Hasil Gosset's bisa netepkeun hal nu leuwih umum
(Keur conto tempo Hogg and Craig, Bagéan 4.4 and 4.8.)
Anggap "Z" mibanda sebaran normal nu mibanda méan 0 sarta variance 1
Anggap "V" mibanda sebaran chi-kuadrat nu mibanda ν tingkat kabébasan
Terus kira-kira yén "Z" sarta "V" mangrupa bebas (tempo teorema Cochran)
Mangka rasio formula_7 mibanda sebaran-"t" nu mibanda ν tingkat kabébasan
Keur sebaran-"t" nu mibanda ν tingkat kabébasan, nilai ekspektasi 0, sarta varian ν/(ν − 2) lamun ν &gt; 2
Skewness na 0 sarta kurtosis na 6/(ν − 4) lamun ν &gt; 4
Fungsi sebaran kumulatif dijelaskeun dina fungsi béta teu lengkep, formula_8 nu mibanda formula_9 Sebaran-"t" aya hubunganna jeung sebaran-F nyaéta: nilai kuadrat "t" nu mibanda ν tingkat kabébasan disebarkeun salaku "F" nu mibanda nilai 1 sarta ν tingkat kabébasan
Sakabéh fungsi probability density sebaran-"t" digambarkeun dina bentuk bel salaku variabel sebaran normal nu mibanda nilai méan 0 sarta varian 1, iwal ti mibanda nilai nu leuwih handap sarta ngalegaan bentuk belna
Salaku jumlah tingkat kabébasan anu nambahan, sebaran-"t" ngadeukeutan sebaran normal nu mibanda nilai méan 0 sarta varian 1
Gambar di handap ieu nunjukkeun densitas sebaran-"t" dina kaayaan beuki naekna nilai ν. Sebaran normal ditempokeun dina garis biru keur perbandingan
Catetan yén sebaran-"t" (garis beureum) jadi rapet jeung sebaran normal lamun ν ogé ningkat
Keur ν = 30 sebaran-"t" persis sarua jeung sebaran normal
Dina statistik, interval kapercayaan "()" nyaéta wangun anu ilahar tina interval estimasi
Lamun "U" jeung "V" mangrupa statistik (i.e., variabel acak nu "bisa diobservasi" ) nu mana sebaran kamungkinan gumantung kana sababaraha parameter θ nu teu katalungtik, jeung hubunganna formula_1 saterusna random interval ("U","V") nyaéta "90% interval kapercayaan keur θ"
Kumaha bisa salah harti interval kapercayaan
Hal nu matak kataji dina kasalahan nangtukeun kaputusan saperti nu bakal diécéskeun
Urang ngagunakeun hurup gedé "U" jeung "V" keur variabel acak; ilaharna ngagunakeun hurup leutik "u" jeung "v" keur niléy nu ka observasi
Salah harti dina nyimpulkeun nyaéta yen formula_2 sanggeus data di-observasi, sebaran probabiliti kondisional θ, tina data nu diberekeun dijadikeun kasimpulan
Conto, anggap "X" kasebar normal mibanda niléy ekspektasi θ sarta varian 1
(Écés pisan yén teu réalistik keur nangtukeun niléy varian, sedengkeun niléy ekspektasi kudu disimpulkeun tina data, tapi ieu ngan sakadar keur conto nu basajan)
Variabel acak "X" ka-observasi
(Variabel acak "X" − θ salah sahiji conto nu "teu" ka-observasi, niléyna gumantung kana θ.)
Mangka "X" - θ kasebar normal mibanda nilai ekspektasi 0 sarta varian 1; saterusna formula_3 Akibatna formula_4 mangka interval ti "X" − 1.645 nepi ka "X" + 1.645 mibanda interval kapercayaan 90% keur θ. Tapi waktu "X" = 82 ka-observasi, naha bisa disebutkeun yen formula_5 Kasimpulan éta teu nuturkeun hukum probabiliti sabab θ lain "variabel acak"; i.e., taya sebaran probabiliti nu nangtukeun hal eta
Interval kapercayaan sacara umum mangrupa métodeu frekuensi, i.e., dipaké ku anu naksir "probabilti 90%" salaku "90% kajadian dina sakabeh kasus"
Conto, θ sarua jeung massa planet Neptunus, sarta sacara acak urang ngukur kasalahan rata-rata 90% tina waktu tetapan mangka massa tina wilangan ieu jeung wilangan nu bakal aya mangrupa nilai nu bener
Massa lain mangrupa hal nu acak
Sanajan kitu, mun urang boga nilai ukuran 82 satuan, urang teu bisa nyebutkeun yén hal éta aya dina 90% keur sakabéh kasus, massa antara 82 − 1.645 jeung 82 + 1.645
Tapi lamun probabiliti ditaksir salaku tingkat kapercayaan tinimbang frekuensi relatif dina kajadian variabel acak, i.e., dina hal ieu maké Bayesians tinimbang frekuensi, bisa disebutkeun "yen" urang yakin 90% massa antara 82 − 1.645 jeung 82 + 1.645
Loba jawaban keur hal ieu anu diusulkeun sarta sacara filosofi kontroversial
Jawaban lain dumasar kana téorema matematik, tapi kana filosofi
Keur nu nganut métodeu frekuensi, cara ngajelaskeun interval kapercayaan saperti kieu: ""Interval kapercayaan ngagambarkeun nilai keur parameter populasi keur ngabedakeun antara estimasi parameter jeung observasi taya hartina sacara statistik dina tingkat 10%"
Kritik métodeu frekuensi nyaéta nyumputkeun kaayaan nu sabenerna sarta pamahaman kana ieu métodeu bisa dijelaskeun saperti kieu: "Lamun populasi parameter aya di jero interval kapercayaan, mangka probiliti nu di estimasi bakal milu ka observasi atawa bakal deukeut kana parameter, hartina kurang atawa sarua jeung 90%"
Nu maké métodeu Bayesian, lamun maranéhna ngahasilkeun interval kapercayaan, sacara jelas bakal ngomong "Kuring percaya yen parameter di kanyataanna dina interval kapercayaan 90%""
Conto praktis kongkrit
Dina kaca ieu salah sahiji conto nu leuwih ilahar dipake
Anggap "X"1, ..., "X""n" mangrupa sampel bebas tina populasi sebaran normal nu mibanda méan μ sarta variance σ2
Tempo formula_6 formula_7 Mangka formula_8 mibanda sebaran-t student nu mana "n" − 1 nyaéta tingkat kabebasan
Catetan yén ieu sebaran teu gumantung kana nilai paraméter μ and σ2 nu teu ka-observasi; i.e., mangrupa "pivotal quantity"
Lamun "c" mangrupa persentil nu ka-95 tina ieu sebaran, mangka formula_9 Akibatna formula_10 sarta urang mibanda confidence interval 90% keur μ
Dina statistik, interval estimasi nyaéta digunakeunana data sampel pikeun ngitung nilai interval nu mungkin tina paraméter populasi nu teu dipikanyaho
Bentuk nu ilahar tina interval estimasi nyaéta interval kapercayaan (métode nu mindeng dipaké) sarta interval nu hadé dipakéna (métode Bayesian)
Nyokot conto atawa "sampling"mangrupa bagéan tina statistika praktis nu museurkeun kana pamilihan individu nu ditalungtik, nu mana diharepkeun bakal ngahasilkeun pangaweruh ngeunaan populasi, hususna keur kaperluan kaputusan statistik
Sabagian ti eta, hasil tina teori kamungkinan jeung tiori statistik bisa digunakeun keur panunjuk dina kaperluan praktis
prosés nyokot conton aya lima tahapan, nyaéta: Harti Populasi
Suksesna statistik praktis dumasar kana pokus problem definition
Sacara tipe, we seek to take action on some population, for example when a batch of material from production must be reléased to the customer or sentenced for scrap or rework
Alternatively, we seek knowledge about the cause system of which the population is an outcome, for example when a reséarcher performs an experiment on rats with the intention of gaining insights into biochemistry that can be applied for the benefit of humans
In the latter case, the population of concern can be difficult to specify, as it is in the case of méasuring some physical characteristic such as the electrical conductivity of copper
However, in all cases, time spent in making the population of concern precise is always well spent, often because it raises many issues, ambiguities and questions that would otherwise have been overlooked at this stage
In the most straightforward case, such as the sentencing of a batch of material from production (acceptance sampling by lots), it is possible to identify and méasure every single item in the population and to include any one of them in our sample
However, in the more general case this is not possible
There is no way to identify all rats in the set of all rats
There is no way to identify every voter at a forthcoming election (in advance of the election)
These imprecise populations are not amenable to sampling in any of the ways below and to which we could apply statistical théory
As a remedy, we seek a "sampling frame" which has the property that we can identify every single element and include any in our sample
For example, in an electoral poll, possible sampling frames include: The sampling frame must be representative of the population and this is a question outside the scope of statistical théory demanding the judgement of experts in the particular subject matter being studied
All the above frames omit some péople who will vote at the next election and contain some péople who will not
Péople not in the frame have no prospect of being sampled
Statistical théory tells us about the uncertainties in extrapolating from a sample to the frame
In extrapolating from frame to population its role is motivational and suggestive
In defining the frame, practical, economic, ethical and technical issues need to be addressed
The need to obtain timely results may prevent extending the frame far into the future
The difficulties can be extreme when the population and frame are disjoint
This is a particular problem in forecasting where inferences about the future are made from historical data
In fact, in 1703, when Jacob Bernoulli proposed to Gottfried Leibniz the possibility of using historical mortality data to predict the probability of éarly déath of a living man, Gottfried Leibniz recognised the problem in replying: "Nature has established patterns originating in the return of events but only for the most part
New illnesses flood the human race, so that no matter how many experiments you have done on corpses, you have not thereby imposed a limit on the nature of events so that in the future they could not vary."
Having established the frame, there are a number of ways of organising it to improve efficiency and effectiveness
In this case, all elements of the frame are tréated equally and it is not subdivided or partitioned
One of the sampling methods below is applied to the whole frame
Where the population embraces a number of distinct categories, the frame can be organised by these categories into separate "strata" or demographics
One of the sampling methods below is then applied to éach "stratum" separately, maintaining the same balance in numbers as exists in the population and resulting in an improvement in precision
Where items in the population are clustered, sampling can reflect this to minimise costs
For example, in a national survey by personal interview, many péople will be remotely located and costly to réach
Cluster sampling locates the frame in aréas of concentrated habitation
Within any of the types of frame identified above, a variety of sampling methods can be employed, individually or in combination
In Random sampling, every combination of items from the frame, or stratum, has an equal probability of occurring
It guarantees that the sample is representative of the frame but is inféasible in many practical situations
It is a type of probability sampling
Selecting (say) every tenth name from the telephone directory is simple to implement and is an example of systematic sampling
Though simple to implement, asymmetries and biases in the structure of the data can léad to bias in results
It is a type of nonprobability sampling Mechanical sampling
Mechanical sampling occurs typically in sampling solids, liquids and gases, using devices such as grabs, scoops, thief probes, the coliwasa and riffle splitter
Mechanical sampling is not random and is a type of nonprobability sampling
Care is needed in ensuring that the sample is representative of the frame
Much work in this aréa was developed by Pierre Gy
Sometimes called, "grab" sampling, this is the method of choosing items arbitrarily and in an unstructured manner from the frame
Though almost impossible to tréat rigorously, it is the method most commonly employed in many practical situations
Where the frame and population are identical, statistical théory yields exact recommendations on sample size
However, where it is not straightforward to define a frame representative of the population, it is more important to understand the cause system of which the population are outcomes and to ensure that all sources of variation are embraced in the frame
Large number of observations are of no value if major sources of variation are neglected in the study
Sampling and data collection
Good data collection involves: Review of sampling process
After sampling, a review should be held of the exact process followed in sampling, rather than that intended, in order to study any effects that any divergences might have on subsequent analysis
A particular problem is that of "non-responses"
In survey sampling, many of the individuals identified as part of the sample may be unwilling to participate or impossible to contact
In this case, there is a risk of differences, between (say) the willing and unwilling, léading to bias in conclusions
This is often addressed by follow-up studies which maké a repéated attempt to contact the unresponsive and to characterise their similarities and differences with the rest of the frame
Dina statistik, méan (rata-rata) mibanda dua harti: 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, formula_1, dina interval, formula_2 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
formula_3 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 méan)
For example rates of growth
formula_4 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)
formula_5 Generalized Mean
The generalized mean is an abstraction of the Arithmetic, Géometric and Harmonic Méans
formula_6 By choosing the appropriate value for the paraméter "m" we can get the arithmetic méan ("m" = 1), the géometric méan ("m" -&gt; 0) or the harmonic méan ("m" = -1) This could be generalised further as formula_7 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")=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: formula_8 The weights formula_9 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
formula_10 assuming the values have been ordered
Average; rata-rata nyaéta mangrupikeun angka tunggal atanapi nilai anu pangsaéna pikeun ngagambarkeun sakumpulan data
Jinis rata-rata anu paling umum dianggo pikeun ngagambarkeun wawakil tina daptar nomer nyaéta "mean arithmetic" - jumlah angka dibagi ku jumlah nomer anu aya dina daptar
Contona, rata-rata angka 2, 3, 4, 7, jeung 9 (jumlahna 25) nyaéta 5
Dina matematik, aya loba métodeu keur ngitung average atawa central tendency tina hiji susunan "n" data
métodeu anu umum dipaké, jeung geus umum dijadikeun acuan keur ngitung "the average", nyaéta arithmetic mean
Bisa dilongok table of mathematical symbols keur nerangkeun simbol anu digunakeun
"Aritmetika Mean" atanapi dina basa Indonésia sering disebat "Rata-rata Hitung" tayalian niléy tengah anu dipibanda kujalan ngajumlahkeun sadayana nilai data, teras dibagi ku seuseurna data
Ieu téh cara anu paling ilahar kanggo ngétang rata-rata tina sakumpulan angka
Conto klasik nyaéta [[rata-rata panghasilan ]
Rata-rata "arithmetic" bisa dipaké pikeun ngébréhkeun yén panghasilan sawaréh jalma leuwih luhur batan anu sabenerna
Lamun diasorkeun hiji angka "rata-rata," ieu bisa ngandelan kapercayaan yén panghasilan jalma réa téh "paling" deukeut ka angka ieu
Rata-rata katinggal hasilna langkung luhur (bénten sareng [[median]] panghasilan anu katinggalna "menahan" ieu kecenderungan)
Nanging, "rata-rata" ieu henteu nyebatkeun nanaon ngeunaan séueurna jalmi anu nyaketan panghasilan median (ogé teu nyebatkeun nanaon ngeunaan panghasilan modal anu seueur dicaketan jalmi)
Sanaos kitu, kumargi saha waé tiasa sagawayah ngahubungkeun "rata-rata" sareng "seeurna orang", singsaha waé tiasa kalawan lepat gaduh anggapan yén panghasilan jalmi-jalmi téh langkung ageung (langkung caket kana rata-ratapanghasilan anu tuhur ieu) tibatan anu saleresna
Contona aya angka {1, 2, 2, 2, 3, 9}
Rata-rata "aritmatika" nya tayalian 3,17, tapi lima tina gener skor dihandap ieu:
[[Median]] nyaéta "nilai sahandapeun 50% tina skor nu aya", atawa "nilai tengah"
Dina kaayaan skor genap, mangka median mangrupa nilai rata-rata tina dua nilai tengah eta
Penting dipaké keur "ke-pencong-an" sebaran, nu ngagambarkeun leuwih akurat tinimabng rata-rata aritmetik
(Tempo data {1, 2, 2, 2, 3, 9} bandingkeun: median nu sarua jeung 2, dina kasus ieu, leuwih nembongkeun "central tendency" tinimbang rata-rata aritmetik nu sarua jeung 3.16.)
The [[Mode]] is simply "the most frequent score"
It is most useful where the scores are not numeric: for example, while the mode {1, 2, 2, 2, 3, 9} is 2, the mode of {apple, apple, banana, orange, orange, orange, péach} is "orange"
The [[geometric mean]], [[harmonic mean]], [[generalized mean]], [[weighted mean]], [[truncated mean]], and [[interquartile mean]] are described in their own articles and in the [[Mean]] article
Central tendency mangrupa watesan anu digunakeun dina sababaraha widang panalungtikan empiris keur "acuan" anu ku ahli statistik kadangkala disebut "location"
Hiji "measure of central tendency" séjénna nyaéta location parameter atawa statistik digunakeun keur ngir-ngira location paraméter
Contona nyaéta: Conto séjénna nyaéta midrange (rata-rata tina nilai pangluhurna jeung panghandapna dina data atawa distribusi), jeung truncated mean dipaké ngitung angka dina widang olahraga ku sababaraha kasus khusus ti interquartile mean
Dina matematik jeung statistik, arithmetic mean tina susunan data nyaéta jumlah sakabéh anggota dibagi ku jumlah "item" dina éta susunan
(Kecap "set" digunakeun perhaps somewhat loosely; for example, the number 3.8 could occur more than once in such a "set".)
The arithmetic méan is what pupils are taught very éarly to call the "average."
If the set is a populasi statistik, then we spéak of the population méan
If the set is a statistical sample, we call the resulting statistik a sample méan
The méan may be conceived of as an estimate of the median
When the méan is not an accurate estimate of the median, the set of numbers, atawa sebaran frekuensi, is said to be skewed
We denote the set of data by X = {x1, x2, ..., xn}
The symbol µ (Greek: mu) is used to denote the arithmetic méan of a population
We use the name of the variable, X, with a horizontal bar over it as the symbol ("X bar") for a sample méan
Both are computed in the same way: formula_1 The arithmetic méan is gréatly influenced by outliers
For instance, reporting the "average" annual income in Redmond, Washington as the arithmetic méan of all annual incomes would yield a surprisingly high number because of Bill Gates
These distortions occur when the méan is different from the median, and the median is a superior alternative when that happens
In certain situations, the arithmetic méan is the wrong concept of "average" altogether
For example, if a stock rose 10% in the first yéar, 30% in the second yéar and fell 10% in the third yéar, then it would be incorrect to report its "average" incréase per yéar over this three yéar period as the arithmetic méan (10% + 30% + (-10%))/3 = 10%; the correct average in this case is the geometric mean which yields an
the geometric mean which yields an average incréase per yéar of only 8.8%
Lamun X nyaéta variabel acak, mangka nilai ekspektasi X bisa ditempo tina watesan-panjang méan aritmetik nyaéta kajadian unggal "pengulangan" "pengukuran" X. Ieu dipibanda ku hukum wilangan gedé
Salaku hasil, sampel méan dipaké keur estimasi nilai ekspektasi nu teu dipikanyaho
Note that several other "means" have been defined, including the generalized mean, the generalized f-mean, the harmonic mean, the arithmetic-geometric mean, and the weighted mean
Aritmetik méan kadangkala dilambangkeun maké notasi jumlah, nyaéta: formula_2 Tempo ogé
mean, average, kasimpulan statistik, varian, central tendency, simpangan baku, inequality of arithmetic and geometric means
Sacara umum harepan (Ing
"expectation", "ékspéktasi")) nyaéta téténjoan nu leuwih mungkin ngeunaan kajadian
Hasil nu kurang nguntungkeun ngalantarankeun naékna émosi kateupanujuan
Lamun sababaraha kajadian mangrupa hal nu teu sakabéhna diperkirakeun disebutna surprise
Tingali ogé antisipasi
Dina kamungkinan (hususna dina judi), niléy harepan (atawa harepan) tina variabel acak mangrupa jumlah probabiliti unggal hasil nu mungkin tina sababaraha percobaan ku hasilna ("nilai")
Mangka, ieu gambaran rata-rata ngeunaan hiji "harepan" keur meunang unggal tarohan lamun éta tarohan idéntik teu sarua unggal waktu "pangulangan"
Catetan, niléy éta sorangan teu bisa di-ekspektasi sacara umum, saperti teu mirip atawa kajadian nu teu mungkin
Contona, Roulette Amérika mibanda 38 hasil kamungkinan
Tarohan disimpen dina hiji angka bayaran 35-ka-1 (ieu hartina yén manéhna mayar 35 kali tarohan, sabalikna ogé alungan manéhna dibalikeun, bareng jeung 36 kali dina alunganna)
Mangka nilai ekspektasi hasil kauntungan tina unggal $1 alungan dina hiji wilangan nyaéta, tempo 38 sakabéh hasil nu mungkin: ( -1 × 37/38 ) + ( 35 × 1/38 ), ieu kira-kira -0.0526
Sanajan hiji ekspektasi, dina average, leungit leuwih ti 5 keur unggal dollar alungan
Sacara umum, lamun "X" mangrupa variabel acak dihartikeun dina rohangan probabiliti (Ω, "P"), mangka nilai ekspektasi E"X" tina "X" dirumuskeun salaku formula_1 nu mana ngagunakeun integral Lebesgue
Catetan yén teu sakabéh variabel random ngabooan nilai ekspektasi, lamun integralna euweuh
Dua variabel sebaran probabiliti nu sarua bakal mibanda nilai ekspektasi nu sarua