Pearson's chi-square test

'''Karl Pearson chi-square test''' (χ²) is one of a variety of Chi-square test � Statistics procedures whose results are evaluated by reference to the chi-square distribution. It tests a null hypothesis that the relative frequencies of occurrence of observed events follow a specified frequency distribution. The events must be mutually exclusive. One of the simplest examples is the hypothesis that an ordinary six-sided die is "fair", i.e., all six outcomes occur equally often. Chi-square is calculated by finding the difference between each observed and theoretical frequency, squaring them, dividing each by the theoretical frequency, and taking the sum of the results: :

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

where: :''O'' = an observed frequency :''E'' = an expected (theoretical) frequency, asserted by the null hypothesis For example, to test the hypothesis that a random sample of 100 people has been drawn from a population in which men and women are equal in frequency, the observed number of men and women would be compared to the theoretical frequencies of 50 men and 50 women. If there were 45 men in the sample and 55 women: :

\chi^2 = {(45 - 50)^2 \over 50} + {(55 - 50)^2 \over 50} = 1

There is one degrees of freedom in the comparison (since either difference between observed and expected frequencies, once known, dictates the other). Consultation of the chi-square distribution for 1 degree of freedom shows that the probability of observing this difference (or a more extreme difference than this) if men and women are equally numerous in the population is approximately 0.3. This probability is higher than conventional criteria for statistical significance, so normally we would accept the null hypothesis that the number of men in the population is the same as the number of women. Pearson's chi-square is used to assess two types of comparison: tests of goodness of fit and tests of independence. A test of goodness of fit establishes whether or not an observed frequency distribution differs from a theoretical distribution. A test of independence assesses whether paired observations on two variables, expressed in a contingency table, are independent of each other � for example, whether people from different regions differ in the frequency with which they report that they support a political candidate. Pearson's chi-square is the original and most widely-used chi-square test. The null distribution of the Pearson statistic is only approximated as a chi-square distribution. This approximation arises as the true distribution, under the null hypothesis, of the expected value is given by a Binomial distribution: :

E =^d \mbox{Bin}(n,p)

where: :''p'' = probability, under the null hypothesis :''n'' = number of observations in the sample In the above example the hypothesised probability of a male observation is 0.5, with 100 samples. Thus we expect to observe 50 males. When comparing the Pearson test statistic against a chi-squared distribution, the above binomial distribution is approximated as a Gaussian (normal) distribution: :

\mbox{Bin}(n,p) \approx^d \mbox{N}(np, np(1-p))

By definition, a sum of

k

standard normal variates,

Z

, is distributed as chi-square with

k

degrees of freedom: :

\sum_{i=1}^k Z^2_i =^d \chi^2_k

In cases whereby the expected value, E, is found to be small (indicating either a small underlying population probability, or a small number of observations), the normal approximation of the binomial distribution can fail, and in such cases it is found to be more appropriate to use the G-test, a likelihood-ratio test-based test statistic. Where the total sample size is small, it is necessary to use an appropriate exact test, typically either the binomial test or (for contingency tables) Fisher's exact test. ==See also== *Yates' correction for continuity *median test *Nomogram#Chi-squared nomogram ==External links== * [http://stat-www.berkeley.edu/~stark/Java/SampleChi.htm Sampling Distribution of the Sample Chi-Square Statistic] — a Java applet showing the sampling distribution of the Pearson test statistic. Statistics

Pearson's chi-square test

What's wrong with simply chi-square test? Are there more than one that are of encyclopedic interest? --User:Maveric149 :You've got to be kidding!!! There are zillions of them (zillions = at least a dozen or so) that are so different from each other except in sharing a common null distribution that it is astonishing that anyone could wonder about this . Well, maybe not astonishing to the layman, but still.... User:Michael Hardy 00:38 May 14, 2003 (UTC) :[I removed my earlier hasty and ill-considered -- and incorrect -- comment because Michael makes the point better.] User:Jfitzg In that case, then perhaps there should be an antry at chi-square test saying that there are lots of them, that the general principle was developed by A and B in century C, that these three tests are the most commonly used although those 7 are sometimes used for purpose X and purpose Y, and that all have in common the idea ''Z''. As a generality, the maths entries on the 'pedia are dense and forbidding to the non-mathematician. This sort of thing helps a lot to make stuff accessible to the general reader - which ''is'' what we are here for, isn't it? User:Tannin 00:47 May 14, 2003 (UTC) ------ could we have a brief intro in english please? :The current intro is comprehensible. I didn't write it -- it replaces a simpler one I wrote which probably appeared more English but which was not specific enough.User:Jfitzg ''I'' can comprehend it, sure. But I spent a couple of years studying stats, and even so I don't find it exactly easy reading. If I had happened to take a different minor, there is no way I could read and understand that intro, nor would I expect anyone else without at least some specialist training to be able to do so. I appreciate that the maths people want to get the maths entries as precise and strictly correct as possible, and applaud that urge, but we need to make sure that the casual reader is able to look at an entry and, even if he is unable to understand it in detail (or unwilling to put in the half-hour or so of concentrated effort it might take to grasp the detail), at least he should be able to walk away with a rough idea of what it is all about. I suggest changing the first para to something like this: * Karl Pearson chi-square test (χ²)—one of a variety of different chi-square tests—is a Statistics procedure used with category data to decide if experimental results are statistical significance, or else can reasonably be explained by mere chance. Like most statistical tests, it compares observed frequencies (under some kind of test condition) with expected frequency: in general, the greater the difference between the two, the less likely it is that the experimental results are simply the result of luck. * In more detail, Pearson's chi-square is for testing a null hypothesis that states that relative frequencies of occurrence of several specified mutually exclusive events, at least one of which must occur each time a specified experiment is performed, follow a specified frequency distribution. One of the simplest examples ....... etc. User:Tannin 13:03 14 May 2003 (UTC) THing is, I ''am'' a maths person. (I just happen to be allergic to stats). Maybe it's the Chi-square test article which should give an overview of ''what'' they are and why they're useful / interesting -- User:Tarquin 13:10 May 14, 2003 (UTC) ::Hmmm ... OK, but Pearson's chi-square is ''the'' chi-square in a very real sense. Sure, there are others, but this one is the only one that most people are ever going to use. I think it is a special case. The really obscure ones need less contextualising at the start of the entry. User:Tannin :::After further thought I concluded that an introduction like the one suggested by Tannin would be desirable. I'd suggest starting with a more general definition -- no reference to category data, for example. I don't think it's accurate to say that most statistical tests compare observed and expected frequencies, but perhaps I'm missing something or that wasn't what was intended. Anyway, I had thought of combining the current definition with the less detailed one originally posted. I suppose at some point someone's just going to have to grasp the nettle and change it.User:Jfitzg ::::Ahh, I ''knew'' it would be better to put it here than straight into the entry - my stats is very rusty, and I'm not surprised to have been caught in an error. I meant that most statistical tests compare observed and expected scores ''of some kind'' - not always frequencies, obviously. And I should have said "most statistical testing" (i.e., most frequently performed) as oposed to "most tests" (i.e., largest number of different tests). I mentioned the category data because, for the only-knows-a-little-bit statistician (your average social scientist, let's say), that's the key thing you have to remember: chi-square for category data, f-test or t-test or ANOVA for everything else. And if you can't use one of those, ask a ''real'' statistician. :) User:Tannin :::::Slip is a better word than error, I think. That's what I call them when I remove mine from contributions I've made, anyway. I'll log back on later (I think it's about time I made some money) and if no one has taken a stab at modifying the beginning I'll have a go and await comments.User:Jfitzg ---- I agree that it's too dense; I wrote it hastily. It has the advantage over the earlier version, of being correct. The earlier version spoke of differences between observed and theoretical frequencies, but that is trivial: The observed frequences are nearly always obviously different from the ones specified by the null hypothesis, and that is not what is of interest. What is of interest is whether the unobservable population frequencies differ from the theoretical ones. I don't think it's a good idea to say that the purpose of the test is to decide whether the data are statistically significant. Statistical significance is of interest only because it indicates that the null hypothesis is false. Whether the null hypothesis is false is what is of interest; interest in statistical significance is secondary and merely a means to an end. The null and alternative hypotheses should be made clear in any statement of the purpose of this test. I'll probably get to this within a few days. User:Michael Hardy 01:47 15 May 2003 (UTC) :In the meantime I simply shortened the sentences in what you wrote. As I said above, it is thoroughly comprehensible, and if people found it dense it was probably because of the one long sentence. It's only a suggestion -- I didn't make it here first because I wasn't altering the content all that much. User:Jfitzg

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