What is Chi square: Definition and 46 Discussions

In probability theory and statistics, the chi-square distribution (also chi-squared or χ2-distribution) with k degrees of freedom is the distribution of a sum of the squares of k independent standard normal random variables. The chi-square distribution is a special case of the gamma distribution and is one of the most widely used probability distributions in inferential statistics, notably in hypothesis testing and in construction of confidence intervals. This distribution is sometimes called the central chi-square distribution, a special case of the more general noncentral chi-square distribution.
The chi-square distribution is used in the common chi-square tests for goodness of fit of an observed distribution to a theoretical one, the independence of two criteria of classification of qualitative data, and in confidence interval estimation for a population standard deviation of a normal distribution from a sample standard deviation. Many other statistical tests also use this distribution, such as Friedman's analysis of variance by ranks.

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  1. patric44

    What is the correct formula for the reduced Chi square?

    Hi all I want to calculate the reduced Chi square and root mean square deviation RMSD of some data points that i have, but I am confused about the correct formula for each of them, which one is the correct one. I found this formula in a paper where they referred to it as the RMSD : $$...
  2. A

    A The chi square goodness-of-fit test with no degrees of freedom left

    I have an empirical frequency distribution as for example below: ##f_{2} = \, \, \, 21## ##f_{3} = 111## ##f_{4} = \, \, \, 24## The theoretical distribution is determined by two parameters. So for a chi-square goodness-of-fit test there are actually no degrees of freedom left. Yet the...
  3. M

    I Reduced chi square for few data points

    Hello! I need to make a straight line fit to 8 points, with errors on them. The data is like this ##x = [1,2,3,4,5,6,7,8]##, ##y=[377.488 691.191 , 1030.319, 1428.801, 1753.884, 2113.065 , 2398.642, 2797.664]##, ##y_{err}=[97.145, 131.452, 160.492, 188.997, 209.397, 229.840, 244.879...
  4. F

    I Probability-To-Exceed (PTE) and Chi^2 distribution

    I would like to know the difference between the ##\chi^{2}## distribution and the PTE (Probability-To-Exceed) ? I must compare 2 data sets A and B and in the article I am reading, they talk about this PTE : For the moment, I only know the ##\chi^{2}## distribution with ##k=2## degrees of...
  5. Jakub

    I Small Reduced Chi Squared interpretation

    Hello everyone, I would be happy if someone explained the small reduced chi squared value to me. I have fitted a set of measured data with an exponential function, which I need for some sw calculations. The fit seams great, the origin sw also provides the reduced chi squared, but it is very...
  6. C

    Mathematica Chi square minimisation wrt variables in an integral?

    I'm trying to fit a model curve to some data by performing a chi square minimisation wrt three parameters a,b and NN. The trouble I am having is that the variables with which I want to minimise the chi square with respect to appear in an integral. I attach the code I am working with...
  7. R

    Chi-square goodness of fit cannot find expected values

    Homework Statement An article in Business Week reports profits and losses of firms by industry. A random sample of 100 firms is selected, and for each firm in the sample, we record whether the company made money or lost money, and whether or not the firm is a service company. The data are...
  8. T

    I Understanding Yates' correction

    I was studying Biostatistics by Zar, the Yates' correction part and stumbled upon this: My doubts: The conculsions are not taking place exactly at the level of alpha- what does this mean? Do the values of Chi square mentioned in table not correspond to the alpha(probability) indicated? Is...
  9. T

    I Online sourses to learn chi square test of homogeneity

    I want to study chi square test of homogeneity from any authentic source- book / website especially problems where samples are compared for more than one attribute. What are some relevant sources? Relevant background: I was studying examples from random online sources before I saw this book...
  10. T

    I What is Yates' correction of contingency?

    I can't understand a word of https://en.wikipedia.org/wiki/Yates's_correction_for_continuity']Wikipedia.[/PLAIN] P.S. What I know so far is subtracting 0.5 from O-E if df=1 in Goodness of fit (?) is Yates' correction.
  11. T

    I Level of significance and acceptance and rejection of the null hypothesis

    Why do we reject the null hypothesis in Goodness of fit when the Chi square statistic is less than the tabulated value of chi-square at say 5% level of significance and accept when it is more? What does it mean to have a Chi-square value more or less than the value assigned to a certain level...
  12. Jules Winnfield

    I How do I compare a model to logarithmic data?

    I have a model which is quadratic (e.g. ##y = k x^2##). I'm comparing it against a large set of data (galaxy cluster masses) which spans several Log10 decades (e.g. ##10^{11}## to ##10^{15}## solar masses). What is the right way to say how good the data fits the model? Obviously the errors in...
  13. cynnetje

    A Do outliers exist in categorical data and how can they be detected?

    Hello! I am working on a pre-analysis plan and have to specify what I am going to do with outliers. I have two categorical variables (5 levels and 2 levels) and I will be performing a chi-square test for independence. I thought of using a boxplot to detect outliers, but now I am not sure if...
  14. C

    MHB Chi-Square Test: Solving Doubt w/ Kepler

    Good afternoon, I'm glad I've joined this forum. Here's my doubt: I have a serie of values in a table like this: Case 1 34 55 Case 2 23 10 Case 3 55 40 etc... the 34 means the observed value, and the 55 the control group, and so on. It's easy to do the test of course if... The problem is: if...
  15. JulienB

    Reduced chi square test in physics

    Homework Statement Hi everybody! Our experiments teacher asked us to perform a reduced chi square test in order to estimate how good a model fits to our measured data. The experiment was the melde's experiment (vibration of a string) and we measured the frequency ##f_n## for ##n=1## to ##9##...
  16. S

    A Chi squared test for data with error

    Hi everyone. I am totally new to statistics so my question may or may not be simple! I know that for the data fitting we can do a chi squared test like: \begin{equation} \chi^2 = \Sigma \frac{(f_{data}-f_{model})^2}{(error_{data})^2}\end{equation} So I have been doing this for a while, but now...
  17. D

    Goodness of fit, Residual STD, chi square

    Homework Statement Hello, I am using CasaXPS to model synthetic peak models for X-ray photoelectron spectroscopy data. I am fitting. The software has a lot of manuals online but they do not explain how they yield a Residual Standard Deviation, after each fit iteration. Most software use...
  18. S

    Error in summation of spectral components

    HI everyone, Imagine we are sampling of a gaussian signal along time and need to know the power/variance associated with the first N spectral components. So we take our favorite fft algorithm to get the PSD. The error associated with a given estimated spectral component f(w) (w is the...
  19. P

    Can a Chi2 test be used on uniform p test results?

    Not sure that I've phrased the question correctly. If you have a series of p values from a series of tests, and they're all meant to be uniformly distributed, why do you have to do a KS test on that, and not another Chi-squared test? The following is an extract from a test program's output:-...
  20. AlanKirby

    Straight line equation that accounts for known error in X&Y?

    I'm looking for how to mathematically relate X and Y, in such a way that the Chi-Squared per degree of freedom is minimised. However I can't understand how this would work, given that I'm trying to use known X,Y dX and dY (errors) values, to get the related gradient and intercept for the line of...
  21. N

    Chi Square Distribution Problem

    Homework Statement Suppose that X has normal distribution. Find the distribution of n(Sample Mean - μ)2/σ2. The Attempt at a Solution I honestly have no idea where to begin with this problem. Any ideas?
  22. D

    Error bars, Chi square distribution

    Hi, I'm doing a fit using Chi-square distribution. I have a data set and their errors, I found the best estimate minimizing Chi square, as usual, and I like to found the error bars of my best estimates but I don't know how to do that. Which is the standard form to do it?
  23. Mogarrr

    Prove Chi Square is Stochastically Increasing

    Homework Statement Prove that the X^2 distribution is stochastically increasing in its degrees of freedom; that is if p>q, then for any a, P(X^2_{p} > a) \geq P(X^2_{q} > a), with strict inequality for some a. Homework Equations 1.(n-1)S^2/\sigma^2 \sim X^2_{n-1} 2.The Chi squared(p) pdf is...
  24. S

    CHI square independent test (small expected frequency)

    Homework Statement the expected frequency for the age of (21-30, 31-36, more than or equal to 37) for operation is less than 5 , so combination of these numbers are required for the expected frequency to be more than or equal to 5 . my question is which numbers should i combine? would my ans...
  25. G

    CHI square test - finding degree of freedom

    Homework Statement i have problem of finding the degree f freedom for this question. the ans for v is 3 , but my ans is v=n-1 , where n = 6 , so my v=5... Homework EquationsThe Attempt at a Solution
  26. Math Is Hard

    Analyzing Enrollment Campaigns with Chi-Square: Is It the Right Approach?

    Homework Statement I'm trying to test a hypothesis that sending people both an email announcement and direct mail announcement produces significantly more enrollments in a free webinar than email or direct mail alone. I'd like to do an analysis on these groups created from 400 people...
  27. L

    Statistics Chi Square test of normally distributed data

    Homework Statement The following data is hypothesized to possesses a normal distribution, is that hypothesis sustained by a Chi-square test at .05 significance: <66...4 67-68...24 68-70...35 70-72...15 72-74...8 >74...4 The Attempt at a Solution 1. I know that I should...
  28. A

    How do you use a chi square table?

    For example, if the moment generating function is (1-2t)^(-6), then the degree of freedom r=3, right? The question is asking me to find P(X< 5.23)...I want to use the chi-square table, but I'm not sure how... This is what the question states: If (1-2t)^{-6}, t<1/2, is the mgf of the...
  29. M

    Convolution of iid non central Chi square and normal distribution

    Hi, I am doing research and I am stuck at this point I need help to convolute iid non central chi-square with normal distribution.
  30. M

    Inverse Chi Square: Rejecting Null Hypothesis at α=5%, 9DF

    How can I determine what the smallest value of a χ2 statistic must be to reject the null hypothesis at α = 5%, for a distribution with 9 degrees of freedom? Thanks in advance.
  31. B

    Deciding Bins for Chi Square Tests

    Homework Statement I know that for a Chi Square test to adequately describe a distribution, you need each bin to have an estimated frequency > 5. As a rule of thumb though, do I want to pool the bins with less than 5 estimated frequency so as to maximize the number of bins or minimize them? For...
  32. B

    Non central chi square distribution

    I read this article about non central chi square distribution http://en.wikipedia.org/wiki/Noncentral_chi-square_distribution in practice, if I have a sum of X_i^2, where X_i is gaussian with mean \mu_i and std \sigma_i what would be the pdf of the sum? In the article the assume you have...
  33. S

    Help with chi square distribution

    How do i show that the a [X1 has a chi square distribution with n degrees of freedom] + [X2 has a chi square distribution with m degrees of freedom] is a [X1+X2 has a chi square distribution with n+m degrees of freedom]? How can i use moment generating functions to do this?
  34. D

    Central Limit Theorem Variation for Chi Square distribution?

    Central Limit Theorem Variation for Chi Square distribution? If this question fits into Homework Help, please move it over there. I'm not too sure. I encountered the following problem: Now, this problem seems fairly similar to a simple proof the central limit theorem. I am damn sure...
  35. S

    Chi Square Analysis of Phenotype Problem

    Parental Pheotypes Long x ShortF1 All LongF2 Phenotypes 787 long, 277 short1. What is the expected value for the recessive phentotype of the cross Long versus Short 787+277= 1064 I did 787/1064= .739 Rounded up to .74 For Long 277/1064= .26 for Short .74x1064= 787.36 Is the Expected...
  36. S

    Chi square Dominant phenotype problem

    What is the ChiSquare value for the dominant phentotype of the cross Purple versus White. Shouldn't the answer be 696 or 696.75?? Parental Phenotypes Purple X WhiteF1 Phenotype All PurpleF2 Phenotype 705 purple, 224 whiteI must be missing something: Observed Purple=705 Observed White= 224...
  37. Z

    How are degrees of freedom determined in a Chi-square test for genetics?

    In the Chi-square test, my textbook says that degrees of freedom are the number of independent variables minus one ,df = n - 1 does this mean that that n is equal to the number of observed values from the equation aka the number of times I've added or the number of terms? sum [(O-E)^2]/E Is...
  38. M

    Chi Square Test for Gaussian Distribution

    Hi there, I have very naive to statistics. I have a set of data points. that can be like 10, 12, 13, 14 ,15 , 15, 12, 13 17, 18, 19, 12, 19, 20 .... Now i need to know if these days points follows any gaussian distribution / normal distribution or not? IS chi -square test the right...
  39. E

    What is the Generalization of the Chi-square Distribution?

    What is the Multinomial expansion of the CDF of the Chi-square distribution? Thanks in advance
  40. D

    MGF Techniques for Chi-Square Distribution on 2n Degrees of Freedom

    Homework Statement rvX has f(x) = \alpha \exp^{-\alpha x} , and \ W = 2n \alpha \overline {X} defines a random sample from the distribution. Use moment generating function techniques to show that the distribution of W is chi-square on 2n degrees of freedom. Homework Equations The Attempt...
  41. D

    O: "Solving Chi Square Problems: E(s^2) and Var(s^2) Using MLE

    Homework Statement W = \frac{vS^2}{\sigma^2}, distributed as X^2_v Find E(s^2) and Var(s^2) Homework Equations E(W) = v , Var(w)=2vThe Attempt at a Solution Have been trying to figure this out with no luck. Can I use MLE for variance to show Var(s^2)= \sigma^2? Really don't know how to get...
  42. C

    Reviewing Image Retrieval Performance Using Chi-Square

    Hi ll, I have been asked to review a new image retrieval technique that has improved the retrieval performance compared with the old technique, ie, the hypothesis is that retrieval performance is influenced by retrieval techniques. The following table summarises the retrieval results. (They...
  43. S

    Anyone has Idea on Chi Square

    The following table shows brand preferences against gender on the basis of a random sample of size 100 from population. It is intended to examine possible association between gender and brand preferences. Brand A Brand B Men 30 20 Women 20 30 At 5% level of significance will you...
  44. Math Is Hard

    Can Chi Square be Used for a Mixed Design with Repeated Measures?

    My lab partner and I are trying to recreate this experiment: http://www.blackwell-synergy.com/doi/pdf/10.1111/j.1467-9280.2006.01731.x? This appears to be a mixed design. Half the participants were put in a positive mood and half were put into a neutral mood. After the mood induction, each...
  45. J

    Testing Phenotypic Ratios: Results of Chi Square Test

    Hi, My Question: Problem 1. The expected phenotypic ratios for each of the crosses below is 1:1. Determine whether the observed ratios are differ significantly (p < 0.05) from the expected. I used a Chi Square test to help me determine that answer and found the there is no probability of...
  46. A

    Chi square is useful - but why square?

    Hello! While conducting chi square -tests I asked my teacher what the "square" means, and why it was there? Why chi squared, and not just chi? He couldn't give an answer so he told me to find out, as a homework :surprise: ... After a quarter of "Googeling" I gave up. So, is there anyone out...
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