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A chisquared test, also referred to as chisquare test or test, is any statistical hypothesis test in which the sampling distribution of the test statistic is a chisquared distribution when the null hypothesis is true. Also considered a chisquared test is a test in which this is asymptotically true, meaning that the sampling distribution (if the null hypothesis is true) can be made to approximate a chisquared distribution as closely as desired by making the sample size large enough. The chisquare (I) test is used to determine whether there is a significant difference between the expected frequencies and the observed frequencies in one or more categories. Do the number of individuals or objects that fall in each category differ significantly from the number you would expect? Is this difference between the expected and observed due to sampling error, or is it a real difference?
The following are examples of chisquared tests where the chisquared distribution is approximately valid:
Pearson's chisquared test, also known as the chisquared goodnessoffit test or chisquared test for independence. When the chisquared test is mentioned without any modifiers or without other precluding context, this test is usually meant (for an exact test used in place of , see Fisher's exact test).
Using the chisquared distribution to interpret Pearson's chisquared statistic requires one to assume that the discrete probability of observed binomial frequencies in the table can be approximated by the continuous chisquared distribution. This assumption is not quite correct, and introduces some error.
To reduce the error in approximation, Frank Yates, an English statistician, suggested a correction for continuity that adjusts the formula for Pearson's chisquared test by subtracting 0.5 from the difference between each observed value and its expected value in a 2 × 2 contingency table.^{[1]} This reduces the chisquared value obtained and thus increases its pvalue.
One case where the distribution of the test statistic is an exact chisquared distribution is the test that the variance of a normally distributed population has a given value based on a sample variance. Such a test is uncommon in practice because values of variances to test against are seldom known exactly.
1. Quantitative data. 2. One or more categories. 3. Independent observations. 4. Adequate sample size (at least 10). 5. Simple random sample. 6. Data in frequency form. 7. All observations must be used.
If a sample of size n is taken from a population having a normal distribution, then there is a result (see distribution of the sample variance) which allows a test to be made of whether the variance of the population has a predetermined value. For example, a manufacturing process might have been in stable condition for a long period, allowing a value for the variance to be determined essentially without error. Suppose that a variant of the process is being tested, giving rise to a small sample of n product items whose variation is to be tested. The test statistic T in this instance could be set to be the sum of squares about the sample mean, divided by the nominal value for the variance (i.e. the value to be tested as holding). Then T has a chisquared distribution with n − 1 degrees of freedom. For example if the sample size is 21, the acceptance region for T for a significance level of 5% is the interval 9.59 to 34.17.
