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There is low probability that Χ² is very close to or very far from zero. The curve starts out low, increases, and then decreases again. When k is greater than two, the chi-square distribution is hump-shaped. When k is one or two, the chi-square distribution is a curve shaped like a backwards “J.” The curve starts out high and then drops off, meaning that there is a high probability that Χ² is close to zero. A probability density function is a function that describes a continuous probability distribution. We can see how the shape of a chi-square distribution changes as the degrees of freedom ( k) increase by looking at graphs of the chi-square probability density function. If you sample a population many times and calculate Pearson’s chi-square test statistic for each sample, the test statistic will follow a chi-square distribution if the null hypothesis is true. is the summation operator (it means “take the sum of”).Pearson’s chi-square test statistic is: Formula Pearson’s chi-square test was the first chi-square test to be discovered and is the most widely used. Χ 2 k = ( Z 1) 2 + ( Z 2) 2 + … + ( Z k) 2Ĭhi-square tests are hypothesis tests with test statistics that follow a chi-square distribution under the null hypothesis. More generally, if you sample from k independent standard normal distributions and then square and sum the values, you’ll produce a chi-square distribution with k degrees of freedom. If each time you sampled a pair of values, you squared them and added them together, you would have the chi-square distribution with k = 2. Now imagine taking samples from two standard normal distributions ( Z 1 and Z 2). If you squared all the values in the sample, you would have the chi-square distribution with k = 1. Imagine taking a random sample of a standard normal distribution ( Z). The standard normal distribution, which is a normal distribution with a mean of zero and a variance of one, is central to many important statistical tests and theories. Relationship to the standard normal distributionĬhi-square distributions are useful for hypothesis testing because of their close relationship to the standard normal distribution. In contrast, most other widely used distributions, like normal distributions or Poisson distributions, can describe useful things such as newborns’ birth weights or disease cases per year, respectively. The main purpose of chi-square distributions is hypothesis testing, not describing real-world distributions. Very few real-world observations follow a chi-square distribution. The shape of a chi-square distribution is determined by the parameter k, which represents the degrees of freedom. They’re widely used in hypothesis tests, including the chi-square goodness of fit test and the chi-square test of independence.
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