Chi-squared Test

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Chi-Squared Test

How do you know the variation in your data is due to chance or due to one of the variables you are testing? How do we find this out?

In one of our previous posts, we talked about using binomial tests to check if the distribution of tests deviates from a null hypothesis or not in the case of experiments with two categories (like head/tail; Republican/Democrat etc).

However, this does not work if we have more than two categories and we want to test their distribution to some null hypothesis about the proportion of each that we expect to see. In this case, we use the chi-square test.

What is chi-square?

The chi-squared (χ2) test is a statistical tool used to determine whether there is a significant difference between the observed frequencies and the expected frequencies in two or more groups. The chi-squared statistic is calculated by taking the sum of the squared differences between each observed frequency and its corresponding expected frequency. This statistic is then divided by the number of degrees of freedom, which is equal to the number of groups minus one. If this value exceeds a predetermined threshold, then there is evidence that there is a statistically significant difference between the groups.

The chi-squared test can be used to compare two independent groups or two dependent groups. In addition, it can be used to compare three or more independent groups, as well as two or more dependent groups with multiple measures. The chi-squared test has been found to be particularly useful for comparing proportions, such as percentages or odds ratios.

In 1900, Pearson published a paper on the (χ²) test which is considered to be one of the foundations of modern statistics. In this paper, Pearson investigated a test of goodness of fit.

Pearson’s chi square test (goodness of fit) (video) | Khan Academy

Sal uses the chi square test to the hypothesis that the owner’s distribution is correct.

Khan Academy

Degree of freedom

A degree of freedom in a chi-square test is the number of independent pieces of information that are used to calculate the statistic. In most cases, this is the number of rows minus one or the number of columns minus one. This value determines how much information is used to calculate the chi-square statistic and helps to determine whether or not it is significant.

Critical Value

A critical value in a chi-square test is the point at which the significance of the difference between the observed and expected frequencies is determined. The chi-square statistic is used to determine whether this difference is significant, and if it is, then a critical value will be used to determine how likely it was that this difference occurred by chance. If the chi-square statistic falls below the critical value, then it can be concluded that there was not enough evidence to reject the null hypothesis.

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