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Chi-Square (\u03c72) Statistic

Chi-Square (χ2) Statistic

What Is a Chi-Square Statistic?

A chi-square (\u03c72) ^^statistic is a test that measures how a model compares to genuine noticed data. The data utilized in working out a chi-square statistic must be random, raw, mutually exclusive, drawn from independent variables, and drawn from a sufficiently large sample. For instance, the consequences of flipping a fair coin meet these criteria.

Chi-square tests are often utilized in hypothesis testing. The chi-square statistic compares the size of any disparities between the expected outcomes and the genuine outcomes, given the size of the sample and the number of variables in the relationship.

For these tests, degrees of freedom are used to decide whether a certain null hypothesis can be dismissed in light of the total number of variables and samples inside the examination. Likewise with any statistic, the larger the sample size, the more dependable the outcomes.

The Formula for Chi-Square Is

χc2=∑(Oi−Ei)2Eiwhere:c=Degrees of freedomO=Observed value(s)E=Expected value(s)\begin&\chi^2_c = \sum \frac{(O_i - E_i)^2} \&\textbf\&c=\text\&O=\text{Observed value(s)}\&E=\text{Expected value(s)}\end

What Does a Chi-Square Statistic Tell You?

There are two primary sorts of chi-square tests: the trial of independence, which poses an inquiry of relationship, for example, "Is there a relationship between student sex and course decision?"; and the goodness-of-fit test, which asks something like "How well does the coin in my grasp match a hypothetically fair coin?"

Chi-square analysis is applied to clear cut variables and is particularly helpful when those variables are nominal (where order doesn't make any difference, as marital status or orientation).


While considering student sex and course decision, a \u03c72 test for independence could be utilized. To do this test, the researcher would collect data on the two picked variables (sex and courses picked) and afterward compare the frequencies at which male and female students select among the offered classes utilizing the formula given above and a \u03c72 statistical table.

Assuming there is no relationship among sex and course selection (that is, on the off chance that they are independent), the genuine frequencies at which male and female students select each offered course ought to be expected to be around equivalent, or alternately, the extent of male and female students in any chosen course ought to be roughly equivalent to the extent of male and female students in the sample.

A \u03c72 test for independence can perceive us how likely it is that random chance can make sense of any noticed difference between the genuine frequencies in the data and these hypothetical expectations.


\u03c72 gives a method for testing how well a sample of data matches the (known or assumed) qualities of the larger population that the sample is expected to address. This is known as goodness of fit. In the event that the sample data don't fit the expected properties of the population that we are keen on, then we would have no desire to utilize this sample to draw decisions about the larger population.


For instance, think about a nonexistent coin with precisely a 50/50 chance of landing heads or tails and a real coin that you throw 100 times. In the event that this coin is fair, it will likewise have an equivalent likelihood of landing on one or the other side, and the expected consequence of flipping the coin 100 times is that heads will come up 50 times and tails will come up 50 times.

In this case, \u03c72 can see us how well the genuine consequences of 100 coin flips compare to the hypothetical model that a fair coin will give 50/50 outcomes. The genuine throw could come up 50/50, or 60/40, or even 90/10. The farther away the genuine consequences of the 100 throws is from 50/50, the less great the fit of this set of throws is to the hypothetical expectation of 50/50, and the more probable we could infer that this coin isn't really a fair coin.

When to Use a Chi-Square Test

A chi-square test is utilized to help decide whether noticed results are in accordance with expected results, and to rule out that perceptions are due to chance. A chi-square test is proper for this when the data being broke down is from a [random sample](/straightforward random-sample), and when the variable being referred to is an unmitigated variable. An unmitigated variable is one that comprises of selections like type of vehicle, race, instructive fulfillment, male versus female, the amount someone enjoys a political candidate (from particularly to very little), and so forth.

These types of data are often collected by means of survey responses or polls. Hence, chi-square analysis is often most valuable in dissecting this type of data.


  • \u03c72 relies upon the size of the difference among genuine and noticed values, the degrees of freedom, and the sample size.
  • A chi-square (\u03c72) ^^statistic is a measure of the difference between the noticed and expected frequencies of the results of a set of events or variables.
  • Chi-square is valuable for dissecting such differences in clear cut variables, particularly those nominal in nature.
  • It can likewise be utilized to test the goodness-of-fit between a noticed distribution and a hypothetical distribution of frequencies.
  • \u03c72 can be utilized to test whether two variables are connected or independent from each other.


Is Chi-square Analysis Used When the Independent Variable Is Nominal or Ordinal?

A nominal variable is a straight out variable that contrasts by quality, however whose mathematical order could be irrelevant. For example, asking someone their number one tone would deliver a nominal variable. Asking someone's age, then again, would deliver an ordinal set of data. Chi-square can be best applied to nominal data.

Who Uses Chi-Square Analysis?

Since chi-square applies to absolute variables, generally utilized by researchers are concentrating on survey response data. This type of research can go from demography to consumer and marketing research to political science and economics.

What Is a Chi-square Test Used For?

Chi-square is a statistical test used to look at the differences between unmitigated variables from a random sample to judge goodness of fit among expected and noticed results.