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Chi-Square (χ2) Statistic Definition

What Is a Chi-Sq. Statistic?

A chi-square (χ2) statistic is a take a look at that measures how a mannequin compares to precise noticed information. The info utilized in calculating a chi-square statistic have to be random, uncooked, mutually exclusive, drawn from unbiased variables, and drawn from a big sufficient pattern. For instance, the outcomes of tossing a good coin meet these standards.

Chi-square assessments are sometimes utilized in hypothesis testing. The chi-square statistic compares the dimensions of any discrepancies between the anticipated outcomes and the precise outcomes, given the dimensions of the pattern and the variety of variables within the relationship.

For these assessments, degrees of freedom are utilized to find out if a sure null hypothesis could be rejected primarily based on the whole variety of variables and samples throughout the experiment. As with all statistic, the bigger the pattern dimension, the extra dependable the outcomes.

Key Takeaways

  • A chi-square (χ2) statistic is a measure of the distinction between the noticed and anticipated frequencies of the outcomes of a set of occasions or variables.
  • Chi-square is beneficial for analyzing such variations in categorical variables, particularly these nominal in nature.
  • χ2 depends upon the dimensions of the distinction between precise and noticed values, the levels of freedom, and the samples dimension.
  • χ2 can be utilized to check whether or not two variables are associated or unbiased from each other.
  • It may also be used to check the goodness-of-fit between an noticed distribution and a theoretical distribution of frequencies.

The Formulation for Chi-Sq. Is














the place:



Levels of freedom



Noticed worth(s)

beginaligned&chi^2_c = sum frac(O_i – E_i)^2E_i &textbfwhere:&c=textDegrees of freedom&O=textObserved worth(s)&E=textExpected worth(s)endaligned

χc2=Ei(OiEi)2the place:c=Levels of freedomO=Noticed worth(s)

What Does a Chi-Sq. Statistic Inform You?

There are two predominant sorts of chi-square assessments: the take a look at of independence, which asks a query of relationship, comparable to, “Is there a relationship between pupil intercourse and course selection?”; and the goodness-of-fit test, which asks one thing like “How properly does the coin in my hand match a theoretically truthful coin?”

Chi-square evaluation is utilized to categorical variables and is particularly helpful when these variables are nominal (the place order does not matter, like marital standing or gender).


When contemplating pupil intercourse and course selection, a χ2 take a look at for independence may very well be used. To do that take a look at, the researcher would gather information on the 2 chosen variables (intercourse and programs picked) after which examine the frequencies at which female and male college students choose among the many supplied lessons utilizing the system given above and a χ2 statistical desk.

If there is no such thing as a relationship between intercourse and course choice (that’s, if they’re unbiased), then the precise frequencies at which female and male college students choose every supplied course ought to be anticipated to be roughly equal, or conversely, the proportion of female and male college students in any chosen course ought to be roughly equal to the proportion of female and male college students within the pattern.

A χ2 take a look at for independence can inform us how seemingly it’s that random probability can clarify any noticed distinction between the precise frequencies within the information and these theoretical expectations.


χ2 supplies a technique to take a look at how properly a pattern of knowledge matches the (identified or assumed) traits of the bigger inhabitants that the pattern is meant to signify. This is called goodness of fit. If the pattern information don’t match the anticipated properties of the inhabitants that we’re fascinated about, then we might not wish to use this pattern to attract conclusions concerning the bigger inhabitants.


For instance, contemplate an imaginary coin with precisely a 50/50 probability of touchdown heads or tails and an actual coin that you simply toss 100 occasions. If this coin is truthful, then it would even have an equal likelihood of touchdown on both aspect, and the anticipated results of tossing the coin 100 occasions is that heads will come up 50 occasions and tails will come up 50 occasions.

On this case, χ2 can inform us how properly the precise outcomes of 100 coin flips examine to the theoretical mannequin {that a} truthful coin will give 50/50 outcomes. The precise toss might come up 50/50, or 60/40, and even 90/10. The farther away the precise outcomes of the 100 tosses is from 50/50, the much less good the match of this set of tosses is to the theoretical expectation of fifty/50, and the extra seemingly we would conclude that this coin will not be really a good coin.

When to Use a Chi-Sq. Check

A chi-square take a look at is used to assist decide if noticed outcomes are in keeping with anticipated outcomes, and to rule out that observations are as a consequence of probability. A chi-square take a look at is acceptable for this when the info being analyzed is from a random sample, and when the variable in query is a categorical variable. A categorical variable is one which consists of choices comparable to kind of automotive, race, academic attainment, male vs. feminine, how a lot someone likes a politician (from very a lot to little or no), and so forth.

All these information are sometimes collected through survey responses or questionnaires. Due to this fact, chi-square evaluation is usually most helpful in analyzing the sort of information.

What’s a chi-square take a look at used for?

Chi-square is a statistical take a look at used to look at the variations between categorical variables from a random pattern with a view to choose goodness of match between anticipated and noticed outcomes. 

Who makes use of chi-square evaluation?

Since chi-square applies to categorical variables, it’s most utilized by researchers who’re finding out survey response information. The sort of analysis can vary from demography to shopper and advertising analysis to political science and economics.

Is chi-square evaluation used when the unbiased variable is nominal or ordinal?

A nominal variable is a categorical variable that differs by high quality, however whose numerical order may very well be irrelevant. As an example, asking someone their favourite colour would produce a nominal variable. Asking someone’s age, alternatively, would produce an ordinal set of knowledge. Chi-square could be greatest utilized to nominal information.

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