chi-squared statistic

Let X be a discrete random variable with m possible outcomes x1,,xm with probability of each outcome P(X=xi)=pi.

n independent observations are obtained where each observation has the same distributionPlanetmathPlanetmath as X. Bin the observations into m groups, so that each group contains all observations having the same outcome xi. Next, count the number of observations in each group to get n1,,nk corresponding to the outcomes x1,,xk, so that n=ni. It is desired to find out how close the actual number of outcomes ni are to their expected valuesMathworldPlanetmath npi.

Intuitively, this “closeness” depends on how big the sample is, and how large the deviations are between the observed and the expected, for all categories. The value

χ2=i=1m(ni-npi)2npi, (1)

called the χ2 statisticMathworldMathworldPlanetmath, or the chi-squared statistic, is such a measure of “closeness”. It is also known as the Pearson-chi-squared statistic, in honor of the English statistician Karl Pearson, who showed that (1) has approximately a chi-squared distribution ( with m-1 degrees of freedom. The degree of freedom depends on the number of free variablesMathworldPlanetmathPlanetmath in χ2, and is not always m-1, as we will see in Example 3.

Usually, χ2 statistic is utilized in hypothesis testingMathworldPlanetmath, where the null hypothesis specifies that the actual equals the expected. A large value of χ2 means either the deviations from the expectations are large or the sample is small, and therefore, either the null hypothesis should be rejected or there is not enough information to give a meaningful interpretationMathworldPlanetmathPlanetmath. How large of a deviation, compared to the sample size, is enough to reject the null hypothesis depends on the degree of freedom of chi-squared distribution of χ2 and the specified critical values.


  1. 1.

    Suppose a coin is tossed 10 times and 7 heads are observed. We would like to know if the coin is fair based on the observations. We have the following hypothesisMathworldPlanetmath:

    H0:p=12  H1:p12.

    Break up the observations into two groups: heads and tails. Then, according to H0,


    Checking the table of critical values of chi-squared distributions, we see that at degree of freedom =1, there is a 0.100 chance that the χ2 value is higher than 2.706. Since 1.600<2.706, we may not want to reject the null hypothesis. However, we may not want to outrightly accept it either simply because the sample size is not very large.

  2. 2.

    Now, a coin is tossed 100 times and 70 heads are observed. Using the same null hypothesis as above,


    Even at p-value =0.005, the corresponding critical value of 7.879 is quite a bit smaller than 16. So we will reject the null hypothesis even at confidence level 99.5%(=1-p-value).

  3. 3.

    χ2 statistic can be used in non-parametric situations as well, particularly, in contingency tablesMathworldPlanetmath. Three dice of varying sizes are each tossed 100 times and the top faces are recorded. The results of the count of each possible value of the top face, for each die is summarized in the following table:

    Die\top face 1 2 3 4 5 6 all
    Die 1 16 19 17 15 19 14 100
    Die 2 17 18 14 13 22 16 100
    Die 3 12 20 19 18 20 11 100
    All dice 45 57 50 46 61 41 300

    Let Xi= count of top face=i, and Yj= Die j. Next, we want to test the following hypotheses:

    H0:Xi is independent of Yj  H1:otherwise.

    Since we do not know the exact distribution of the top faces, we approximate the distribution by using the last row. For example, the (marginal) probability that top face = 1 is 45300=0.15. This says that the probability that top face = 1 in Die i = 0.15×13=0.05. Then, based on the null hypothesis, we have the following table of “expected count”:

    Die\top face 1 2 3 4 5 6
    Die 1 15.0 19.0 16.7 15.3 20.3 13.7
    Die 2 15.0 19.0 16.7 15.3 20.3 13.7
    Die 3 15.0 19.0 16.7 15.3 20.3 13.7

    For each die, we can compute the χ2. For instance, for the first die,

    χ2 = (16-15.0)215.0+(19-19.0)219.0+(17-16.7)216.7+
    = 0.176

    The results are summarized in the following

    χ2 degrees of freedom
    Die 1 0.176 5
    Die 2 1.636 5
    Die 3 1.969 0
    All dice 3.781 10

    Note that the degree of freedom for the last dice is 0 because the expected counts in the last row are completely determined by those in the first two rows (and the totals). Looking up the table, we see that there is a 90% that the value of χ2 will be greater than 4.865, and since 3.781<4.865, we accept the null hypothesis: the outcomes of the tosses have no bearing on which die is tossed.

Remark. In general, for a p×q 2-way contingency table, the χ2 statistic is given by

χ2=i=1pj=1q(nij-mij)2mij, (2)

where nij and mij are the actual and expected counts in Cell (i,j). When the sample is large, χ2 has a chi-squared distribution with (p-1)(q-1) degrees of freedom. In particular, when testing for the independence between two categorical variables, the expected count mij is

mij=ni*n*jn, where ni*=j=1qnij, n*j=i=1pnij, and n=i=1pj=1qnij.
Title chi-squared statistic
Canonical name ChisquaredStatistic
Date of creation 2013-03-22 15:10:00
Last modified on 2013-03-22 15:10:00
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 9
Author CWoo (3771)
Entry type Definition
Classification msc 62F03
Classification msc 62G10
Classification msc 62H17
Synonym χ2 statistic
Synonym chi-square statistic
Synonym Pearson-chi-squared statistic
Synonym Pearson-chi-square statistic
Related topic ChiSquaredRandomVariable
Related topic HypothesisTesting