What is  Degrees of Freedom : While comparing the calculated value of X2 with the table value we have to determine the degrees of freedom. By degrees of freedom we mean the number of classes to which the values can be assigned arbitrarily or at will without violating the restrictions or limitations placed. For example, if we are to choose nay five numbers whose total is 100? We can exercise our independent choice for any four numbers only. The fifth number is the total is the total being 100 as it must be equal to 100 minus the total of the four numbers selected. For example, if the four numbers are 20, 35, 15, 10, the fifth number must be [100 - (20 + 35 + 15 + 10] 20. Thus though we were to choose any five numbers we could choose any four only. Our choice was reduced by one because of one condition placed in the data. That of total being 100, thus there was only one restraint on our freedom - the degrees of freedom were only foot. If more restraint ions are placed our freedom to choose will be still curtailed. For example, if there are 10 classes and we want out frequencies to be distributed in such a manner that the number of cases. The mean and the standard deviation agree with original distribution, we have three constraints (restrictions) and so three degrees of freedom are lost. Hence in this case the degrees of freedom will be 10 - 3 = 7. Thus the number of degrees of freedom is obtained by subtracting form the number of classes the number of degrees of freedom lost in fitting. Symbolically, the degrees of freedom are denoted by the symbol (pronounced nu) or by D.F. and are obtained as follows:

Where k refers to the number of independent constraints. We have a constraint or restriction whenever observed or theoretical frequencies are made to agree with one another in some respect in the operations that lead to the calculation of X2. Thus a constraint is imposed by the condition ∑fo = ∑fe. In general when we fit a binomial distribution, the number of degrees of freedom is one less than the number of classes; then we fit a poison distribution. The degrees of freedom are 2 less than the number of degrees of freedoms is one less than the number of classes: (∴ we use total frequency and arithmetic mean), and when we fit normal curve, the number of degrees of freedom is small by 3 than the number of classes (because in the fitting we use total frequency, mean and standard deviation).

In a contingency table the degrees of freedom are calculated in a slightly different manner. The marginal total or frequencies place the limit on our choice of selecting cell frequencies. The cell frequencies of all columns but on e(c -1) and of all rows but one (r - 1) can be assign arbitrarily and so the number of degrees of freedom for all the cell frequencies = (c - 1) (r - 1) where c refers to column and r refers to rows. Thus in a2 X2 table the degrees of freedom = (2 - 1) (2 - 1) = 1. Having filed up one cell in such a table the rest of the frequencies automatically follow - there is no choice for them. Similarly in a3 x3contingency tale, the number of degrees of freedom is (3 - 1) 3 - 1) = 4, and so on. It means only 4 expected frequencies need be computed. The others are obtained by subtraction from normal totals.

Points to note

The following points about the X2 test are worth noting:
    
The sum of the observed and expected frequencies is always zero, symbolically.

∑ (0-E) = ∑ 0 - ∑ E = N - N = 0

Needless to say that this provides an important check on the calculation in the computation of X2,
    
The X2 test depends only on the set of observed and expected frequencies and no degrees of freedom. It is a non-parametric test, also known as distribution-free test, since no assumptions are made about the parameters of the populations.

X2 distribution is a limiting approximation of the multinomial distribution. 

Though X2 distribution is essentially a continuous distribution the X2 test can be applies to discrete random variables whose frequencies can be counted and tabulated with or without grouping. 

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