Statistical Methods

7.1.Introduction to Chi-square distribution

Unit 7 - Chi-square distribution

7.1.Introduction to Chi-square () distribution
Theoretically, Chi-square () distribution can be defined as the sum of squares of independent normal variates. If X^₁, X^₂ ………….X^_n are n independent standard normal variates, then sum of squares of these variates X₁²+X₂² +…………………………+X_n² follows the distribution with n degrees of freedom. Alternatively if a sample of size n, is drawn from a normal population with variance σ², the quantity (n-1) s² / σ² follows distribution with (n-1) degrees of freedom where s² is the sample variance.

The shape of distribution depends on the degrees of freedom which is also its mean (Fig.1). When n is small, the distribution is markedly different from normal distribution but as n increases the shape of the curve becomes more and more symmetrical and for n > 30, it can be approximated by a normal distribution. The values of have been tabulated for different degrees of freedom at different levels of probability. (Fisher and Yates, 1963). The is always greater than or equal to zero i.e. ≥ 0.

Most data on biological investigations can be classified either as quantitative or qualitative (attribute) data. The statistical procedures discussed so far apply mostly to quantitative data. There are many instances in fisheries research, wherein attribute data describe the phenomenon under investigations more adequately than quantitative data. The chi-square test based on distribution is commonly used for analysis of attribute data.