# Chi-Square Test

The Chi-Square method relies upon the difference between the observed value (O) of a data-point obtained from some practical measurement and what its expected value (E) would be for some
theoretically possible distribution of that same data-point. For each point of interest, the quotient of the square of that difference (which will necessarily be positive) and the value expected for that point in some known theoretical probability distribution (catalogued in reference handbooks on mathematical statistics), is summed over some portion of data-points already collected. This procedure yields a non-zero numerical result known as the X2-value. The theoretical probability distribution whose X2-value shows the smallest variance from the observed values tabulated as actual data is then selected as the most appropriate for modelling the distribution of the rest of these actual data. The following equation computes the X2-value:

(3.39)

where

0 =an observed frequency

E. = an expected (theoretical) frequency

1 = the number of data-points from the table or portion thereof

From this equation, note that when distribution resulting from the practical test matches the probability distribution, the equation reduces to X2 = 0.

The Chi-Square Test is highly reliable for small data samples of less than 30 values, and there is no requirement that the observed data be continuously generated. Many hand-held engineering calculators provide statistical routines that compute x2 values at the press of a button after the inputting pairs of 0; and E. values.