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For determining chi-square or goodness of fit, the size of the population must be considered. Suppose in one cross of tall and dwarf plants, out of the 100 plants of F2, 70 are tall and 30 dwarf instead of 75 and 25 as expected from a 3: 1 ratio. Obviously there is a deviation of 5 from the normal.
In a second cross where 1000 plants of F2 progeny showed the same numerical deviation of 5, there were 745 tall plants and 255 dwarf ones. In the third cross the 1000 plants of F2 generation appeared in the same proportion as in the first cross (70: 30 or 7: 3) so that 700 plants were tall and 300 dwarf.
The chi-square formula given below will show whether the differences in the observed results of the three crosses are significantly different from the 3: 1 ratio or not (Table 7.1).
where d represents deviation from expected ratio, e the expected ratio, and Σ is the sum. The smaller the chi-square value the more likely it is that deviation has occurred due to chance.
Now to find out whether the differences between expected and observed results are due to chance alone or not, we must be familiar with two more concepts namely degree of freedom and level of significance.
Degrees of Freedom:
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The number of degrees of freedom is calculated as the number of classes whose value is required to describe the outcome from all classes. The concept of degrees of freedom is important in experiments and genetic ratios because one must consider the total number of observed individuals in the experiment as a fixed or given quantity. This fixed quantity is composed of one or more classes some of which are variable.
In the experiment between tall and dwarf pea plants there are only two classes, tall and dwarf. As soon as the number of one class is set, the other can be determined. Thus when two classes are scored, there is one degree of freedom.
In an experiment where three classes are scored, there are two degrees of freedom, and so on. The rule states that for the kind of genetic experiments described, the degrees of freedom are equal to one less than the number of classes.
Level of Significance:
In the experiment described the actual ratio departs from that which is expected. We must now determine how significant is this discrepancy so that we can decide to accept or reject the results.
Small discrepancies are not significant; large discrepancies are significant and lead to rejection of a result or hypothesis. Therefore values are assigned to these two kinds of discrepancies—the large discrepancies are the largest 5% and small discrepancies are remaining 95%.
On this basis if the discrepancy lies in the large class it is significant and the result may be discarded. The 5% frequency value that enables us to reject the result is called the 5% level of significance. The level of significance can be changed.
If 5% is too high we can decide on a low level of significance say 1%. In this case it is not so easy to reject a result. Contrarily, if we decide on a high level of significance say 10%, it is easier to reject a result. Usually the accepted level of significance is between the two extremes, that is 5%.
After determining the degrees of freedom in an experiment and deciding on the level of significance, the actual size of the discrepancy between expected and observed is found by chi-square.
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Statisticians have prepared tables that relate the number of degrees of freedom with the probability that particular groups of chi-square values will be found (Table 2). For a more detailed table refer to Table IV in Fisher and Yates, 1963.
We can now examine the results of the experiment described in Table 1. The chi-square values of the first two crosses are 1.33 and 0.133. Both are acceptable discrepancies because these values are smaller than the chi-square value for one degree of freedom given as 3.84 in Table 2.
The results of the first two crosses therefore may be considered to be consistent with Mendel’s hypo thesis, the difference between expected and observed being due to chance.
