The way to identify significant difference by statistics

If we want to know whether the experimental result conforms a certain hypothesis, we can use chi-square method to test if. When we perform the chi-square test, we need to follow the following steps.

Step 1 ĄG Calculation of the Chi-square value
The value of chi-square is the sum of the variants of all items. The variant of an item is the square of the difference between the observed frequency and the expected frequency over the expected value.

.	(Observed frequency - Expected frequency)²
Variant =	------------------------------------------------------
.	Expected frequency

In a cross between a purebred round seed coat pea is crossed with a purebred wrinkled seed coat pea, 85 round peas and 31 wrinkled pea are obtained. Then, there are a total of 116 peas. According to Mendel's Laws, 3 quarters of them should be round-seeded and 1 quarter wrinkled seeded. That means the expected frequency for round seeds should be : 116 X 3/4 = 87, and wrinkled seeds, 116 X 1/4 = 29 pieces. So the data is as follow.

Class	Trait	Observed frequency	Expected frequency
Class 1	Round seeded	85	(85 + 31) X 3/4 = 87
Class 2	Wrinkled seeded	31	(85 + 31) X 1/4 = 29

The variant of the round seeded class = (85 - 87)²/87 = 0.046

The variant of the wrinkled seeded class = (31 - 29)²/29 = 0.138

Chi-square = variant of round seeded class + variant of wrinkled seeded class = 0.046 + 0.138 = 0.184

Step 2 ĄG Find out the degree of freedomĄC
The degree of freedom equals to the number of classes -1. In this case, there are two classes, the round seeded and the wrinkled seeded. So, the degree of freedom is (2 - 1) = 1 .

Step 3 ĄG To find out the critical value of chi-square at least at the confidence level 95% (or 0.95), or at the error level smaller than 0.05 (or 5%). This confidence level (or error level) is generally accepted in statistics.

The following is the chi-square table.

Number of classes	Degree of freedom \ Confidence level	95%	98%	99%
2	1	3.84	5.41	6.64
3	2	5.99	7.82	9.21
4	3	7.82	9.84	11.34
5	4	9.49	11.67	13.28
6	5	11.07	13.39	15.09

If the chi-square value is greater than the value at 95%, we say that we have greater than 95% of confidence to say that the event has a significant difference from the hypothesis. 95% is the basic requirement to say that there is a significant difference. If the chi-square value is smaller than the value at 95%, then, it is not qualified to be said to have significant difference.

The claim of confidence level greater than 99% means to have a greater confidence than that claiming a significant difference at 98%. And, 98% level of confidence means a higher confidence level than that at 95%. If there is no significant difference from that obtained from Mendel's Laws, that means the event conforms the Mendel's Laws.

Exercise

A farmer grew peas and a vines. He crossed the true-bred tall stem pea with the true-bred dwarf stem pea, in F₂, 31 out of 40 gave tall stems and 9 gave dwarf stems. When he did the same to the vines, in F₂, 3100 out of 4000 gave tall stems and 900 gave dwarf stems. Did both of them follow Mendel's Laws ?
Answer : No. Why ?

(15.09.2008)