The chi-squared (χ 2 ) test
The chi-squared (χ²) test is a statistical test used to analyze data mathematically, allowing you to draw conclusions from the data with greater confidence. It is specifically used to determine if observed results are statistically different from expected results. In essence, it assesses how well observed results "fit" with theoretical expected results.
Purpose and Null Hypothesis
The χ² test is applied when you have categorical (grouped) data.
Before performing the test, a null hypothesis must be formulated. This hypothesis always states that there is no significant difference between the observed and expected results, or no correlation between the things being investigated.
The purpose of the test is to determine if any difference between observed and expected results is simply due to chance, or if it's a significant difference indicating that the underlying theory might be wrong.
Calculation of the Chi-Squared Value
The χ² value is calculated using the formula: χ² = Σ (O – E)² / E.
O represents the observed result (the actual number recorded from the experiment).
E represents the expected result (predicted by the theory). For genetic crosses, 'E' is worked out using the total number of offspring divided by the ratio total, multiplied by the predicted ratio for each phenotype.
Σ means "the sum of".
The calculation involves several steps:
Determine the expected number (E) for each category.
Calculate the difference between observed and expected (O - E) for each category.
Square this difference (O - E)² to eliminate negative signs.
Divide each squared difference by the expected result (O - E)² / E.
Sum all these values to get the final χ² value.
Degrees of Freedom
To interpret the χ² value, you need to determine the degrees of freedom (d.f.).
For the χ² test, the degrees of freedom are calculated as the number of classes (phenotypes or categories) minus one. For example, if there are two phenotypes, d.f. = 2 - 1 = 1.
Critical Value and Interpretation
The calculated χ² value is compared to a critical value from a chi-squared table.
Biologists typically use a probability (P) value of 0.05 (5%) as the critical level. This means there's a 5% chance that the observed differences are due to chance alone.
Decision Rule:
If the calculated χ² value is larger than or equal to the critical value at P = 0.05, then there is a significant difference between the observed and expected results. In this case, the null hypothesis is rejected. This implies something other than chance is causing the difference.
If the calculated χ² value is smaller than the critical value at P = 0.05, then there is no significant difference between the observed and expected results. The null hypothesis cannot be rejected. This means the difference is likely due to chance.
It is important to note that you can never "prove" a null hypothesis is true; you can only "fail to reject" it, meaning the evidence does not give a reason to believe it is wrong.
Applications
The χ² test is commonly used in genetics to test theories about the inheritance of characteristics. For example, it can determine if observed phenotypic ratios from monohybrid or dihybrid crosses align with expected Mendelian ratios (e.g., 3:1 or 9:3:3:1). If the χ² value is large, it suggests that factors like epistasis or sex linkage might be affecting the outcome.
It is also used in ecology to compare observed and expected results in categorical data.
Considerations
The exact formula for χ² and critical value tables are typically provided in exams.
While calculation is important for understanding, interpreting the results is often the focus of examination questions.
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