Chi-Square Test
IMPORTANT TERMS
- Observed Frequency (O): The actual data collected from an experiment.
- Expected Frequency (E): The frequency we expect to get if our hypothesis is correct.
- Degree of Freedom (df): The number of independent comparisons possible, usually calculated as (Number of categories – 1).
- Null Hypothesis (H₀): Assumes no difference between observed and expected results.
- Alternative Hypothesis (H₁): Assumes a significant difference exists between observed and expected results.
INTRODUCTION
The Chi-Square (χ²) Test is a statistical method used to check whether the difference between observed data and expected data is due to chance or due to some real factor. In genetics, it is widely used to test if the observed ratios of offspring (for example, 3:1, 9:3:3:1) agree with Mendelian inheritance patterns.
CHARACTERISTICS OF THE TEST
- It is a non-parametric test (does not depend on population parameters like mean or variance).
- It compares observed vs. expected frequencies.
- The test value is always positive (since it is based on squared values).
- It is widely used for categorical data (not for continuous data).
CHI SQUARE DISTRIBUTION
- The distribution of χ² values is positively skewed (tail on the right).
- As the sample size increases, the distribution becomes more symmetrical.
- The shape depends on the degree of freedom.
APPLICATIONS OF CHI SQUARE TEST
- Genetics: To test if offspring ratios match Mendelian laws.
- Goodness of Fit: To check if observed data fit a theoretical distribution.
- Test of Independence: To find if two variables are related (example: blood group vs. gender).
- Homogeneity Test: To check if samples come from the same population.
CALCULATION OF THE CHI SQUARE
The formula is:
χ² = Σ (O – E)² / E
Where O = Observed frequency, E = Expected frequency
Steps:
- Write observed (O) and expected (E) values.
- Find difference (O – E).
- Square the difference.
- Divide by expected value.
- Add all values to get χ².
CONDITION FOR THE APPLICATION OF THE TEST
- Data should be in frequency form (not percentages).
- Observations should be independent.
- Expected frequency should be at least 5 in each category.
- Sample size should be reasonably large.
EXAMPLE (Genetics)
Suppose we cross two heterozygous pea plants (Tt × Tt) for tallness.
Expected ratio = 3 Tall : 1 Dwarf.
Suppose in an experiment, we get:
Tall = 62, Dwarf = 38 (Total = 100).
Step 1: Expected values
- Tall = 3/4 × 100 = 75
- Dwarf = 1/4 × 100 = 25
Step 2: Table Calculation
| Category | Observed (O) | Expected (E) | O – E | (O – E)² | (O – E)² / E |
|---|---|---|---|---|---|
| Tall | 62 | 75 | -13 | 169 | 2.25 |
| Dwarf | 38 | 25 | +13 | 169 | 6.76 |
| Total χ² | 9.01 | ||||
Step 3: Compare with χ² table
df = (2 categories – 1) = 1
At 5% level, critical χ² = 3.84
Since 9.01 > 3.84 → The difference is significant.
Conclusion: The observed ratio does not follow Mendelian 3:1 perfectly.
YATE’S CORRECTION FOR CONTINUITY
Applied when df = 1 and sample size is small.
Adjusted formula:
χ² = Σ (|O – E| – 0.5)² / E
This correction reduces the overestimation of significance in small samples.
LIMITATIONS OF THE TEST
- Cannot be used if expected frequency is very small (<5).
- Only works for frequency data, not for means or percentages.
- Sensitive to sample size – large samples may give significant results even for small differences.
- Does not show the direction of deviation (only that difference exists).
USE OF CHI-SQUARE TEST IN GENETICS (Simple Words)
In genetics, scientists use the χ² test to check whether the number of offspring with different traits matches the expected Mendelian ratios.
For example, if Mendel expected a 3:1 ratio (Tall : Dwarf) but got 62:38, the χ² test helps decide:
- Is this difference just by chance?
- Or does it mean that Mendel’s ratio doesn’t work here?
👉 In short: Chi-square test tells us whether our experimental genetic data agree with theoretical laws of inheritance or not.
