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Chi-Square Goodness of Fit Test

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Chi-Square Goodness of Fit Test

Definition

Chi-Square Goodness of Fit Test is used to determine whether sample data is consistent with a hypothesized probability distribution. It compares observed frequencies with expected frequencies to test if the differences are statistically significant.

Formula

Test Statistic:

χ2=∑i=1k(Oi−Ei)2Ei\chi^2 = \sum_{i=1}^k \frac{(O_i - E_i)^2}{E_i}

Where:

  • OiO_i = observed frequency for category ii
  • EiE_i = expected frequency for category ii
  • kk = number of categories
  • df=k−1df = k - 1 (degrees of freedom)

Key Assumptions

Random Sample: Data must be randomly sampled
Independence: Observations must be independent
Sample Size: Expected frequency should be ≥ 5 for each category
Categorical Data: Data must be categorical

Practical Example

Step 1: State the Data

Die roll frequencies from 60 rolls:

FaceObserved (O)Expected (E)(O-E)²/E
110100.000
28100.400
312100.400
410100.000
515102.500
65102.500
Step 2: State Hypotheses
  • H0H_0: The die is fair (equal probabilities)
  • HaH_a: The die is not fair
  • α=0.05\alpha = 0.05
Step 3: Calculate Test Statistic

Chi-square statistic:

χ2=∑(Oi−Ei)2Ei=5.800\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i} = 5.800

Degrees of freedom = 6−1=56 - 1 = 5

Step 4: Determine Critical Value

At α=0.05\alpha = 0.05 with df=5df = 5:

χ52=11.070\chi^2_5 = 11.070
Step 5: Calculate P-value

Using chi-square distribution:

p-value=0.326p\text{-value} = 0.326
Step 6: Draw Conclusion

Since χ2=5.800<χ52=11.070\chi^2 = 5.800 < \chi^2_5 = 11.070 and pp-value = 0.326>0.050.326 \gt 0.05, we fail to reject H0H_0. There is insufficient evidence to conclude that the die is unfair.

Effect Size

Cramer's V for goodness of fit test:

V=χ2n(k−1)V = \sqrt{\frac{\chi^2}{n(k-1)}}

Where:

  • χ2\chi^2 = chi-square statistic
  • nn = total sample size
  • kk = number of categories

For our example:

V=5.80060(6−1)=0.139V = \sqrt{\frac{5.800}{60(6-1)}} = 0.139

Interpretation guidelines:

  • Small effect: V≈0.10V \approx 0.10
  • Medium effect: V≈0.30V \approx 0.30
  • Large effect: V≈0.50V \approx 0.50

With V = 0.139, this indicates a small to medium effect size, suggesting that while there are some deviations from the expected frequencies, they are relatively modest in practical terms.

Code Examples

R
1# Chi-Square Goodness of Fit Test
2# Observed frequencies
3observed <- c(10, 8, 12, 10, 15, 5)
4
5# Perform chi-square test
6result <- chisq.test(
7  observed,
8  p = rep(1/6, 6)  # Equal probabilities for each face
9)
10
11print(result)
Python
1# Chi-Square Goodness of Fit Test
2from scipy.stats import chisquare
3
4# Observed frequencies
5observed = [10, 8, 12, 10, 15, 5]
6
7# Expected frequencies (equal probabilities)
8n = sum(observed)  # total observations
9p = 1/6  # probability for each face
10expected = [n * p] * 6
11
12# Perform chi-square test
13stat, pvalue = chisquare(observed, expected)
14
15print(f'Chi-square statistic: {stat:.4f}')
16print(f'p-value: {pvalue:.4f}')
17
18# Calculate degrees of freedom
19df = len(observed) - 1
20
21# Calculate critical value
22from scipy.stats import chi2
23critical_value = chi2.ppf(0.95, df)
24print(f'Critical value (α=0.05): {critical_value:.4f}')

Alternative Tests

Consider these alternatives:

  • G-test: Alternative to chi-square for categorical data
  • Exact Multinomial Test: For small sample sizes

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