Chi-Square (χ²) Test for Goodness of Fit

The Chi-Square (χ²) Test for Goodness of Fit is a non-parametric hypothesis test used to assess whether an observed frequency distribution differs significantly from an expected theoretical frequency distribution across categorical groups.

Chi-Square Test Statistic Formula:

\chi^2 = \sum_{i=1}^{k} \frac{(O_i - E_i)^2}{E_i} \quad (df = k - 1)

Where:

$O_i$ is the observed frequency in category $i$ .
$E_i$ is the expected theoretical frequency under $H_0$ .
$k$ is the number of categories.

Example of Chi-Square Test

Observed Counts (k = 4)

O = [25, 19, 16, 20], \quad E_i = 20

Test at $\alpha = 0.05$

Step 1 & 2: Calculate Component Residuals

\frac{(25-20)^2}{20} = 1.25, \quad \frac{(19-20)^2}{20} = 0.05

\frac{(16-20)^2}{20} = 0.80, \quad \frac{(20-20)^2}{20} = 0.00

Step 3 & 4: Compute Chi-Square & Compare

\chi^2 = 1.25 + 0.05 + 0.80 + 0.00 = 2.10

\chi^2_{\text{calc}} = 2.10 < 7.815

Conclusion

\boxed{\text{Fail to Reject } H_0 \text{ at } \alpha = 0.05}

Therefore, because $\chi^2_{\text{calc}} = 2.10 < 7.815$ , we fail to reject the null hypothesis and conclude that there is no statistically significant difference between observed and expected frequencies.

Interactive Chi-Square Calculator

Chi-Square ( $\\chi^2$ ) Goodness of Fit Solver

PreviousStudent's t-Test

Unit 5•27 of 28

NextF-Test

Chi-Square (χ²) Test for Goodness of Fit

Chi-Square Test Statistic Formula:

\chi^2 = \sum_{i=1}^{k} \frac{(O_i - E_i)^2}{E_i} \quad (df = k - 1)

Where:

$O_i$ is the observed frequency in category $i$ .
$E_i$ is the expected theoretical frequency under $H_0$ .
$k$ is the number of categories.

Example of Chi-Square Test

Observed Counts (k = 4)

O = [25, 19, 16, 20], \quad E_i = 20

Test at $\alpha = 0.05$

Step 1 & 2: Calculate Component Residuals

\frac{(25-20)^2}{20} = 1.25, \quad \frac{(19-20)^2}{20} = 0.05

\frac{(16-20)^2}{20} = 0.80, \quad \frac{(20-20)^2}{20} = 0.00

Step 3 & 4: Compute Chi-Square & Compare

\chi^2 = 1.25 + 0.05 + 0.80 + 0.00 = 2.10

\chi^2_{\text{calc}} = 2.10 < 7.815

Conclusion

\boxed{\text{Fail to Reject } H_0 \text{ at } \alpha = 0.05}

Interactive Chi-Square Calculator

Chi-Square ( $\\chi^2$ ) Goodness of Fit Solver

PreviousStudent's t-Test

Unit 5•27 of 28

NextF-Test

Chi-Square (χ²) Test for Goodness of Fit

Example of Chi-Square Test

Step 1 & 2: Calculate Component Residuals

Step 3 & 4: Compute Chi-Square & Compare

Conclusion

Interactive Chi-Square Calculator

Chi-Square (chi2\\chi^2chi2) Goodness of Fit Solver

NETZ Engine

Chi-Square (χ²) Test for Goodness of Fit

Example of Chi-Square Test

Step 1 & 2: Calculate Component Residuals

Step 3 & 4: Compute Chi-Square & Compare

Conclusion

Interactive Chi-Square Calculator

Chi-Square (chi2\\chi^2chi2) Goodness of Fit Solver

Chi-Square ( $\\chi^2$ ) Goodness of Fit Solver

Chi-Square ( $\\chi^2$ ) Goodness of Fit Solver