G-Test Goodness of Fit Calculator

Introduction

The G-test goodness of fit calculator evaluates whether an observed frequency distribution aligns with a theoretical expected distribution. This likelihood-ratio test is utilised in statistical analysis to determine if the discrepancies between observed counts $O_{i}$ and expected counts $E_{i}$ are significant or merely due to random variation, resulting in a calculated p-value $p$ .

What this calculator does

This tool processes datasets containing observed and expected frequencies across multiple categories. It automatically scales expected values if totals differ, then computes the G-statistic, degrees of freedom, and the associated probability density. The output provides a clear determination of statistical significance based on a selected alpha level $α$ , accompanied by a step-by-step breakdown of the summation process and a visual probability density curve.

Formula used

The G-statistic is calculated by summing the products of observed counts and the natural logarithm of the ratio between observed and expected frequencies for each category $i$ . The degrees of freedom $df$ are determined by subtracting one from the total number of categories $n$ .

G = 2 \sum O_{i} ln (\frac{O_{i}}{E_{i}})

df = n - 1

How to use this calculator

1. Enter the observed counts as a comma-separated list.
2. Input the corresponding expected counts for each category.
3. Select the desired significance level and decimal precision.
4. Execute the calculation to view the G-statistic and p-value results.

Example calculation

Scenario: A biological researcher examines the distribution of four distinct phenotypes in a population study to verify if they follow a specific predicted genetic ratio.

Inputs: Observed counts $O_{i}$ are 45, 55, 60, 70; Expected counts $E_{i}$ are 50, 50, 60, 60.

Working:

Step 1: $G = 2 \sum O_{i} ln (\frac{O_{i}}{E_{i}})$

Step 2: $2 \times [45 ln (\frac{45}{50}) + 55 ln (\frac{55}{50}) + 60 ln (\frac{60}{60}) + 70 ln (\frac{70}{60})]$

Step 3: $2 \times [- 4.74 + 5.24 + 0 + 10.79]$

Step 4: $2 \times 11.29$

Result: G = 22.58

Interpretation: The resulting G-statistic is compared against the Chi-square distribution to determine if the p-value falls below the significance threshold.

Summary: The high G-statistic suggests a discrepancy between the data and the model.

Understanding the result

The primary output is the p-value, which indicates the probability of obtaining the observed data if the null hypothesis is true. If the p-value is less than the alpha level, the difference is statistically significant, suggesting the observed distribution does not fit the expected model within expected random limits.

Assumptions and limitations

The G-test assumes that observations are independent and that all expected frequencies are greater than zero. It is generally recommended for sample sizes where the total count is sufficiently large to provide a stable approximation of the Chi-square distribution.

Common mistakes to avoid

Users should ensure that the number of observed values exactly matches the number of expected values. Another common error is including non-numeric characters or scientific notation, which the system rejects to ensure computational stability and accuracy during the logarithmic summation process.

Sensitivity and robustness

The calculation is sensitive to categories with zero expected frequencies where observations exist, as this creates an undefined ratio. Small variations in counts generally result in stable shifts in the G-statistic, though the logarithmic nature of the formula means that large deviations in individual categories can disproportionately influence the final outcome.

Troubleshooting

If an error appears regarding invalid characters, check for non-numeric symbols or spaces. If the G-statistic is zero, it indicates a perfect match between observed and scaled expected values. Ensure the total dataset size does not exceed 1,000 points to remain within the memory limits of the processor.

Frequently asked questions

What happens if my totals do not match?

The calculator applies an adjustment scalar to the expected values, scaling them proportionally so their sum matches the total observed count before performing the calculation.

Can I use negative numbers?

No, counts represent frequencies and must be zero or positive. Negative inputs will trigger a validation error as they are mathematically invalid for this statistical test.

Why is scientific notation restricted?

To maintain strict input validation and prevent potential precision errors in the logarithmic steps, the calculator requires standard decimal or integer formats for all data points.

Where this calculation is used

The G-test is widely used in social research to analyse survey responses against demographic benchmarks and in environmental science to compare species distributions against habitat models. In population studies, it serves as an alternative to the Pearson Chi-square test, particularly in complex experimental designs. It is also frequently employed in sports analysis to evaluate if team performance aligns with historical probability models. This method is fundamental in probability theory for testing the adequacy of various mathematical models in representing real-world frequency data.

Results are based on standard mathematical and statistical methods and may involve rounding or approximation. If precise accuracy is required, please verify results independently. See full disclaimer.