Calculator
Example Data Table
| Category | Observed Count | Expected Proportion | Expected Count For N = 120 |
|---|---|---|---|
| Red | 45 | 0.40 | 48 |
| Blue | 32 | 0.30 | 36 |
| Green | 18 | 0.20 | 24 |
| Yellow | 25 | 0.10 | 12 |
Formula Used
The calculator uses the chi squared goodness of fit statistic:
χ² = Σ((Oᵢ − Eᵢ)² / Eᵢ)
Oᵢ is the observed count. Eᵢ is the expected count. The degrees of freedom are calculated as:
df = k − 1 − m
k is the number of categories. m is the number of estimated parameters. The p value comes from the upper tail of the chi squared distribution.
How To Use This Calculator
- Enter category labels in the first box.
- Enter observed counts in the same order.
- Select the expected pattern type.
- Enter expected proportions or expected counts when needed.
- Set alpha, such as 0.05.
- Enter estimated parameters if the expected pattern was fitted from data.
- Press Calculate Test.
- Review the p value, decision, residuals, and contribution table.
Understanding The Chi Squared GOF Test
A chi squared goodness of fit test checks one simple question. Do observed category counts match an expected pattern? The pattern may be equal shares, known proportions, or planned expected counts. This calculator keeps those choices in one place. It also shows each category contribution, so the final statistic is easier to audit.
When The Test Helps
Use this test when data is counted in categories. Examples include survey choices, color defects, page clicks, package sizes, or grade bands. Each observation should belong to one category only. The sample should be collected without bias. Expected counts should be large enough for a stable approximation. A common rule is that most expected counts should be five or more.
What The Output Means
The chi squared statistic grows when observed and expected counts move apart. Small values suggest a closer match. The p value estimates how surprising the gap is, assuming the expected pattern is true. If the p value is less than alpha, the calculator reports rejection of the null hypothesis. That means the data gives evidence against the expected distribution.
Residuals And Contributions
Advanced users should inspect residuals, not only the final decision. A positive residual means the observed count is higher than expected. A negative residual means it is lower. Large absolute residuals identify categories that drive the result. Contributions show how much each row adds to the statistic. These details are useful for reports, dashboards, and quality reviews.
Practical Notes
The degrees of freedom equal the number of categories minus one, then minus any parameters estimated from the data. If you estimate one parameter before testing, reduce the degrees of freedom by one. Do not use percentages as observed counts. Enter real counts. For expected proportions, enter ratios such as 0.25 values or percentages such as 25 values. The calculator normalizes proportions before building expected counts.
Reporting The Test
A clear report names the test, sample size, degrees of freedom, statistic, p value, alpha, and decision. Add the table of expected counts and residuals. Explain any small expected counts. This makes the conclusion transparent, reproducible, and useful for statistical communication. Keep exports for review and shared team decisions after testing ends.
FAQs
What is a chi squared GOF test?
It is a statistical test that compares observed category counts with expected counts. It checks whether the sample follows a planned or theoretical distribution.
Can I use percentages as observed values?
No. Observed values should be real counts. Percentages can be used only for expected proportions, because the calculator converts them into expected counts.
What does the p value mean?
The p value shows how unusual the observed differences are when the expected distribution is assumed true. A smaller p value gives stronger evidence against that distribution.
What alpha should I use?
Many reports use 0.05. You may choose 0.01 for stricter evidence or 0.10 for a more sensitive screening test.
Why are expected counts important?
Expected counts are the baseline for comparison. Very small expected counts can make the chi squared approximation less reliable and should be reviewed carefully.
What are degrees of freedom?
Degrees of freedom show how many category values can vary after totals and estimated parameters are considered. They affect the p value calculation.
What are residuals?
Residuals show the direction and size of each category difference. Positive values are above expected. Negative values are below expected.
Why download CSV or PDF reports?
CSV files help with spreadsheet review. PDF reports are useful for sharing results, saving records, and documenting statistical decisions.