Calculator Inputs
Example Data Table
| Category | Observed Count | Expected Count | Contribution Formula |
|---|---|---|---|
| Group A | 18 | 20 | (18 - 20)² / 20 |
| Group B | 22 | 20 | (22 - 20)² / 20 |
| Group C | 20 | 20 | (20 - 20)² / 20 |
| Group D | 25 | 20 | (25 - 20)² / 20 |
| Group E | 15 | 20 | (15 - 20)² / 20 |
Formula Used
The main chi square statistic is calculated as:
χ² = Σ (O - E)² / E
O means observed count. E means expected count. The calculator adds every category contribution. For an independence test, expected count equals row total times column total divided by grand total.
Expected cell = (Row Total × Column Total) / Grand Total
Goodness of fit degrees freedom equal categories minus one minus estimated parameters. Independence test degrees freedom equal rows minus one times columns minus one. The p value is found from the upper tail of the chi square distribution. The critical value is matched to the selected alpha level.
How to Use This Calculator
- Select goodness of fit or independence mode.
- Enter observed values for your categories or table cells.
- Enter expected values, or choose equal expected counts.
- Set the alpha level for your hypothesis test.
- Add estimated parameters when using a fitted distribution.
- Choose optional scaling or Yates correction when needed.
- Press the calculate button to show results above the form.
- Download the CSV or PDF report for records.
Chi Square Statistic Guide
What the Statistic Measures
A chi square statistic measures distance between observed counts and expected counts. It is useful when data comes as frequencies. The method does not compare averages. It compares count patterns. A larger statistic shows a larger gap from the expected pattern. A small statistic suggests the counts are close to expectation. This calculator supports two common uses. Goodness of fit checks one categorical variable. Independence testing checks relationships between two categorical variables.
Why Expected Counts Matter
Expected counts are the baseline for the test. They may come from theory. They may come from a fair split. They may also come from row and column totals. The calculator can scale expected counts. This helps when expected proportions are entered instead of final counts. Expected counts should not be zero. Very small expected counts can weaken the test. Many courses prefer expected values of five or more.
Reading the P Value
The p value gives upper tail probability. It answers a direct question. How unusual is this statistic under the null hypothesis? A p value below alpha usually rejects the null. A p value above alpha does not prove the null. It only shows weak evidence against it. The calculator also gives a critical value. This supports traditional table based decisions.
Residuals and Effect Size
Contributions show which cells create the statistic. Residuals show direction and size. Positive residuals mean observed counts are higher than expected. Negative residuals mean counts are lower. Goodness of fit uses Cohen's W. Independence tests use Phi or Cramer's V. These values help explain practical strength. Always report context with the number. Statistics are stronger when design and data quality are sound.
FAQs
1. What is a chi square statistic?
It is a sum of squared differences between observed and expected counts. Each squared difference is divided by its expected count.
2. When should I use goodness of fit mode?
Use it when you have one categorical variable. It checks whether observed category counts match expected counts or expected proportions.
3. When should I use independence mode?
Use it for a contingency table. It checks whether two categorical variables appear related through their count distribution.
4. What does the p value mean?
The p value is the chance of getting this statistic, or a larger one, if the null hypothesis is true.
5. What are degrees of freedom?
Degrees of freedom describe how many count values can vary after totals and model restrictions are considered.
6. What is Yates correction?
Yates correction is a continuity adjustment often used for 2 by 2 tables. It can make the statistic more conservative.
7. Why are expected counts important?
Expected counts define the comparison baseline. Zero or very small expected counts can make chi square results unreliable.
8. Can I export the results?
Yes. Use the CSV button for spreadsheet work. Use the PDF button for a simple printable report.