Updated Chi-Square Test Suite

Chi-Square Calculator

Run chi-square goodness-of-fit tests, chi-square tests of independence, compute p-values from a given \(\chi^2\) and degrees of freedom, and look up right-tail critical values – all in one unified calculator.

Goodness-of-Fit Test Test of Independence p-Value from \(\chi^2\) Right-Tail Critical Values

Chi-Square Tests, p-Values & Critical Values

This Chi-Square Calculator provides four tools in one interface. You can test whether an observed distribution matches an expected pattern, check independence between categorical variables, convert a chi-square statistic and degrees of freedom into an approximate p-value, and find right-tail chi-square critical values for a chosen significance level \(\alpha\). All formulas in the content are written in LaTeX and rendered with MathJax.

Chi-square hypothesis tests use a right-tail region by default: \(p = P(\chi^2 \ge \chi^2_{\text{obs}})\). The calculator reports this p-value and compares it to \(\alpha\) for a simple “reject / do not reject” decision.

Use the chi-square goodness-of-fit test to compare observed frequencies in \(k\) categories with a set of expected frequencies under a null model. The test statistic is \[ \chi^2 = \sum_{i=1}^{k} \frac{(O_i - E_i)^2}{E_i}, \] with degrees of freedom \(df = k - 1\) when all expected counts are fixed in advance.

Enter the observed and expected counts as comma, space or line-break separated lists. Both lists must have the same length, with all expected values \(E_i > 0\).

Use the chi-square test of independence to check whether two categorical variables in an \(r \times c\) contingency table are associated. For each cell, \[ E_{ij} = \frac{(\text{row total}_i)(\text{column total}_j)}{\text{grand total}}, \] and the test statistic is \[ \chi^2 = \sum_{i=1}^{r} \sum_{j=1}^{c} \frac{(O_{ij} - E_{ij})^2}{E_{ij}}, \] with degrees of freedom \(df = (r - 1)(c - 1)\).

Enter each row on a new line. Within a row, separate cell counts by spaces, commas or semicolons. All rows must have the same number of columns and all counts must be non-negative.

Use this tool when you already have a chi-square statistic \(\chi^2_{\text{obs}}\) and degrees of freedom \(df\), and you want the right-tail p-value \[ p = P(\chi^2 \ge \chi^2_{\text{obs}} \mid df). \]

Use this tool to find a right-tail chi-square critical value \( \chi^2_{\text{crit}} \) such that \[ P(\chi^2 \ge \chi^2_{\text{crit}} \mid df) = \alpha. \] This is useful when you want to construct a rejection region or reproduce critical values from standard chi-square tables.

Chi-Square Calculator – Complete Guide to \(\chi^2\) Tests, p-Values and Critical Values

The Chi-Square Calculator on MyTimeCalculator brings together the most common chi-square tools in a single interface. You can run chi-square goodness-of-fit tests, chi-square tests of independence, convert a chi-square statistic and degrees of freedom into a right-tail p-value, and find chi-square critical values for a chosen significance level \(\alpha\). All key formulas are written using LaTeX and rendered by MathJax for clarity.

Chi-square methods are widely used in statistics to analyze categorical data, compare observed counts to theoretical expectations and test for associations between variables in contingency tables. The core idea is to measure how far the observed frequencies deviate from the expected frequencies under a null hypothesis.

1. Chi-Square Goodness-of-Fit Test

The chi-square goodness-of-fit (GOF) test checks whether an observed frequency distribution across \(k\) categories is consistent with an expected distribution. Suppose you have observed counts \(O_1, O_2, \dots, O_k\) and expected counts \(E_1, E_2, \dots, E_k\) under the null hypothesis \(H_0\). The test statistic is

\[ \chi^2 = \sum_{i=1}^{k} \frac{(O_i - E_i)^2}{E_i}. \]

In the simplest case where the expected counts come from a fully specified model (for example equal probabilities across categories or a fixed theoretical distribution), the degrees of freedom are

\[ df = k - 1. \]

Large values of \(\chi^2\) indicate that the observed and expected frequencies differ more than would be expected from random variation alone. The right-tail p-value \(p = P(\chi^2 \ge \chi^2_{\text{obs}} \mid df)\) is compared to \(\alpha\) to decide whether to reject \(H_0\).

2. Chi-Square Test of Independence

The chi-square test of independence is used with a two-way contingency table to test whether two categorical variables are independent. Suppose you arrange the data in an \(r \times c\) table with observed counts \(O_{ij}\) in row \(i\) and column \(j\). Let the row totals be \(R_i\), the column totals be \(C_j\) and the grand total be \[ N = \sum_{i=1}^{r} \sum_{j=1}^{c} O_{ij}. \]

Under the null hypothesis of independence, the expected count in cell \((i,j)\) is

\[ E_{ij} = \frac{R_i C_j}{N}. \]

The chi-square statistic for the test of independence is then

\[ \chi^2 = \sum_{i=1}^{r} \sum_{j=1}^{c} \frac{(O_{ij} - E_{ij})^2}{E_{ij}}, \]

with degrees of freedom

\[ df = (r - 1)(c - 1). \]

If the right-tail p-value \(P(\chi^2 \ge \chi^2_{\text{obs}} \mid df)\) is less than the chosen significance level \(\alpha\), you reject the null hypothesis of independence and conclude that there is an association between the variables.

3. p-Values and Critical Values for the Chi-Square Distribution

The chi-square distribution with \(df\) degrees of freedom arises as the distribution of the sum of squared independent standard normal variables. For hypothesis testing, two related quantities are important:

  • The right-tail p-value for an observed statistic \(\chi^2_{\text{obs}}\), \[ p = P(\chi^2 \ge \chi^2_{\text{obs}} \mid df), \] which quantifies how extreme the observed value is under the null hypothesis.
  • The right-tail critical value \(\chi^2_{\text{crit}}\) for a given significance level \(\alpha\), defined by \[ P(\chi^2 \ge \chi^2_{\text{crit}} \mid df) = \alpha. \]

The p-value tool in this calculator converts (\(\chi^2_{\text{obs}}, df\)) into an approximate right-tail p-value, while the critical-value tool computes the corresponding \(\chi^2_{\text{crit}}\) for given \(\alpha\) and \(df\). The numerical values are based on standard approximations to the chi-square distribution and are typically accurate enough for many applied and educational settings.

4. How to Use the Chi-Square Calculator

  1. Set the significance level \(\alpha\): use the global \(\alpha\) input (for example \(\alpha = 0.05\)) that applies to the goodness-of-fit and independence tabs.
  2. Goodness-of-fit test: on the Goodness-of-Fit Test tab, enter the observed counts \(O_i\) and the expected counts \(E_i\). Click the calculate button to see \(\chi^2\), \(df\), the right-tail p-value and a decision about \(H_0\).
  3. Independence test: on the Independence Test tab, paste the contingency table with rows separated by new lines and cells separated by spaces or commas. The calculator constructs the expected frequencies, computes \(\chi^2\), \(df = (r - 1)(c - 1)\), the p-value and a decision.
  4. p-value from \(\chi^2\) and \(df\): on the Chi-Square p-Value tab, enter an observed statistic and degrees of freedom to obtain the right-tail p-value.
  5. Critical value: on the Chi-Square Critical Value tab, enter \(df\) and a right-tail \(\alpha\) to find the approximate critical value \(\chi^2_{\text{crit}}\).
  6. Interpret results: for hypothesis tests, compare the p-value to \(\alpha\). If \(p < \alpha\), the result is statistically significant and the calculator indicates that there is evidence to reject \(H_0\).

5. Assumptions and Practical Considerations

Like all statistical procedures, chi-square tests rest on several assumptions that should be checked as closely as possible in practice:

  • Independence of observations: each counted case should contribute to only one cell of the table, and the counts should represent independent observations.
  • Expected counts not too small: a common rule of thumb is that all expected frequencies \(E_i\) should be at least 1 and no more than a small fraction (for example < 20%) of them should be less than 5. If many expected counts are small, exact or simulation-based methods may be more appropriate.
  • Fixed margins (for some designs): in many classical applications, the row and column totals are considered fixed by the design. The chi-square approximations can still work more broadly, but it is good to understand how the data were collected.

For very small sample sizes or sparse tables, Fisher’s exact test or other exact procedures can provide more reliable inference than the chi-square approximation.

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Chi-Square Calculator FAQs

Frequently Asked Questions

Quick answers to common questions about chi-square goodness-of-fit tests, independence tests, p-values and chi-square critical values.

Use a goodness-of-fit test when you have one categorical variable and you want to know whether its distribution across categories matches an expected pattern (for example equal proportions or a known theoretical distribution). Use a test of independence when you have a two-way table crossing two categorical variables and you want to check whether they are associated or independent.

Type each row of the table on a separate line, with cell counts separated by spaces, commas or semicolons. For example, a 2 × 3 table could be entered as:

10 20 30
15 25 35

All rows must have the same number of columns and all entries should be non-negative counts. The calculator automatically computes row totals, column totals, expected frequencies and the chi-square statistic.

The p-value from a chi-square test is the probability, assuming the null hypothesis is true, of obtaining a chi-square statistic at least as large as the one observed. A small p-value (for example \(p < 0.05\)) indicates that such a large discrepancy between observed and expected counts would be unlikely under the null model, providing evidence against \(H_0\). A large p-value suggests the observed pattern is compatible with random variation around the expected frequencies.

The chi-square distribution used for p-values is an approximation that works best when expected counts are not too small. If many \(E_i\) values are near zero, the approximation can become inaccurate. A common guideline is that all expected frequencies should be at least 1, and that only a small proportion should be below about 5. When this condition is violated, exact tests or simulation-based methods are often recommended instead of the standard chi-square test.

Yes. You can enter the same observed and expected frequencies or contingency tables from your exercise and compare the calculator’s \(\chi^2\), degrees of freedom and p-values to your own work. This is especially helpful for verifying multiple-choice answers or intermediate computations. For written work, however, you should still show the underlying formulas and reasoning, since the calculator provides numerical results rather than step-by-step algebraic derivations.

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