Chi Squared Distribution Calculator

Chi-Squared Statistic Calculator

Use this calculator to compute the Chi-Squared (χ²) statistic for goodness-of-fit tests, independence tests, or homogeneity tests. Enter your observed and expected frequencies as comma-separated lists, and specify the degrees of freedom.

function calculateChiSquared() { var observedStr = document.getElementById("observedFrequencies").value; var expectedStr = document.getElementById("expectedFrequencies").value; var dfInput = document.getElementById("degreesOfFreedom").value; var resultDiv = document.getElementById("chiSquaredResult"); resultDiv.innerHTML = ""; // Clear previous results var observed = observedStr.split(',').map(Number); var expected = expectedStr.split(',').map(Number); var df = parseInt(dfInput); // — Input Validation — if (observed.some(isNaN) || expected.some(isNaN)) { resultDiv.innerHTML = "Error: Please enter valid numbers for observed and expected frequencies."; return; } if (observed.length === 0 || expected.length === 0) { resultDiv.innerHTML = "Error: Please enter at least one observed and expected frequency."; return; } if (observed.length !== expected.length) { resultDiv.innerHTML = "Error: The number of observed frequencies must match the number of expected frequencies."; return; } if (expected.some(val => val <= 0)) { resultDiv.innerHTML = "Error: Expected frequencies must be positive numbers."; return; } if (isNaN(df) || df <= 0) { resultDiv.innerHTML = "Error: Degrees of Freedom must be a positive integer."; return; } // — Chi-Squared Calculation — var chiSquaredStatistic = 0; for (var i = 0; i < observed.length; i++) { chiSquaredStatistic += Math.pow((observed[i] – expected[i]), 2) / expected[i]; } // — Display Results — resultDiv.innerHTML = "Calculated Chi-Squared (χ²) Statistic: " + chiSquaredStatistic.toFixed(4) + "" + "Degrees of Freedom (df): " + df + "" + "To find the p-value associated with this Chi-Squared statistic, you would typically compare it to a Chi-Squared distribution table or use statistical software with the given degrees of freedom."; }

Understanding the Chi-Squared (χ²) Distribution

The Chi-Squared (χ²) distribution is a fundamental concept in inferential statistics, widely used for hypothesis testing. It's a non-symmetrical, positively skewed distribution that arises when summing the squares of independent standard normal random variables. Its shape is determined by a single parameter: the degrees of freedom (df).

What is the Chi-Squared Statistic?

The Chi-Squared statistic (χ²) is a measure of the difference between observed frequencies and expected frequencies in one or more categories. In simpler terms, it quantifies how much your observed data deviates from what you would expect if a certain hypothesis were true. A larger Chi-Squared value indicates a greater discrepancy between observed and expected data.

The formula for the Chi-Squared statistic is:

χ² = Σ [(Oᵢ – Eᵢ)² / Eᵢ]

Where:

• Oᵢ = Observed frequency in category i
• Eᵢ = Expected frequency in category i
• Σ = Summation across all categories

Common Uses of the Chi-Squared Test

The Chi-Squared test is versatile and is primarily used for:

1. Goodness-of-Fit Test: To determine if a sample data matches a population or a theoretical distribution. For example, testing if the observed number of candies of different colors in a bag matches the manufacturer's stated proportions.
2. Test of Independence: To assess whether there is a statistically significant association between two categorical variables. For instance, testing if there's a relationship between gender and preference for a certain political candidate.
3. Test of Homogeneity: To determine if two or more independent samples come from the same population or have the same distribution for a single categorical variable. For example, comparing the distribution of opinions on a new policy across different age groups.

Degrees of Freedom (df)

The degrees of freedom (df) represent the number of independent values that can vary in a data set. In the context of Chi-Squared tests, df is crucial because it dictates the shape of the Chi-Squared distribution and, consequently, the critical value used for hypothesis testing.

• For a Goodness-of-Fit Test: df = (number of categories) – 1 – (number of parameters estimated from the data). Often, if no parameters are estimated, it's simply (number of categories) – 1.
• For a Test of Independence/Homogeneity (contingency table): df = (number of rows – 1) × (number of columns – 1).

Interpreting the Results

Once you have the Chi-Squared statistic and the degrees of freedom, the next step is to determine the p-value. The p-value tells you the probability of observing a Chi-Squared statistic as extreme as, or more extreme than, the one calculated, assuming the null hypothesis is true.

• If p-value < α (significance level, e.g., 0.05): You reject the null hypothesis. This suggests a statistically significant difference between observed and expected frequencies, or a significant association between variables.
• If p-value ≥ α: You fail to reject the null hypothesis. This suggests that any observed differences or associations could be due to random chance.

This calculator provides the Chi-Squared statistic and degrees of freedom, which are the necessary inputs to find the p-value using a Chi-Squared distribution table or statistical software.

Example Calculation: Goodness-of-Fit

Imagine a company claims its product comes in three colors: Red, Green, and Blue, in equal proportions. You buy a sample of 150 products and observe the following:

• Observed Frequencies (Oᵢ): Red = 50, Green = 60, Blue = 40
• Expected Frequencies (Eᵢ): If proportions are equal, then 150 / 3 = 50 for each color. So, Red = 50, Green = 50, Blue = 50
• Degrees of Freedom (df): Number of categories – 1 = 3 – 1 = 2

Using the calculator with these values:

• Observed Frequencies: 50,60,40
• Expected Frequencies: 50,50,50
• Degrees of Freedom: 2

The calculator would compute the Chi-Squared statistic as:

• For Red: (50 – 50)² / 50 = 0 / 50 = 0
• For Green: (60 – 50)² / 50 = 10² / 50 = 100 / 50 = 2
• For Blue: (40 – 50)² / 50 = (-10)² / 50 = 100 / 50 = 2
• Total Chi-Squared (χ²) = 0 + 2 + 2 = 4

With χ² = 4 and df = 2, you would then consult a Chi-Squared table or software to find the p-value. At a 0.05 significance level, the critical value for df=2 is approximately 5.991. Since 4 < 5.991, the p-value would be greater than 0.05, leading to a failure to reject the null hypothesis. This suggests that the observed distribution of colors is not significantly different from the expected equal distribution.