Calculate Chi Square Online

Chi-Square Goodness-of-Fit Calculator

Use this calculator to determine the Chi-Square (χ²) statistic and degrees of freedom for a goodness-of-fit test. This test helps you compare observed frequencies from a sample to expected frequencies from a theoretical distribution.

Understanding the Chi-Square Goodness-of-Fit Test

The Chi-Square (χ²) goodness-of-fit test is a statistical hypothesis test used to determine if a sample of data matches a population with a known distribution. In simpler terms, it helps you decide if your observed data differs significantly from what you would expect to see based on a specific theory or hypothesis.

When to Use It?

• When you have one categorical variable from a single population.
• When you want to compare the observed frequencies of categories with the expected frequencies.
• Examples: Testing if the distribution of M&M colors in a bag matches the manufacturer's stated proportions, or if the number of car accidents per day follows a Poisson distribution.

The Chi-Square Formula

The Chi-Square statistic is calculated using the following formula:

χ² = Σ [(Oᵢ – Eᵢ)² / Eᵢ]

Where:

• Σ (Sigma) means "sum of"
• Oᵢ is the observed frequency (the actual count) for category i
• Eᵢ is the expected frequency (the count you would expect based on your hypothesis) for category i

The formula essentially measures the discrepancy between what you observed and what you expected. A larger Chi-Square value indicates a greater difference between observed and expected frequencies.

Degrees of Freedom (df)

Degrees of Freedom (df) is a crucial component for interpreting the Chi-Square statistic. For a goodness-of-fit test, it is calculated as:

df = (Number of Categories) – 1

The degrees of freedom tell you how many values in the final calculation of a statistic are free to vary. You will use the Chi-Square value and the degrees of freedom to look up a p-value in a Chi-Square distribution table.

Interpreting the Results

Once you have the Chi-Square value and the degrees of freedom, you compare your calculated χ² to a critical value from a Chi-Square distribution table (or use statistical software to get a p-value).

• P-value: The p-value tells you the probability of observing a Chi-Square statistic as extreme as, or more extreme than, the one calculated, assuming the null hypothesis (that there is no significant difference between observed and expected frequencies) is true.
• Significance Level (α): This is typically set at 0.05 (5%).
• Conclusion:
  • If p < α (e.g., p < 0.05), you reject the null hypothesis. This suggests there is a statistically significant difference between your observed and expected frequencies.
  • If p ≥ α (e.g., p ≥ 0.05), you fail to reject the null hypothesis. This suggests there is no statistically significant difference between your observed and expected frequencies.

Example Scenario: Coin Toss

Imagine you toss a coin 100 times and observe 60 heads and 40 tails. You want to test if the coin is fair. A fair coin would be expected to yield 50 heads and 50 tails.

• Category 1 (Heads): Observed = 60, Expected = 50
• Category 2 (Tails): Observed = 40, Expected = 50

Using the calculator:

• For Heads: (60 – 50)² / 50 = 10² / 50 = 100 / 50 = 2
• For Tails: (40 – 50)² / 50 = (-10)² / 50 = 100 / 50 = 2

Chi-Square (χ²) = 2 + 2 = 4
Degrees of Freedom (df) = 2 categories – 1 = 1

With χ² = 4 and df = 1, if you look up a Chi-Square table, the p-value is less than 0.05 (specifically, around 0.045). This means you would reject the null hypothesis that the coin is fair, suggesting the coin is likely biased.

Chi-Square (χ²) Value:

Degrees of Freedom (df):

To interpret these results, compare your Chi-Square value to a critical value from a Chi-Square distribution table using ${degreesOfFreedom} degrees of freedom, or use statistical software to find the p-value.