Chi Square Calculate

The Chi-Square ($\chi^2$) test is a fundamental statistical hypothesis test used to determine if there is a significant difference between the observed frequencies and the expected frequencies in one or more categories. It's a non-parametric test, meaning it doesn't assume a specific distribution for the population data.

It is commonly applied in two main scenarios:

• Goodness-of-Fit Test: This test determines if a sample distribution matches a hypothesized population distribution. For example, you might use it to test if the observed frequencies of different colors in a bag of candies match the manufacturer's stated proportions, or if the distribution of grades in a class follows a normal curve.
• Test of Independence: This test determines if there is a significant association between two categorical variables in a contingency table. For example, testing if there's a relationship between gender and preference for a certain political candidate, or between smoking status and the incidence of a particular disease.

This calculator focuses on the goodness-of-fit application or situations where you have pre-defined observed and expected frequencies for several categories. The formula for the Chi-Square statistic is:

$\chi^2 = \sum \frac{(O_i – E_i)^2}{E_i}$

Where:

• $O_i$ = The observed frequency (count) for category $i$. This is what you actually counted or measured.
• $E_i$ = The expected frequency (count) for category $i$. This is what you would expect to see if your null hypothesis were true.
• $\sum$ = Summation across all categories. This means you calculate the $(O_i – E_i)^2 / E_i$ term for each category and then add them all together.

The calculated Chi-Square value is then compared to a critical value from a Chi-Square distribution table, using the appropriate degrees of freedom (df) and a chosen significance level (e.g., 0.05). The degrees of freedom for a goodness-of-fit test are typically calculated as the number of categories minus 1 (df = k – 1).

How to Use the Calculator

Enter the observed and expected frequencies for each category. You can use up to 5 pairs of values. If you have fewer categories, leave the unused rows blank. The calculator will sum the contributions from each valid pair to give you the total Chi-Square value and the degrees of freedom.

Understanding the Results

After calculating the Chi-Square value and degrees of freedom, you would typically compare your calculated $\chi^2$ value to a critical value from a Chi-Square distribution table. This comparison helps you decide whether to reject or fail to reject the null hypothesis.

• Null Hypothesis ($H_0$): There is no significant difference between observed and expected frequencies (or no association between variables). Any differences are due to random chance.
• Alternative Hypothesis ($H_1$): There is a significant difference between observed and expected frequencies (or there is an association). The differences are too large to be attributed to random chance.

If your calculated Chi-Square value is greater than the critical value for your chosen significance level (e.g., 0.05 or 5%) and degrees of freedom, you would reject the null hypothesis. This suggests that the observed frequencies are significantly different from the expected frequencies, or that there is a statistically significant association between the variables.

Conversely, if your calculated Chi-Square value is less than or equal to the critical value, you would fail to reject the null hypothesis. This means there isn't enough evidence to conclude a significant difference or association.

Example Scenario: Mendelian Genetics

A geneticist is studying the inheritance pattern of a certain trait in pea plants. According to Mendelian genetics, they expect a phenotypic ratio of 3:1 (dominant:recessive) in the offspring. They observe 120 plants in total, with 95 exhibiting the dominant trait and 25 exhibiting the recessive trait.

1. Observed Frequencies:

• Dominant Trait: 95 plants
• Recessive Trait: 25 plants

Total Observed: 95 + 25 = 120 plants

2. Expected Frequencies (based on 3:1 ratio for 120 plants):

• Dominant Trait: (3/4) * 120 = 90 plants
• Recessive Trait: (1/4) * 120 = 30 plants

Using the calculator, you would input:

• Category 1 (Dominant): Observed = 95, Expected = 90
• Category 2 (Recessive): Observed = 25, Expected = 30

The calculator would perform the following calculation:

• $\chi^2 = \frac{(95-90)^2}{90} + \frac{(25-30)^2}{30}$
• $\chi^2 = \frac{5^2}{90} + \frac{(-5)^2}{30}$
• $\chi^2 = \frac{25}{90} + \frac{25}{30}$
• $\chi^2 \approx 0.2778 + 0.8333$
• $\chi^2 \approx 1.1111$

The Degrees of Freedom (df) would be calculated as: Number of categories – 1 = 2 – 1 = 1.

Interpretation: With a calculated $\chi^2 \approx 1.11$ and df = 1, if we were to look up a critical value for a common significance level (e.g., $\alpha = 0.05$), the critical value is 3.841. Since our calculated $\chi^2$ value (1.11) is less than the critical value (3.841), we would fail to reject the null hypothesis. This means there is no statistically significant difference between the observed plant ratios and the expected 3:1 Mendelian ratio, suggesting the trait follows the expected inheritance pattern.

Calculated Chi-Square ($\chi^2$):

Degrees of Freedom (df):

Note: For interpretation, compare this $\chi^2$ value to a critical value from a Chi-Square distribution table using the calculated degrees of freedom and your chosen significance level.