Chi test in statistics.
Provide more and simple example clearly step by step
A chi-squared (χ²) test is a statistical test used to determine if there is a significant association between two categorical variables. It is often used to compare observed frequencies with expected frequencies and evaluate whether the observed frequencies deviate significantly from what would be expected under a given hypothesis.
Let's consider a simple example to understand chi-square test step by step:
Example: A researcher wants to examine whether there is a relationship between gender (male or female) and favorite ice cream flavor (chocolate, vanilla, or strawberry). The researcher surveyed 100 individuals and recorded their responses.
Step 1: Set up hypotheses:
- Null Hypothesis (H₀): There is no association between gender and favorite ice cream flavor.
- Alternative Hypothesis (H₁): There is an association between gender and favorite ice cream flavor.
Step 2: Set the significance level (α), which determines the probability of rejecting the null hypothesis when it is true. Let's assume α = 0.05.
Step 3: Create a contingency table:
Construct a contingency table to summarize the observed frequencies of each combination of gender and ice cream flavors. It might look like this:
Chocolate Vanilla Strawberry Total
Male 20 30 10 60
Female 40 15 25 80
Total 60 45 35 140
Step 4: Calculate expected frequencies:
Calculate the expected frequencies for each combination under the assumption that the null hypothesis is true (i.e., no association between gender and ice cream flavor). This is done by multiplying the row total by the column total and dividing by the grand total:
Chocolate Vanilla Strawberry Total
Male 25.71 19.29 14.00 60
Female 34.29 25.71 20.00 80
Total 60 45 35 140
Step 5: Calculate the chi-square test statistic:
The chi-square test statistic is calculated as the sum of the squared differences between the observed and expected frequencies divided by the expected frequencies for each cell. The formula is:
χ² = ∑ ((O - E)² / E)
where O is the observed frequency and E is the expected frequency for each cell. In our example, we have three cells for chocolate, vanilla, and strawberry flavors. So the computation will look like this:
χ² = ((20 - 25.71)² / 25.71) + ((30 - 19.29)² / 19.29) + ((10 - 14)² / 14) + ...
Step 6: Determine the degrees of freedom (df):
The degrees of freedom is calculated as (r - 1) * (c - 1), where r is the number of rows and c is the number of columns in the contingency table. In our example, r = 2 (male and female) and c = 3 (chocolate, vanilla, and strawberry), so df = (2 - 1) * (3 - 1) = 2.
Step 7: Look up the critical value:
Using the chi-square distribution table or a statistical software, find the critical value associated with the degrees of freedom and the chosen significance level (α = 0.05). In our example, with df = 2 and α = 0.05, the critical value is approximately 5.99.
Step 8: Compare the calculated chi-square statistic with the critical value:
If the calculated chi-square statistic is greater than the critical value, we reject the null hypothesis and conclude that there is evidence of an association between gender and favorite ice cream flavor. Otherwise, if the calculated chi-square statistic is less than or equal to the critical value, we do not have enough evidence to reject the null hypothesis.
Step 9: Interpret the results:
Based on the calculated chi-square statistic and comparison with the critical value, we draw our conclusion. If we reject the null hypothesis, we can state that there is an association between gender and favorite ice cream flavor. If we fail to reject the null hypothesis, we cannot conclude that there is an association between the two variables.
Remember that this example provided a basic step-by-step explanation of a chi-square test. In practice, it is always recommended to consult a statistics textbook or statistical software for accurate calculations and to understand the limitations and assumptions of the test.
The chi-square test is a statistical test used to determine whether there is a significant association between two categorical variables. It compares the observed frequencies in each category with the expected frequencies that would occur if there was no association.
Let's say we are interested in studying the association between gender (male or female) and smoking habits (smoker or non-smoker). We have collected data from a random sample of 200 individuals. Here is the step-by-step process to conduct a chi-square test:
Step 1: State the hypotheses
- Null hypothesis (H0): There is no association between gender and smoking habits.
- Alternative hypothesis (Ha): There is an association between gender and smoking habits.
Step 2: Set the significance level
Choose a significance level (often denoted as α), which is the probability of rejecting the null hypothesis when it is true. For example, let's set α = 0.05, which is a common choice.
Step 3: Create a contingency table
Construct a contingency table that displays the observed frequencies for each combination of categories. In our example, the contingency table looks like this:
Smoking Habits
Smoker Non-Smoker
Male 20 60
Female 40 80
Step 4: Calculate expected frequencies
Calculate the expected frequencies for each cell if there is no association between the variables. The expected frequency for each cell is obtained by multiplying the row total by the column total and dividing by the overall sample size. In our case:
Expected Frequency = (Row Total * Column Total) / Overall Sample Size
For example, the expected frequency for the top-left cell would be (80 * 60) / 200 = 24.
Step 5: Calculate the test statistic
Calculate the chi-square test statistic using the formula:
χ² = ∑ ((Observed Frequency - Expected Frequency)² / Expected Frequency)
For each cell, subtract the expected frequency from the observed frequency, square the difference, divide it by the expected frequency, and sum up these values across all cells. In our example:
χ² = ((20-24)²/24) + ((60-56)²/56) + ((40-36)²/36) + ((80-84)²/84)
Step 6: Determine the critical value
Based on the degrees of freedom, look up the critical value from the chi-square distribution table. Degrees of freedom (df) are calculated as (number of rows - 1) * (number of columns - 1). In our example, df = (2-1) * (2-1) = 1. For α = 0.05 and df = 1, the critical value is approximately 3.84 from the table.
Step 7: Compare the test statistic with the critical value
If the test statistic is greater than the critical value, we reject the null hypothesis and conclude that there is a significant association between the variables. Otherwise, we fail to reject the null hypothesis. In our example, if the calculated χ² value is greater than 3.84, we reject the null hypothesis.
That's it! These steps provide a basic overview of conducting a chi-square test using a simple example. Keep in mind that there are additional considerations, such as assumptions and alternative chi-square test variations, that may be relevant for specific scenarios.
A chi-squared test is a statistical method used to determine whether there is a significant association between two categorical variables. It helps to determine whether the observed differences between the observed and expected frequencies are statistically significant or simply due to chance.
Let's walk through a simple example step by step to understand how a chi-squared test works:
Example: Imagine a study is conducted to determine if there is a relationship between smoking habits and the development of respiratory diseases. Researchers surveyed 200 participants and obtained the following data:
Respiratory Disease
Yes No
Smoker 40 110
Non-Smoker 30 20
Step 1: Set up the null and alternative hypotheses:
- Null hypothesis (H₀): There is no association between smoking habits and respiratory diseases.
- Alternative hypothesis (H₁): There is an association between smoking habits and respiratory diseases.
Step 2: Set the significance level:
- Commonly, we use a significance level of 0.05 (5%) to determine statistical significance.
Step 3: Calculate the expected frequencies:
- To determine the expected frequencies, we need to assume that there is no association between the variables and calculate what the frequencies would be if this assumption were true.
- Start by calculating the row and column totals:
Respiratory Disease
Yes No Total
Smoker 40 110 150
Non-Smoker 30 20 50
Total 70 130 200
- Next, calculate the expected frequency for each cell by multiplying the row total by the column total and dividing it by the overall total:
- Expected Frequency = (Row Total * Column Total) / Overall Total
- For example, the expected frequency for Smoker-Respiratory Disease Yes is (150 * 70) / 200 = 52.5.
Step 4: Calculate the observed and expected values:
- Compare the observed frequencies (the actual counts) with the expected frequencies (calculated in the previous step). In our example, we have the observed and expected frequencies as follows:
Respiratory Disease
Yes No
Smoker 40 (O₁) 110 (O₂)
Non-Smoker 30 (O₃) 20 (O₄)
Respiratory Disease
Yes No
Smoker 52.5 (E₁) 97.5 (E₂)
Non-Smoker 17.5 (E₃) 32.5 (E₄)
Step 5: Calculate the chi-squared statistic:
- The chi-squared statistic measures the difference between the observed and expected frequencies.
- Chi-squared statistic = Σ((Oᵢ - Eᵢ)² / Eᵢ)
- Compute this for each cell and sum the values.
Step 6: Determine the degrees of freedom:
- The degrees of freedom are determined by the number of categories minus one. In our example, the degrees of freedom would be (2-1) * (2-1) = 1.
Step 7: Find the critical value:
- Look up the critical value in the chi-squared distribution table based on your chosen significance level and degrees of freedom. For example, with a significance level of 0.05 and 1 degree of freedom, the critical value is 3.841.
Step 8: Determine statistical significance:
- Compare the obtained chi-squared statistic with the critical value from the table. If the obtained statistic is greater than the critical value, we reject the null hypothesis and conclude that there is a significant association between smoking habits and respiratory diseases. If it's smaller, we fail to reject the null hypothesis and conclude that there is not enough evidence to suggest an association.
That's it! By following these steps, you can conduct a chi-squared test to determine the relationship between two categorical variables in your own data.