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Chi-square Test: The chi-squared test (χ²) is a fundamental tool in statistics for analysing data on the basis of observations. In this test, two statistical data sets are compared. It was introduced by Karl Pearson in 1900. Therefore, it is also known as Pearson’s chi-squared test. This statistical method is used to compare observed data with what we would expect under a specific hypothesis, particularly focusing on categorical variables.
What is the Chi-Squared Test?
The chi-squared test evaluates whether there is a visible difference between the observed frequencies and the expected frequencies in one or more categories. It helps to determine if the patterns observed in the data are consistent with the null hypothesis, i.e., the assumption that has no effect or existing difference.
Chi-Square Formula
The Chi-Square statistic (χ²) is used to assess how well-observed data fit expected data under a specific hypothesis. The formula is given by:
χ² = (Oi – Ei)2Ei
where:
- Oi = Observed value (the actual data collected)
- Ei = Expected value (the value you would expect under the null hypothesis)
Also Check: T-test Table – Formula, Chi-Squared Distribution
Calculating the P-Value
The p-value in a chi-squared test represents the probability of obtaining a chi-squared statistic at least as extreme as the one observed. We assume that the null hypothesis is true in this case. Here’s how to interpret the p-value:
- P ≤ 0.05: If the p-value is less than or equal to 0.05, it indicates that there is a significant difference between the observed and expected frequencies. In this case, you reject the null hypothesis, suggesting that the observed frequencies are unlikely to have occurred by chance.
- P > 0.05: If the p-value is greater than 0.05, it means that the differences between observed and expected frequencies could be due to random variation. Therefore, you can not reject the null hypothesis. It means that there isn’t enough evidence to say there is a notable difference.
Types of Chi-Squared Tests
- Chi-Squared Test of Independence
This test assesses whether two categorical variables are independent of each other. For example, it can determine if there is an association between gender and voting preference. - Chi-Squared Test of Goodness of Fit
This test checks if sample data fits a distribution from a specified population. For example, it can test if a die is fair or not based on the frequency of its outcomes.
Why to Use the Chi-Squared Test?
The chi-squared test is widely used because:
- It works well with data that can be divided into distinct categories.
- Chi-Squared Tests identify if there are significant associations between variables.
- It checks if data fits a specific distribution.
Limitations of the Chi-Squared Test
The chi-squared test has the following limitations:
- It requires a sufficiently large sample size for reliable results.
- The expected frequency in each category should be at least 5 to ensure accuracy.
Also Check: Hypothesis Testing
What is the Chi-Square Distribution?
When you perform a chi-squared test, you’re essentially comparing the observed data to what would be expected under the null hypothesis. The chi-square distribution is the theoretical distribution of the test statistic under the null hypothesis.
Chi-square distribution helps determine whether any observed deviations from the expected frequencies are significant or just due to random variation.
Chi-Square Distribution and Its Role in Statistical Analysis
The chi-square distribution is a fundamental concept in statistics, especially when dealing with categorical data. It plays an important role in determining how likely it is that the observed frequencies in a data set are due to random chance, under the assumption that the null hypothesis is true.
Key Features of the Chi-Square Distribution
The chi-square distribution is positively skewed, especially with a low number of degrees of freedom. As the degrees of freedom increase, the distribution becomes more symmetric and approaches a normal distribution.
The shape of the chi-square distribution depends on the degrees of freedom (df). In a chi-squared test, the degrees of freedom are typically calculated based on the number of categories or groups involved.
The chi-square statistic can only take non-negative values (0 and above). Negative values are not possible since it is based on squared differences.
Application of the Chi-Square Test
The chi-squared test is used primarily for categorical data. Here are some practical examples:
Chi-Square test is used to determine if a sample data fits a population with a known distribution.
Chi-Square test is used to examine if two categorical variables are independent.
Relationship with Probability and Statistics
Probability and statistics are closely related concepts used in the chi-squared test:
Probability measures the likelihood of an event occurring. With respect to the chi-squared test, it helps in assessing the chance of observing the data under the null hypothesis. Whereas, statistics is the science of collecting, analysing, interpreting, and presenting data. The chi-squared test is a statistical method used to analyse categorical data and assess hypotheses about frequencies and distributions.
Solved Questions on Chi-Square
Question 1: A researcher wants to test if a die is fair. After rolling the die 60 times, the results are as follows:
Observed: 10 ones, 8 twos, 12 threes, 11 fours, 9 fives, 10 sixes.
Expected: Since the die is fair, each number should appear 10 times (since 606 = 10)
Calculate the Chi-Square value.
Solution: To find the Chi-Square value, we use the formula: χ² = (Oi – Ei)2Ei
where:
Oi = Observed value (the actual data collected)
Ei = Expected value (the value you would expect under the null hypothesis)
χ² = (10 – 10)210 + (8 – 10)210 + (12 – 10)210 + (11 – 10)210 + (9 – 10)210 + (10 – 10)210
χ² = 010 + 410 + 410 + 110 + 110 + 010 =1010 = 1
Therefore, χ² = 1
Question 2: A shop owner believes that the number of customers visiting her shop on different days of the week is uniform. Over a week, she records the following numbers of visitors:
Observed: Monday: 50, Tuesday: 60, Wednesday: 55, Thursday: 45, Friday: 40, Saturday: 50, Sunday: 55.
Expected: With an assumption of uniform distribution, each day should have 3557 =50.7 customers.
Calculate the Chi-Square value.
Solution: to find the Chi-Square value, we use the formula: χ² = (Oi – Ei)2Ei
where:
Oi = Observed value (the actual data collected)
Ei = Expected value (the value you would expect under the null hypothesis)
χ² = (50 – 50.7)250.7 + (60 – 50.7)250.7 + (55 – 50.7)250.7 + (45 – 50.7)250.7 + (40 – 50.7)250.7 + (50 – 50.7)250.7
On solving further, we get:
χ² =5.35
Which is the required answer.
Practice Questions on Chi-Square
Question 1: A survey was conducted to find out if people have a preference for different colours of cars. The observed data are: Red: 40, Blue: 35, Black: 45, and White: 30. The expected number of cars for each colour is 37.5. Calculate the Chi-Square value.
Question 2: A teacher wants to know if students have a preference for different types of learning materials. She observes the following preferences: Books: 25, Videos: 20, Online Articles: 15, and Hands-on Activities: 30. The expected number of preferences for each type is 22.5. Calculate the Chi-Square value
Question 3: In a factory, the production manager suspects that the number of defective items produced varies by machine. The observed defects from three machines are: Machine A: 8, Machine B: 10, and Machine C: 12. The expected number of defects for each machine is 10. Calculate the Chi-Square value.
Question 4: A biologist is studying the distribution of different types of plants in a region. The observed counts are Type A: 50, Type B: 60, Type C: 40, and Type D: 50. The expected count for each type is 50. Calculate the Chi-Square value.
Chi-Square Test FAQs
What is the purpose of a Chi-Square test?
The Chi-Square test is used to determine whether there is a significant difference between the expected and observed frequencies in categorical data. It helps assess if observed results match expected outcomes under a specific hypothesis.
When should I use a Chi-Square test?
Use a Chi-Square test when you want to analyse the relationship between two categorical variables or test if an observed distribution fits an expected distribution. Common applications include goodness-of-fit tests and tests of independence.
How do I interpret the P-value in a Chi-Square test?
The P-value in a Chi-Square test indicates the likelihood that the observed differences occurred by chance. A low P-value (typically less than 0.05) suggests that the differences are statistically significant, leading to the rejection of the null hypothesis.
