Chi Square Test

One-Way Chi-Square Test

• The one-way Chi-Square test, also known as the goodness of fit Chi-Square, is used to compare observed frequencies to expected frequencies.
• It is used to test the significance of the Chi-Square value, which is compared to the critical value in the Chi-Square distribution.

Null Hypothesis

• The null hypothesis is often a hypothesis of equal probable distribution, meaning that the data are distributed according to some theory or previously observed frequencies.
• The null hypothesis can be that the data are normally distributed.

Calculating Chi-Square Value

• To calculate the Chi-Square value, the square differences between the observed and expected frequencies are calculated, divided by the expected frequency for each category.
• The Chi-Square value is then compared to the critical value in the Chi-Square distribution.

Degrees of Freedom and Chi-Square

• The degrees of freedom depend on the number of categories, not the sample size.
• The degrees of freedom are calculated as the number of categories minus one.

Chi-Square Distribution

• The Chi-Square distribution is a family of distributions that depend on the degrees of freedom.
• Chi-Square values are always positive and can increase with sample size, but the critical Chi-Square remains the same as the degrees of freedom don't change.

Interpreting Chi-Square Results

• If the obtained Chi-Square value is larger than the critical value, the null hypothesis is rejected, indicating a significant difference between the observed and expected frequencies.
• If the obtained Chi-Square value is smaller than the critical value, the null hypothesis is not rejected, indicating no significant difference between the observed and expected frequencies.

Chi-Square Test

• The chi-square test is used to compare the observed frequencies with the expected frequencies to determine if there is a significant difference.

Calculation of Chi-Square (c2)

• The chi-square formula is: χ² = Σ [(fo - fe)² / fe]
• fo = observed frequency in the data
• fe = expected frequency under the null hypothesis
• Σ means sum over all the categories

One-Way Chi-Square Example

• The example involves a study on the preferred flavor of Chupa Chup among 284 participants
• The research question is: Are some flavors more popular than others?
• The chi-square test is used to determine if the observed frequencies match the expected frequencies

Null Hypothesis

• H0: In the population, the distribution across categories = expected frequencies
• Any discrepancy between observed and expected frequencies is attributed to chance

Sampling Distribution of c2

• The sampling distribution of c2 is a family of distributions dependent on degrees of freedom (df)
• df = k - 1, where k is the number of categories
• c2 values are always positive and can increase with sample size

Critical Values of Chi-Square

• Critical values of chi-square are used to determine the significance of the test
• The critical value marks the .05 region under the null hypothesis
• Critical values are obtained from a chi-square distribution table

Degrees of Freedom

• df = k - 1, where k is the number of categories
• In the example, k = 4, so df = 3

Measurement Levels

• Nominal level: categorical data with no underlying order (e.g., gender)
• Ordinal level: categorical data with an underlying order (e.g., 1st/2nd place)
• Interval level: continuous data with equal intervals between scores (e.g., temperature in Celsius)
• Ratio level: continuous data with a true zero point (e.g., length)