Chi Square

Chi-Square Test

A nonparametric statistical test used to determine if there is a significant difference between expected and observed frequencies in one or more categories.

Goodness of Fit Test

A chi-square test used to determine if the observed sample distribution matches an expected distribution.

Test of Independence

A chi-square test used to determine whether two categorical variables are independent.

Nonparametric Test

A statistical test that does not assume a specific distribution in the population; often used with nominal or ordinal data.

Parametric Test

A test that involves assumptions about parameters of the population distribution from which the sample is drawn.

Observed Frequency (fo)

The actual count or frequency in each category as recorded from the data.

Expected Frequency (fe)

The frequency that would be expected in each category if the null hypothesis were true.

Chi-Square Statistic (χ²)

A measure of the discrepancy between observed and expected frequencies, calculated as: χ² = Σ((fo

Degrees of Freedom (df)

The number of values in the final calculation of a statistic that are free to vary.

Degrees of Freedom for Goodness of Fit

df = C

Degrees of Freedom for Test of Independence

df = (R

Significance Level (α)

The threshold for rejecting the null hypothesis; common levels are 0.05, 0.01.

Null Hypothesis (H0) for Goodness of Fit

The population is distributed in a specified way, often that all categories have equal proportions or match a known distribution.

Alternative Hypothesis (H1) for Goodness of Fit

The population proportions are not equal to those specified in the null hypothesis.

Null Hypothesis (H0) for Test of Independence

There is no relationship between the two variables; they are independent.

Alternative Hypothesis (H1) for Test of Independence

There is a relationship between the two variables; they are not independent.

Critical Region

The range of values for which the null hypothesis is rejected.

Chi-Square Distribution

A positively skewed distribution used to determine the significance of a chi-square statistic.

Phi Coefficient (Φ)

A measure of effect size for a 2x2 chi-square test, calculated as: Φ² = χ² / n

Cramer's V

A measure of effect size for larger than 2x2 matrices: V = sqrt(χ² / (n * df)) where df is the smaller of (R-1) or (C-1)

Effect Size

A quantitative measure of the strength of a phenomenon; in chi-square tests, this is often reported using Phi or Cramer’s V.

Independence of Observations

An assumption that each observed frequency is generated by a different individual; required for chi-square tests.

Minimum Expected Frequency

Each expected frequency should be at least 5 for the chi-square test to be valid.

Nominal Scale

A scale of measurement that uses labels or names to categorize variables without any quantitative value.

Ordinal Scale

A scale that depicts the order of values but not the difference between them.

Contingency Table

A matrix used in the test of independence to display the frequency distribution of variables.

Row Total (fr)

The sum of frequencies in a given row of a contingency table.

Column Total (fc)

The sum of frequencies in a given column of a contingency table.

Total Sample Size (n)

The total number of observations in the sample.

Expected Frequency for a Cell

(fr * fc) / n

Chi-Square vs. One Sample t Test

Chi-square: nonparametric, used for categorical data; One sample t test: parametric, used for numerical scores

Chi-Square vs. Pearson Correlation

Chi-square evaluates relationships between two categorical variables; Pearson correlation evaluates relationships between two continuous variables

Expected Frequencies

Can include decimals (not always whole numbers)

Observed Frequencies

Must be whole numbers

Large Chi-Square Value

Leads to rejecting the null hypothesis

Degrees of Freedom

Based on the number of categories or dimensions, not the sample size

Chi-Square Values

Always non-negative and don’t show direction of correlation

Example Reporting Format

χ² (df, n = sample size) = test statistic, p < .05

Use Chi-Square When

Data are frequencies (counts); You're testing distribution shape (Goodness of Fit); You're testing variable relationships (Test of Independence)

Do Not Use Chi-Square When

Expected frequency in any cell is < 5; Observations are not independent

Chi-Square Formula

χ² = Σ((fo

Expected Frequency Formula

fe = (fr * fc) / n

Phi Formula

Φ = sqrt(χ² / n)

Cramer's V Formula

V = sqrt(χ² / (n * df*))