Chi-Square Concepts
What is the main purpose of a chi-square test?
To test whether two categorical variables are STATISTICALLY SIGNIFICANTLY related, or whether the pattern observed could be due to random chance.
Chi-square answers: Is the relationship real, or is it just sampling variation?
A bivariate table has 3 rows and 2 columns.
Calculate the degrees of freedom.
df = (rows − 1)(columns − 1) = (3 − 1)(2 − 1) = 2 × 1 = 2
Which measure of association is used for two nominal variables?
LAMBDA (or Cramer's V when Lambda = 0).
Lambda ranges from 0.0 to 1.0; it measures strength only, not direction.
What does ANOVA compare?
ANOVA compares the MEANS of three or more groups to determine whether the differences between them are statistically significant.
It tests whether group differences are larger than we would expect from chance.
In the regression equation Ŷ = a + b(X), what does the slope (b) represent?
The slope (b) represents the change in Y for every 1-unit increase in X.
In plain language: how much the predicted outcome goes up or down each time the predictor increases by one unit.
If two variables are statistically independent, what does that mean?
The two variables are UNRELATED in the population.
Knowing the value of one variable tells you nothing about the other.
Row total = 40, Column total = 50, N = 200
Calculate the expected frequency for this cell.
fₑ = (row total × column total) / N
= (40 × 50) / 200
= 2,000 / 200
= 10
Which measure of association is used for two ordinal variables?
GAMMA (or Kendall's Tau-b for square tables or when there are ties).
Gamma ranges from −1.0 to +1.0; it measures both strength AND direction.
What does the null hypothesis state in ANOVA?
H₀: All group means are equal in the population.
Written as: μ₁ = μ₂ = μ₃ = …
The research hypothesis (H₁) states that at least one group mean is significantly different from the others.
A regression equation has b = −3.
What does this tell you about the relationship between X and Y?
For every 1-unit increase in X, Y is predicted to DECREASE by 3.
The slope is NEGATIVE — as X goes up, Y goes down.
This indicates a negative relationship between the two variables.
What does an expected frequency represent in a chi-square test?
The count we would EXPECT in a cell if the two variables were completely unrelated (statistically independent).
Expected frequencies show what the table would look like if there were no relationship.
Row total = 60, Column total = 30, N = 180
Calculate the expected frequency for this cell.
fₑ = (row total × column total) / N
= (60 × 30) / 180
= 1,800 / 180
= 10
A researcher reports Gamma = −0.50.
Interpret this value. What does the sign and the number tell you?
Sign (negative): As one ordinal variable increases, the other tends to DECREASE.
There is a negative relationship between the two variables.
Number (0.50): MODERATE strength.
SSB = 45, SST = 300
Calculate eta squared and classify the effect size.
(~0.01 = small ~0.06 = moderate ~0.14+ = large)
η² = SSB / SST = 45 / 300 = 0.15
Effect size: LARGE (0.15 > 0.14)
The independent variable explains 15% of the variation in the dependent variable.
Ŷ = 6 + 2X
X = 5
Calculate the predicted value of Y.
Ŷ = 6 + 2(5)
= 6 + 10
= 16
The predicted value of Y is 16.
If observed frequencies and expected frequencies are very different, what does that suggest about the relationship?
A relationship likely EXISTS between the two variables.
Large differences between observed (fₒ) and expected (fₑ) frequencies produce a large chi-square statistic.
The larger the chi-square, the more evidence of a relationship.
χ² obtained = 2.50
χ² critical = 3.84 (df = 1, α = 0.05)
What is the decision?
FAIL TO REJECT H₀
χ² obtained (2.50) < χ² critical (3.84).
The result is not statistically significant; the relationship could be due to chance.
Lambda = 0.20.
Interpret this value using PRE language.
Using the independent variable to predict the dependent variable reduces our prediction errors by 20%.
Lambda is a PRE measure - Proportional Reduction of Error.
Lambda = 0.20 indicates a weak-to-moderate relationship.
MSB = 24, MSW = 6
Calculate the F statistic.
F = MSB / MSW = 24 / 6 = 4
F = 4 means the between-group variance is 4 times larger than the within-group variance, the signal is bigger than the noise.
r² = 0.25
Interpret this value in one plain-language sentence.
The independent variable (X) explains 25% of the variation in the dependent variable (Y).
The remaining 75% of the variation is explained by other factors not included in this model.
A chi-square test returns p = 0.08 at α = 0.05.
What is the decision, and what does it mean in plain language?
Decision: FAIL TO REJECT H₀
p (0.08) > α (0.05) — the result is not statistically significant.
Plain language: There is not sufficient evidence of a relationship between the two variables. The pattern observed in the table could reasonably be due to random chance rather than a real association in the population.
χ² = 6.20, df = 1, α = 0.05
The critical value at df = 1, α = 0.05 is 3.84.
Make a decision and write a one-sentence plain-language interpretation.
Decision: REJECT H₀
χ² obtained (6.20) > χ² critical (3.84). Statistically significant.
Interpretation: There is a statistically significant relationship between the two variables; the pattern observed is unlikely to be due to chance.
Lambda = 0, but column percentages in the table differ by 28%.
Should the researcher conclude there is no relationship?
Explain why or why not, and what they should do instead.
NO — the researcher should NOT conclude there is no relationship.
A 28% percentage difference signals a real pattern in the data.
Lambda equals zero when the modal DV category is the SAME across all IV columns, the IV never changes the researcher's best prediction, so lambda calculates as zero even when clear differences exist.
What to do: Use CRAMER'S V instead, it correctly captures the strength of nominal relationships and does not have this limitation.
F = 5.10, p = 0.02, α = 0.05
Group means: Group A = 8.2 (highest) Group B = 6.5 Group C = 4.9 (lowest)
Make a decision and write a full one-sentence interpretation.
Decision: REJECT H₀
p (0.02) < α (0.05) — the result is statistically significant.
Interpretation: There is a statistically significant difference in scores across the three groups (F = 5.10, p = 0.02); Group A reports the highest mean (8.2) and Group C the lowest (4.9).
Ŷ = 10 + 1.5(X)
r² = 0.30 p = 0.01 α = 0.05
Write a complete interpretation covering direction, significance, and r².
Direction: POSITIVE relationship - as X increases, Y is predicted to increase.
For every 1-unit increase in X, Y is predicted to increase by 1.5.
Significance: The model IS statistically significant.
p (0.01) < α (0.05) - Reject H₀.
r²: The independent variable explains 30% of the variation in Y.
The remaining 70% is explained by factors not included in the model.