Correlation measures the strength and direction of the relationship between two variables.
It tells us whether high values of one variable go with high (or low) values of another.

Pearson’s r

The most common measure is Pearson’s correlation coefficient, $$r$$.
It ranges from –1 to +1.

• $$r = +1$$ → perfect positive correlation (as X increases, Y increases).
• $$r = –1$$ → perfect negative correlation (as X increases, Y decreases).
• $$r = 0$$ → no linear relationship.

Symbolic formula:
$$r = \frac{\sum (X - \bar{X})(Y - \bar{Y})}{\sqrt{\sum (X - \bar{X})^2 , \sum (Y - \bar{Y})^2}}$$

Formula in words:
$$r = \frac{\text{sum of the cross-products of deviations from the mean}}{\text{square root of (sum of squared deviations in X × sum of squared deviations in Y)}}$$

Example

Suppose study hours (X) and test scores (Y) are:

• X = [2, 4, 6]
• Y = [50, 60, 80]

Means:

• $$\bar{X} = 4$$
• $$\bar{Y} = 63.3$$

Deviations:

• (2–4)(50–63.3) = (–2)(–13.3) = 26.6
• (4–4)(60–63.3) = (0)(–3.3) = 0
• (6–4)(80–63.3) = (2)(16.7) = 33.4

Sum cross-products = 60

Sum squares X = (–2)² + 0² + 2² = 8
Sum squares Y = (–13.3)² + (–3.3)² + 16.7² ≈ 466.7

So:
$$r = \frac{60}{\sqrt{8 \times 466.7}} = \frac{60}{\sqrt{3733}} = \frac{60}{61.1} = 0.98$$

A very strong positive correlation.

Coefficient of Determination

The square of correlation is $$r^2$$.
It represents the proportion of variance in Y explained by X.

Example above:
$$r^2 = (0.98)^2 = 0.96$$

So about 96% of the variation in scores is explained by study hours.

Definition

• Correlation: degree of linear relationship between two variables.
• Pearson’s r: ranges from –1 to +1.
• Coefficient of determination (r²): proportion of explained variance.

Visual Placeholders

Figure 9.1 — Scatterplot with positive correlation (points rising, line upward).

Figure 9.2 — Scatterplots showing r ≈ +1, r ≈ 0, r ≈ –1.

Why This Matters

Correlation is the first step in studying relationships.
It helps identify whether variables move together, setting the stage for regression analysis.

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Welcome to Part 5 — Statistical Tests (Cookbook Style) of this free online high school statistics textbook. This practical quick-reference section provides concise, cookbook-style guides to major parametric and non-parametric statistical tests, including detailed formulas, assumptions, degrees of freedom, step-by-step procedures, and real-world examples. High school students and teachers can quickly review when to use each test—perfect for AP Statistics exam preparation, homework help, or reinforcing concepts from earlier parts.

Ideal for quick lookups on ANOVA variants, non-parametric alternatives, and multi-group comparisons, Part 5 delivers clear explanations of one-way ANOVA, factorial ANOVA, repeated-measures ANOVA, mixed ANOVA, Mann-Whitney U, Wilcoxon, Kruskal-Wallis, and Friedman tests in an accessible format with worked examples.

Statistical Tests Covered in Part 5

1. One-Way ANOVA – Comparing means across three or more independent groups, with formula, degrees of freedom, and example.
2. Factorial ANOVA (Two-Way) – Analyzing main effects and interactions in 2×2 or larger designs, including df partition and example.
3. Repeated-Measures ANOVA – Handling multiple measurements on the same subjects, with formula and example.
4. Mixed (Split-Plot) ANOVA – Combining between-subjects and within-subjects factors, with formula and example.
5. Mann-Whitney U Test – Non-parametric alternative for two independent samples, with formula and example.
6. Wilcoxon Signed-Rank Test – Non-parametric option for paired or one-sample data, with procedure and example.
7. Kruskal-Wallis Test – Non-parametric one-way ANOVA for three or more groups, with formula and example.
8. Friedman Test – Non-parametric repeated-measures ANOVA, with formula and example.

A practice self-test quiz is available to test your understanding (optional signup for full interactive access). Use this free high school statistics resource as your go-to cookbook for statistical tests formulas, ANOVA examples, non-parametric tests guides, and quick reference during hypothesis testing!

One-way ANOVA

When to Use:

• Compare means across 3 or more independent groups.
• Interval/ratio data, groups independent, variances roughly equal.

Formula:
$$F = \frac{MS_{\text{between}}}{MS_{\text{within}}}$$

In words:
$$F = \frac{\text{mean square between groups}}{\text{mean square within groups}}$$

Example:
Three groups with means = 70, 75, 85.

• $$SS_{\text{between}} = 300, , df_{\text{between}} = 2, , MS_{\text{between}} = 150$$
• $$SS_{\text{within}} = 200, , df_{\text{within}} = 12, , MS_{\text{within}} = 16.7$$

$$F = \frac{150}{16.7} = 9.0, \quad df = (2, 12)$$

Factorial ANOVA (Two-way)

When to Use:

• Two or more factors studied at once.
• Tests main effects and interactions.

Formula (df partition):

• $$df_A = a - 1, \quad df_B = b - 1$$
• $$df_{A \times B} = (a-1)(b-1)$$
• $$df_{\text{within}} = N - ab$$

Example:
2 × 2 design (Method: Lecture, Online × Time: Morning, Afternoon).

• Lecture: Morning = 70, Afternoon = 90
• Online: Morning = 80, Afternoon = 80

Interaction: Lecture improves over time, Online flat → non-parallel lines.

Repeated-Measures ANOVA

When to Use:

• Same participants tested under multiple conditions.
• Controls for subject variability.

Formula:
$$F = \frac{MS_{\text{conditions}}}{MS_{\text{error}}}$$

Degrees of Freedom:

• $$df_{\text{rows}} = n - 1$$
• $$df_{\text{columns}} = k - 1$$
• $$df_{\text{error}} = (n-1)(k-1)$$

Example:
Five students tested across 3 conditions. Mean scores rise steadily from 70 → 75 → 80.

Mixed (Split-Plot) ANOVA

When to Use:

• Combines a between-subjects factor with a within-subjects factor.
• Common in psychology and education.

Formula (general):
$$F = \frac{MS_{\text{effect}}}{MS_{\text{error}}}$$

Degrees of Freedom:

• $$df_{\text{between}} = a - 1$$
• $$df_{\text{subjects}} = N - a$$
• $$df_{\text{within}} = b - 1$$
• $$df_{A \times B} = (a-1)(b-1)$$

Example:
Two groups (Drug, Placebo) × three weeks (repeated).
Drug scores rise each week, while Placebo rises more slowly → interaction.

Mann–Whitney U Test

When to Use:

• Compare two independent groups when data are ordinal or not normally distributed.
• Non-parametric alternative to independent t-test.

Formula:
$$U = n_1 n_2 + \frac{n_1 (n_1 + 1)}{2} - R_1$$

Where $$R_1$$ = sum of ranks for group 1.

Example:
Two classrooms ranked by teacher ratings. Test whether distributions differ.

Wilcoxon Signed-Rank Test

When to Use:

• Compare the same group measured twice (before vs. after).
• Ordinal or non-normal data.
• Non-parametric alternative to paired t-test.

Procedure:

1. Compute differences (After – Before).
2. Rank absolute differences.
3. Assign signs.
4. Test statistic = smaller of the two signed sums.

Example:
Five students’ skill ranks before vs. after training. Test whether median rank improved.

Kruskal–Wallis Test

When to Use:

• Compare 3+ independent groups when data are ordinal or non-normal.
• Non-parametric alternative to one-way ANOVA.

Formula:
$$H = \frac{12}{N(N+1)} \sum \frac{R_j^2}{n_j} - 3(N+1)$$

Where:

• $$R_j$$ = sum of ranks for group j
• $$n_j$$ = number of observations in group j
• $$N$$ = total number of observations

Example:
Three therapy groups (n = 10 each) ranked by improvement scores.

Friedman Test

When to Use:

• Compare 3+ related groups (repeated measures, ordinal data).
• Non-parametric alternative to repeated-measures ANOVA.

Formula:
$$Q = \frac{12}{nk(k+1)} \sum R_j^2 - 3n(k+1)$$

Where:

• $$R_j$$ = sum of ranks for each condition
• $$n$$ = number of subjects
• $$k$$ = number of conditions

Example:
Ten students ranked across 3 types of training tasks.

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Case 1 — Independent t-test (Two Groups)

Scenario: A teacher wants to compare math test scores between students taught with traditional lectures and those taught with interactive software.

Question: Are the two teaching methods different in average test score?

Design/Test: Independent-samples t-test.

Worked Example:

• Group A (Lecture): mean = 78, SD = 10, n = 20
• Group B (Software): mean = 85, SD = 12, n = 20

Formula:
$$t = \frac{\bar{X}_1 - \bar{X}_2}{\sqrt{\tfrac{s_1^2}{n_1} + \tfrac{s_2^2}{n_2}}}$$

In words:
$$t = \frac{\text{mean}_1 - \text{mean}_2}{\sqrt{\tfrac{\text{variance}_1}{n_1} + \tfrac{\text{variance}_2}{n_2}}}$$

Plugging in values:
$$t = \frac{78 - 85}{\sqrt{\tfrac{100}{20} + \tfrac{144}{20}}} = \frac{-7}{\sqrt{5 + 7.2}} = \frac{-7}{\sqrt{12.2}} = \frac{-7}{3.49} = -2.01$$

Degrees of freedom = 38.

Case 2 — Paired t-test (Before and After)

Scenario: Students take a memory test before and after a week of practice.

Question: Did memory scores improve after training?

Design/Test: Paired-samples t-test.

Worked Example:

Differences (After – Before): 2, 4, 3, 5, 6

• Mean difference:
  $$\bar{D} = \frac{2+4+3+5+6}{5} = 4$$
• Standard deviation of differences: $$s_D = 1.58$$

Formula:
$$t = \frac{\bar{D}}{s_D / \sqrt{n}}$$

Plugging in values:
$$t = \frac{4}{1.58/\sqrt{5}} = \frac{4}{0.71} = 5.63$$

Degrees of freedom = 4.

Case 3 — One-way ANOVA (Three Groups)

Scenario: A psychologist tests three methods of stress reduction: meditation, exercise, and music.

Question: Do the methods differ in average stress score?

Design/Test: One-way ANOVA.

Worked Example (summary):

• Group means: Meditation = 65, Exercise = 70, Music = 80
• $$SS_{\text{between}} = 300, , df_{\text{between}} = 2, , MS_{\text{between}} = 150$$
• $$SS_{\text{within}} = 200, , df_{\text{within}} = 12, , MS_{\text{within}} = 16.7$$

Formula:
$$F = \frac{MS_{\text{between}}}{MS_{\text{within}}}$$

Plugging in values:
$$F = \frac{150}{16.7} = 9.0$$

df = (2, 12).

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Welcome to Part 4 — Applications (Cases and Examples) of this free online high school statistics textbook. This hands-on section brings statistical concepts to life through detailed, worked-out case studies and real-world examples. High school students explore complete applications of hypothesis testing—including t-tests, ANOVA designs, chi-square tests, and non-parametric methods—covering everything from formulating research questions and selecting the appropriate test to performing calculations, interpreting results, and drawing meaningful conclusions.

Ideal for AP Statistics practice and pre-college preparation, Part 4 features 10 comprehensive cases with step-by-step explanations, formulas, data examples, and practical scenarios (e.g., comparing teaching methods, stress reduction programs, and categorical associations). These worked examples reinforce descriptive statistics, inferential statistics, and critical statistical thinking in an engaging, example-driven format.

Case Studies in Part 4: Applications

1. Case 1: Independent t-Test – Comparing two independent groups (e.g., different teaching methods).
2. Case 2: Paired t-Test – Analyzing before-and-after data in the same subjects.
3. Case 3: One-Way ANOVA – Testing differences across three or more groups.
4. Case 4: Factorial ANOVA (2×2 Design) – Examining main effects and interactions.
5. Case 5: Repeated-Measures ANOVA – Handling multiple measurements on the same subjects.
6. Case 6: Mixed ANOVA – Combining between-subjects and within-subjects factors.
7. Case 7: Chi-Square Goodness-of-Fit – Assessing observed vs. expected frequencies.
8. Case 8: Chi-Square Test of Independence – Exploring relationships in categorical data.
9. Case 9: Mann-Whitney U Test – Non-parametric alternative for two independent samples.
10. Case 10: Wilcoxon Signed-Rank Test – Non-parametric option for paired data.

A practice self-test quiz is also available to reinforce learning (optional signup for full interactive access). Dive into these free high school statistics applications for real-world insight into hypothesis testing, statistical analysis examples, and building confidence with data interpretation!

Case 1 — Independent t-test (Two Groups)

Scenario: A teacher compares math scores of students taught by lecture vs. interactive software.

Question: Are the two teaching methods different in average score?

Design/Test: Independent-samples t-test.

Worked Example:

• Group A (Lecture): mean = 78, SD = 10, n = 20
• Group B (Software): mean = 85, SD = 12, n = 20

Formula:
$$t = \frac{\bar{X}_1 - \bar{X}_2}{\sqrt{\tfrac{s_1^2}{n_1} + \tfrac{s_2^2}{n_2}}}$$

In words:
$$t = \frac{\text{mean}_1 - \text{mean}_2}{\sqrt{\tfrac{\text{variance}_1}{n_1} + \tfrac{\text{variance}_2}{n_2}}}$$

Plugging in values:
$$t = \frac{78 - 85}{\sqrt{\tfrac{100}{20} + \tfrac{144}{20}}} = \frac{-7}{\sqrt{12.2}} = \frac{-7}{3.49} = -2.01$$

For Welch’s test, df ≈ 36.80; for the pooled equal-variance test, df = 38. With equal group sizes, both give the same standard error here.

Case 2 — Paired t-test (Before and After)

Scenario: Students take a memory test before and after a week of practice.

Question: Did scores improve after training?

Design/Test: Paired-samples t-test.

Worked Example:

Differences (After – Before): 2, 4, 3, 5, 6

• Mean difference:
  $$\bar{D} = \frac{2+4+3+5+6}{5} = 4$$
• Standard deviation of differences: $$s_D = 1.581$$

Formula:
$$t = \frac{\bar{D}}{s_D / \sqrt{n}}$$

Plugging in values:
$$t = \frac{4}{1.581/\sqrt{5}} = \frac{4}{0.707} = 5.66$$

Degrees of freedom = 4.

Case 3 — One-way ANOVA (Three Groups)

Scenario: A psychologist tests meditation, exercise, and music as stress-reduction methods.

Question: Do the methods differ in mean stress score?

Design/Test: One-way ANOVA.

Worked Example:

• Group means: Meditation = 65, Exercise = 70, Music = 80
• $$SS_{\text{between}} = 300, , df_{\text{between}} = 2, , MS_{\text{between}} = 150$$
• $$SS_{\text{within}} = 200, , df_{\text{within}} = 12, , MS_{\text{within}} = 16.7$$

Formula:
$$F = \frac{MS_{\text{between}}}{MS_{\text{within}}}$$

$$F = \frac{150}{16.7} = 9.0, \quad df = (2,12)$$

Case 4 — Factorial ANOVA (2 × 2 Design)

Scenario: A researcher studies teaching method (Lecture vs. Online) × Time of Day (Morning vs. Afternoon).

Question: Do method, time, or their interaction affect performance?

Design/Test: Two-way (factorial) ANOVA.

Worked Example (summary):

• Lecture: Morning = 70, Afternoon = 90
• Online: Morning = 80, Afternoon = 80

Interaction: Lecture scores rise with time, Online stays flat.

Formulas:

• $$df_A = a - 1, , df_B = b - 1, , df_{A \times B} = (a-1)(b-1), , df_{\text{within}} = N - ab$$

Case 5 — Repeated-Measures ANOVA

Scenario: Five students are tested across three conditions.

Question: Do scores differ across conditions?

Design/Test: Repeated-measures ANOVA.

Worked Example (summary):

• Means increase steadily: 70 → 75 → 80
• df:
  $$df_{\text{rows}} = n - 1, \quad df_{\text{columns}} = k - 1, \quad df_{\text{error}} = (n-1)(k-1)$$

Formula:
$$F = \frac{MS_{\text{columns}}}{MS_{\text{error}}}$$

Case 6 — Mixed ANOVA

Scenario: Two groups (Drug, Placebo) tested across three weeks.

Question: Is there an effect of group, time, or interaction?

Design/Test: Mixed (split-plot) ANOVA.

Worked Example (summary):

• Drug: 70 → 80 → 90
• Placebo: 70 → 72 → 74
• Drug improves by 20 points over time; Placebo improves by 4 points. The unequal changes indicate an interaction.

Formula:
$$F = \frac{MS_{\text{effect}}}{MS_{\text{error}}}$$

Case 7 — Chi-square Goodness-of-Fit

Scenario: A survey asks students to choose a favorite subject: Math, Science, or English.

Question: Is the distribution of responses different from equal chance?

Design/Test: Chi-square goodness-of-fit test.

Formula:
$$\chi^2 = \sum \frac{(O - E)^2}{E}$$

In words:
$$\chi^2 = \frac{\text{(Observed - Expected)}^2}{\text{Expected}}, , \text{summed across categories}$$

Case 8 — Chi-square Test of Independence

Scenario: A researcher tests whether gender (Male, Female) is related to sport preference (Soccer, Basketball, Tennis).

Question: Is there an association between gender and sport?

Design/Test: Chi-square test of independence.

Formula:
$$\chi^2 = \sum \frac{(O - E)^2}{E}$$

Case 9 — Mann–Whitney U Test

Scenario: Students in two different schools are ranked by teacher ratings.

Question: Do the two groups differ in median rank?

Design/Test: Mann–Whitney U test (non-parametric).

Formula:
$$U = n_1 n_2 + \frac{n_1 (n_1 + 1)}{2} - R_1$$

Where $$R_1$$ = sum of ranks for group 1.

Case 10 — Wilcoxon Signed-Rank Test

Scenario: The same students are ranked before and after training.

Question: Did the ranks change?

Design/Test: Wilcoxon signed-rank test (non-parametric).

Formula (summary):

• Compute differences (After – Before).
• Rank the absolute differences.
• Assign signs and sum.
• Test statistic = smaller of the two signed sums.

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Case 1

Scenario: A teacher compares test scores of students in two different classrooms (Class A vs. Class B).
Question: Are the two groups significantly different in mean score?
Answer: Independent-samples t-test.

Case 2

Scenario: A researcher tests the same group of students before and after tutoring.
Question: Did their scores improve after the program?
Answer: Paired-samples t-test (dependent t-test).

Case 3

Scenario: Three groups of students use different study methods: flashcards, highlighting, and practice tests.
Question: Do the study methods lead to different mean scores?
Answer: One-way ANOVA.

Case 4

Scenario: A psychologist measures anxiety scores in patients given three different drugs.
Question: Do the drugs produce different mean anxiety scores?
Answer: One-way ANOVA.

Case 5

Scenario: A study compares two groups of athletes: runners vs. swimmers, on reaction time.
Question: Are the two sports groups different in mean reaction time?
Answer: Independent-samples t-test.

Case 6

Scenario: Students are tested at three times: beginning, middle, and end of the semester.
Question: Did their scores change over time?
Answer: Repeated-measures ANOVA.

Case 7

Scenario: Two teaching methods (Lecture, Online) are tested across two times of day (Morning, Afternoon).
Question: What are the effects of method, time, and their interaction?
Answer: Two-way (factorial) ANOVA.

Case 8

Scenario: A company compares productivity of three work shifts (Day, Evening, Night) across two departments (Sales, Service).
Question: Are there main effects of shift and department, and is there an interaction?
Answer: Two-way (factorial) ANOVA.

Case 9

Scenario: Students are randomly assigned to a control or experimental group, and both groups are measured three times (Weeks 1, 2, 3).
Question: Is there an effect of group, time, and interaction?
Answer: Mixed (split-plot) ANOVA.

Case 10

Scenario: A survey asks students to choose their favorite subject: Math, Science, or English.
Question: Is the distribution of responses different from chance?
Answer: Chi-square goodness-of-fit test.

Case 11

Scenario: A researcher studies whether gender (Male, Female) is related to preference for sports (Soccer, Basketball, Tennis).
Question: Is there an association between gender and sport preference?
Answer: Chi-square test of independence.

Case 12

Scenario: Students are ranked by teacher ratings: 1st, 2nd, 3rd, etc. Two different teaching methods are compared on these ranks.
Question: Do the groups differ in median ranks?
Answer: Mann–Whitney U test (non-parametric).

Case 13

Scenario: The same students are ranked before and after a training program.
Question: Did the ranks change after training?
Answer: Wilcoxon signed-rank test (non-parametric).

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Welcome to Part 3 — Statistical Design of this free online high school statistics textbook. This practical section focuses on one of the most important skills in statistics: choosing the right statistical test for a given research question or scenario. Through decision-oriented lessons, real-world examples, and guided exercises, high school students learn how to evaluate study designs, identify appropriate descriptive and inferential methods, and avoid common pitfalls in statistical planning—essential preparation for AP Statistics, college-level courses, and real data analysis.

Part 3 bridges theory from earlier sections to applications, emphasizing experimental design, hypothesis formulation, variable types, sample size considerations, and test selection. Ideal for learners seeking to master statistical test selection, research design, and study planning in an intuitive, step-by-step format with free resources and interactive decision tools.

Lessons in Part 3: Statistical Design

1. Formulating Research Questions and Hypotheses – Crafting clear statistical questions and null/alternative hypotheses with practical examples.
2. Identifying Variables and Scales – Determining independent/dependent variables, scales of measurement, and their impact on test choice.
3. Choosing Between Descriptive and Inferential Statistics – When to summarize data vs. draw conclusions about populations.
4. Selecting the Right Test for One or Two Samples – Decision guide for t-tests, z-tests, and non-parametric alternatives.
5. Tests for Multiple Groups and Relationships – Flowcharts and criteria for ANOVA, correlation, regression, and chi-square tests.
6. Experimental vs. Observational Designs – Randomization, control groups, confounding variables, and ethical considerations.
7. Sample Size and Power Basics – Understanding why sample size matters and simple rules for adequate power.
8. Common Design Pitfalls and How to Avoid Them – Real-world cases highlighting errors in planning and interpretation.

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A mixed design combines a between-subjects factor (different groups of participants) with a within-subjects factor (the same participants measured repeatedly).
It is also called a split-plot design.

This design is common in psychology, education, and medicine.
Example: groups of patients (between factor) measured at different time points (within factor).

Structure of the Design

• Between-subjects factor: separate groups of participants (e.g., Drug vs. Placebo).
• Within-subjects factor: repeated measures on each participant (e.g., Week 1, Week 2, Week 3).
• Interaction: tests whether the effect of the within factor depends on the between factor.

Degrees of Freedom

For a design with:

• $$a$$ levels of the between-subjects factor
• $$b$$ levels of the within-subjects factor
• $$n$$ subjects in total
• Between: $$df_{\text{between}} = a - 1$$
• Subjects (within groups): $$df_{\text{subjects}} = N - a$$
• Within: $$df_{\text{within}} = b - 1$$
• Interaction: $$df_{A \times B} = (a-1)(b-1)$$
• Error terms depend on design partitioning.

Example

Two groups of students (Drug, Placebo) are tested across three weeks.

• Between factor (Group): Drug vs. Placebo
• Within factor (Time): Weeks 1–3
• Interaction: Drug improves by 20 points over time; Placebo improves by 4 points. The unequal changes indicate an interaction

Symbolic Formula

$$F = \frac{MS_{\text{effect}}}{MS_{\text{error}}}$$

Where $$\text{effect}$$ may be between, within, or interaction, depending on the hypothesis.

Definition

• Mixed (split-plot) ANOVA: combines a between factor (different groups) and a within factor (repeated measures).
• Use: tests real-world designs where groups are compared across time or conditions.

Visuals

Figure L9.1 — Mixed ANOVA Layout. Two groups (Drug, Placebo) × three repeated measures (Weeks 1–3).

Figure L9.2 — Mixed ANOVA Interaction Plot. Drug group line rises sharply; Placebo line flat.

Figure L9.3 — ANOVA Summary Table for mixed design.

Why This Matters

Mixed designs are realistic and powerful.
They reflect how experiments are often run: groups compared across time.
This design unites the logic of between- and within-subjects testing.

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In a repeated-measures design, the same participants are tested under multiple conditions.
This reduces error, because each person serves as their own control.
It is more powerful than a one-way ANOVA with independent groups.

Structure of the Design

• Rows (subjects): variation due to individual differences
• Columns (conditions): variation due to treatments
• Error: leftover variability after accounting for rows and columns

Degrees of Freedom

• $$df_{\text{rows}} = n - 1$$
• $$df_{\text{columns}} = k - 1$$
• $$df_{\text{error}} = (n - 1)(k - 1)$$

Where:

• $$n$$ = number of subjects
• $$k$$ = number of conditions

Example

Five students are tested under three conditions:

• Means increase steadily across conditions.
• ANOVA will partition the variance into Rows, Columns (treatments), and Error.

Symbolic Formula

$$F = \frac{MS_{\text{columns}}}{MS_{\text{error}}}$$

Formula in words:
$$F = \frac{\text{mean square for conditions}}{\text{mean square for error}}$$

Definition

• Repeated-measures ANOVA: compares means of the same group measured under different conditions.
• Advantage: controls for subject differences, increases statistical power.

Visuals

Figure L8.1 — Repeated-Measures Profile Plot. Each subject shown as a line across conditions.

Figure L8.2 — ANOVA Summary Table for repeated measures. Rows | Columns | Error.

Why This Matters

Repeated-measures designs are common in psychology, neuroscience, and medicine.
They allow researchers to detect changes over time or across treatments with fewer subjects and greater sensitivity.

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A factorial design includes two or more factors studied at once.
This allows us to test not only the effect of each factor separately, but also whether the factors interact.

Example: 2 × 2 Design

• Factor A: Teaching method (Lecture, Online)
• Factor B: Time of day (Morning, Afternoon)

This design has 4 groups (2 levels of A × 2 levels of B).

We can test:

1. The main effect of Factor A (method).
2. The main effect of Factor B (time).
3. The interaction between method and time.

The ANOVA Partition

For a 2 × 2 design:

• Main effect A: $$df_A = a - 1$$
• Main effect B: $$df_B = b - 1$$
• Interaction A × B: $$df_{A \times B} = (a - 1)(b - 1)$$
• Error (within): $$df_{\text{within}} = N - ab$$

Where $$a$$ = levels of Factor A, $$b$$ = levels of Factor B, $$N$$ = total number of observations.

Interaction

An interaction occurs when the effect of one factor depends on the level of the other factor.

• If lines in a plot are parallel, there is no interaction.
• If lines cross or diverge, there is an interaction.

Example

Suppose means are:

• Lecture: Morning = 70, Afternoon = 90
• Online: Morning = 80, Afternoon = 80

Here:

• Main effect of method: none (Lecture and Online both have marginal mean 80)
• Main effect of time: Afternoon > Morning overall (85 vs. 75)
• Interaction: Lecture scores rise with time, Online scores stay flat → non-parallel lines.

Visuals

Figure L7.1 — Factorial Layout (2 × 2). A 2 × 2 grid: Method × Time.

Figure L7.2 — Interaction Plot. Lecture line slopes upward, Online line flat. Caption: “Lines not parallel = interaction.”

Figure L7.3 — ANOVA Summary Table for 2 × 2 design. Source | SS | df | MS | F | p.

Why This Matters

Factorial designs let us test more than one factor at a time.
They are efficient and powerful, and the concept of interaction is central in science.
Two-way ANOVA is the foundation for more complex designs, including repeated measures and mixed ANOVA.

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The t-test compares two means. But what if we have three or more groups?
We could run multiple t-tests, but that inflates the chance of error.

The solution is the Analysis of Variance (ANOVA).
ANOVA partitions the variability into two parts: between groups and within groups.

Partitioning the Variance

Total variability = variability between groups + variability within groups.

• Between groups: differences due to the factor (treatment).
• Within groups: differences due to chance or individual variation.

Symbolic formula:
$$F = \frac{MS_{\text{between}}}{MS_{\text{within}}}$$

Formula in words:
$$F = \frac{\text{mean square between groups}}{\text{mean square within groups}}$$

Where:

• $$MS_{\text{between}} = \tfrac{SS_{\text{between}}}{df_{\text{between}}}$$
• $$MS_{\text{within}} = \tfrac{SS_{\text{within}}}{df_{\text{within}}}$$

Degrees of Freedom

• $$df_{\text{between}} = k - 1$$
• $$df_{\text{within}} = N - k$$
• $$df_{\text{total}} = N - 1$$

Where $$k$$ = number of groups, $$N$$ = total number of observations.

Example (One-way ANOVA)

Three groups of students use different study techniques:

• Group A: mean = 70
• Group B: mean = 75
• Group C: mean = 85

Suppose calculations give:

• $$SS_{\text{between}} = 300, , df_{\text{between}} = 2 \Rightarrow MS_{\text{between}} = 150$$
• $$SS_{\text{within}} = 200, , df_{\text{within}} = 12 \Rightarrow MS_{\text{within}} = 16.7$$

Then:

$$F = \frac{150}{16.7} = 9.0$$

This F value is compared to the F table at df = (2, 12).

Definition

• ANOVA: compares means across three or more groups.
• F ratio: signal-to-noise ratio (treatment effect vs. error).

Visual Placeholders

Figure L6.1 — Partitioning Variance. Total variability divided into Between vs. Within.

Figure L6.2 — One-way ANOVA Layout. Bar graph with three groups (A, B, C).

Figure L6.3 — ANOVA Summary Table. Source | SS | df | MS | F | p.

Why This Matters

ANOVA generalizes the t-test to multiple groups.
It is one of the most widely used tools in psychology, education, and medicine.
Understanding the F ratio is key: a large F means treatment differences are greater than chance variation. 

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