effect-size

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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This lecture emphasizes conceptual understanding of the t-test, its logic, and how it fits into the broader structure of statistical reasoning.

The t-test is one of the most widely used statistical tools.
It compares two means and asks: Is the difference between them real, or could it be due to chance?

The t-test is closely related to the z-test.
When the population standard deviation is unknown and the sample size is small, we use t instead of z.

Types of t-Tests

• One-sample t-test: compares a sample mean to a known or hypothesized population mean.
• Independent-samples t-test: compares means from two separate groups.
• Paired-samples t-test: compares two scores from the same group (before vs. after).

Symbolic Formulas

One-sample t-test
$$t = \frac{\bar{X} - \mu_0}{s / \sqrt{n}}$$

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

Paired-samples t-test
$$t = \frac{\bar{D}}{s_D / \sqrt{n}}$$

Degrees of Freedom

• One-sample: $$df = n - 1$$
• Independent-samples: $$df = n_1 + n_2 - 2$$
• Paired-samples: $$df = n - 1$$

Example (Independent t-Test)

Two groups of students try different study methods:

• Group A: \(n = 10\), mean = 80, SD = 10
• Group B: \(n = 10\), mean = 90, SD = 10

$$t = \frac{80 - 90}{\sqrt{\tfrac{10^2}{10} + \tfrac{10^2}{10}}} = \frac{-10}{\sqrt{10 + 10}} = \frac{-10}{\sqrt{20}} = \frac{-10}{4.47} = -2.24$$

Degrees of freedom = 18.
Compare this t-value to the critical value in the t-table at \(df = 18\).

Example (Paired t-Test)

Students take a test before and after tutoring.
Differences (After − Before): 4, 6, 5, 3, 2.

Mean difference:
$$\bar{D} = \frac{4 + 6 + 5 + 3 + 2}{5} = 4$$

Standard deviation of differences:
$$s_D = 1.58$$

$$t = \frac{4}{1.58 / \sqrt{5}} = \frac{4}{0.71} = 5.63$$

Degrees of freedom = 4.
This large t-value indicates strong evidence of improvement.

Definition

• Independent t-test: compares two separate groups.
• Paired t-test: compares the same group measured twice.
• Degrees of freedom (df): number of independent pieces of information.

Visuals

Figure L5.1 — Independent t-Test. Bar graph of two groups (A and B) with means and SEM error bars.

Figure L5.2 — Paired t-Test. Line plot showing before vs. after scores for each student.

Figure L5.3 — t vs. z Distribution. Overlay of the normal (z) curve and t curves with df = 5 and 20.

Why This Matters

The t-test is the workhorse of statistics.
It forms the foundation for many other methods (ANOVA, regression, mixed models).
Understanding t means understanding how we compare signal (mean difference) to noise (variability).

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