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Parametric Statistical Tests: t-test, F-test, ANOVA, and Pearson's r

Introduction​

A researcher at IGNOU wants to know whether male and female psychology students differ in their attitude towards teaching. Another wants to compare anxiety scores across three different therapy groups. A third wants to know if two correlations computed from different samples are significantly different from each other.

Each of these questions requires a different parametric test β€” and understanding which test to use, and why, is the core practical skill in inferential statistics. This unit covers the key parametric tests: the t-test (for comparing two means), ANOVA/F-test (for three or more means), and tests involving Pearson's r (correlation).

πŸ“„ Source PDF: Block-1/Unit-1.pdf
πŸ“š MPC-006 Statistics in Psychology β€” Block 1: Introduction to Statistics

Foundation: Sampling Distribution and Standard Error​

Before any inferential test, we need to understand sampling distributions and standard error β€” the conceptual pillars underlying all parametric inference.

Sampling Distribution of Means​

When multiple large random samples (N > 30) are drawn from a population, their means form a distribution β€” the sampling distribution of means. The Central Limit Theorem tells us this distribution is:

  • Normally distributed (regardless of the original population's shape, for large N)
  • Centred on the population mean (ΞΌ)
  • Has a standard deviation called the Standard Error of the Mean (SEM)

Standard Error of the Mean (SEM)​

The SEM tells us how much sample means are expected to deviate from the population mean:

SE_M = Οƒ / √N

Where:

  • Οƒ = Standard deviation of the population
  • N = Sample size

Interpretation: A smaller SEM means sample means cluster tightly around the population mean β€” more reliable estimation.

For small samples (N ≀ 30), the sampling distribution follows the t-distribution (developed by William Sealy Gosset, ~1815), which is broader and flatter than the normal curve, with heavier tails.

The t-Test: Comparing Two Means​

The t-test tests whether two group means differ significantly β€” whether the observed difference is larger than what we'd expect from sampling error alone.

There are three scenarios, each with a different formula:

1. Independent Large Samples (z-test / CR)​

When comparing means from two independent groups with large samples (N > 30 each):

  • Groups are "independent" when drawn from totally different, unrelated populations
  • The formula uses the Critical Ratio (CR) β€” essentially a z-test

The Standard Error of the Difference between Means (Οƒ_dM) is:

Οƒ_dM = √(Οƒ_M1Β² + Οƒ_M2Β²)

Where Οƒ_M1 and Οƒ_M2 are the standard errors of the means of Sample 1 and Sample 2 respectively.

Worked example: Male M.A. Psychology students at IGNOU score a mean of 55 on an attitude scale; female students score 59. The difference is 4 β€” but is this real or sampling error? The CR/z-test determines whether this 4-point difference exceeds what chance would produce.

2. Independent Small Samples (t-test)​

When N is small (≀ 30 per group) and groups are independent, use Student's t-test. The t-distribution accounts for the extra uncertainty introduced by small samples.

3. Dependent/Correlated Samples (Paired t-test)​

Means are dependent/correlated when obtained from:

  • The same participants tested twice (pre-test/post-test)
  • Matched pairs β€” participants matched on key attributes

The formula for correlated means:

t = (M₁ - Mβ‚‚) / √(Οƒ_M1Β² + Οƒ_M2Β² - 2r₁₂σ_M1Οƒ_M2)

Where:

  • M₁, Mβ‚‚ = Means of initial and final testing
  • Οƒ_M1, Οƒ_M2 = Standard errors of the two means
  • r₁₂ = Correlation between scores on initial and final testing

The r₁₂ term adjusts for the non-independence β€” if scores at two time points are correlated, the effective variance is reduced.

When to use paired t-test:

  • Pre/post intervention study (measure anxiety before and after mindfulness training)
  • Repeated measures design (same participants in two conditions)
  • Matched subjects design (matched for age, IQ, or other variables)

ANOVA: Comparing Three or More Means​

When we have more than two groups, using multiple t-tests inflates the Type I error rate (false positive risk). Analysis of Variance (ANOVA) solves this by testing all group differences simultaneously, producing a single F-statistic.

F-statistic​

F = Variance Between Groups / Variance Within Groups

  • If the null hypothesis is true (all means equal), F β‰ˆ 1
  • If group means differ meaningfully, the numerator increases β†’ F > 1
  • Statistical significance is determined by comparing computed F to a critical F value in tables

Assumptions of ANOVA​

  1. Normal distribution in the population (though robust to violations with large N)
  2. Random sampling from subpopulations β€” randomisation is the "keystone" of ANOVA
  3. Homogeneity of variances β€” tested using F-max test:

F_max = Largest Variance / Smallest Variance

If computed F-max < critical F-max at p = .05, variances are homogeneous.

IGNOU example: Suppose we compare the teaching aptitude of M.A. Psychology students from three different zones (North, South, East). ANOVA tests whether any of the three group means differ significantly from one another.

Why Pearson's r Needs Special Treatment​

The sampling distribution of Pearson's r is not normal in most practical situations:

  • When r is very high (β‰₯ 0.80) and N is small β†’ distribution is negatively skewed
  • When r is very low (≀ 0.20) β†’ distribution is positively skewed
  • Only when r is near zero and N is large (β‰₯ 30) is the distribution approximately normal

Fisher's Z Transformation​

The solution is Fisher's Z transformation β€” converting r to a Z coefficient whose sampling distribution is normal regardless of sample size and population r:

  • Use a conversion table (Fisher's r-to-Z table) to find Z for any r
  • The SE of Z depends only on N:

SE_Z = 1 / √(N - 3)

This is elegant: the standard error of the Fisher Z coefficient is a simple function only of sample size.

Testing Significance of Pearson's r​

To test whether a single correlation is significantly different from zero, convert r to Z and compute:

CR = (Z₁ - Zβ‚‚) / SE_DZ

Comparing Two Correlations (Independent Samples)​

When testing whether two r values from different, independent samples differ significantly:

SE_DZ = √(1/(N₁ - 3) + 1/(Nβ‚‚ - 3))

CR = (Z₁ - Zβ‚‚) / SE_DZ

Example: A researcher finds r = .65 between anxiety and performance in Sample 1 (N = 50) and r = .42 in Sample 2 (N = 48). Are these correlations significantly different from each other?

Step 1: Convert both r values to Fisher's Z using tables
Step 2: Compute SE_DZ using the formula above
Step 3: Compute CR
Step 4: Compare CR to critical z-value (1.96 for p = .05)

Summary of Parametric Tests​

TestPurposeSample TypeN Requirement
z-test / CRCompare two independent meansIndependent largeN > 30 per group
t-test (independent)Compare two independent meansIndependentAny (especially small)
t-test (paired)Compare two correlated meansDependent/matchedAny
ANOVA (F-test)Compare three or more meansIndependentLarge preferred
Pearson's rMeasure linear correlationAnyN > 30 for normal approximation
Fisher's Z testTest/compare correlationsIndependent/singleAny (recommended for all r)

🌐 External Resources​

Wikipedia​

Educational Videos​

Research Papers​

Additional Resources​

🧠 Memory Aid: "2 = t, 3+ = F, Corr = rβ†’Z"​

2 groups β†’ t-test
3+ groups β†’ F-test (ANOVA)
Correlation significance β†’ Pearson's r β†’ Fisher's Z

For t-test types:

Independent groups β†’ Independent samples t-test
Dependent (same/matched) groups β†’ Dependent samples (paired) t-test

βœ… Self-Assessment Questions​

Level 1 β€” Recall​

  1. What is the formula for the Standard Error of the Mean?
  2. Why is Fisher's Z transformation used instead of directly testing Pearson's r?

Level 2 β€” Understanding​

  1. A researcher compares anxiety scores between two independent groups (N₁ = 35, Nβ‚‚ = 40). She uses an independent samples t-test. Later she realises participants in both groups completed both conditions (it was a within-subjects design). What test should she have used, and why?
  2. What is the F-max test? When and why is it used before ANOVA?

Level 3 β€” Application & Analysis​

  1. A researcher finds that a mindfulness intervention reduces depression scores from a pre-test mean of 24.6 to a post-test mean of 18.2 (N = 28 matched pairs, r₁₂ = .72, Οƒ_M1 = 1.8, Οƒ_M2 = 1.6). Using the paired t-test formula provided in this unit, calculate the denominator of the t-ratio. Is the denominator larger or smaller than if you had assumed the groups were independent (r₁₂ = 0)? What does this tell you about the effect of matching?

Summary​

Parametric statistical tests form the core toolkit of psychological research. The t-test handles two-group comparisons (independent or paired), while ANOVA extends this to three or more groups using the F-ratio. Pearson's r measures correlation, but because its sampling distribution is non-normal, Fisher's Z transformation is used for significance testing and comparison. All these tests rest on the foundation of the Central Limit Theorem, standard error, and the assumptions explored in the previous unit. Mastery of these tests β€” including their logic, formulas, and assumptions β€” is essential for both conducting and critically reading psychological research.

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