t-Tests for Independent and Correlated Samples

"To call in the statistician after the experiment is done may be no more than asking him to perform a post-mortem examination." — R.A. Fisher

The four variants of the t-test for two means cover every practical research scenario in psychology — small or large samples, independent or paired groups. Mastering these formulas and the step-by-step testing process is essential for IGNOU MAPC exams and real-world research.

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📄 Source Material

Source PDFs:

• 📄 Block-3/Unit-2.pdf — Pages 60–76
• 📚 MPC-006 Statistics in Psychology, Block 3

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1. The Four Situations: Overview

Situation	Sample Type	Size	Formula Used
1	Independent (Uncorrelated)	Large (N > 30)	CR with Z distribution
2	Independent (Uncorrelated)	Small (N ≤ 30)	t formula with pooled SD
3	Correlated (Dependent)	Large (N > 30)	t with r₁₂ correction
4	Correlated (Dependent)	Small (N ≤ 30)	Difference method or r₁₂ method

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2. Situation 1: Two Large Independent Samples (N > 30)

Already covered in the previous section (CR formula). Use: $CR = \frac{M_1 \sim M_2}{\sqrt{\frac{\sigma_1^2}{N_1} + \frac{\sigma_2^2}{N_2}}}$

Compare CR to ±1.96 (at .05) or ±2.58 (at .01). Use t-table for df = N₁ + N₂ – 2 when N is close to 30.

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3. Situation 2: Two Small Independent Samples (N ≤ 30)

For small samples, we use a pooled standard deviation estimate:

Formula (when raw scores are given):

$t = \frac{M_1 \sim M_2}{\sqrt{\frac{\sum d_1^2 + \sum d_2^2}{N_1 + N_2 - 2} \times \frac{N_1 + N_2}{N_1 \times N_2}}}$

Where d₁ = (X₁ – M₁) and d₂ = (X₂ – M₂)

Formula (when statistics are given, no raw data):

$t = \frac{M_1 \sim M_2}{\sqrt{\frac{\sigma_1^2(N_1-1) + \sigma_2^2(N_2-1)}{N_1 + N_2 - 2} \times \frac{N_1 + N_2}{N_1 \times N_2}}}$

df = (N₁ – 1) + (N₂ – 1) = N₁ + N₂ – 2

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Worked Example: Urban vs. Rural Boys (N < 30)

Attitude toward vocational courses test:

Urban Boys (N₁ = 10): Scores = 6, 7, 8, 10, 15, 16, 9, 10, 9, 10
→ M₁ = 10, Σd₁² = 92

Rural Boys (N₂ = 5): Scores = 4, 3, 2, 1, 5
→ M₂ = 3, Σd₂² = 10

H₀: No difference in attitude (M_urban = M_rural)
Level: .05, two-tailed

$t = \frac{10 - 3}{\sqrt{\frac{92 + 10}{10 + 5 - 2} \times \frac{10 + 5}{10 \times 5}}} = \frac{7}{\sqrt{\frac{102}{13} \times 0.30}} = \frac{7}{\sqrt{7.85 \times 0.30}} = \frac{7}{\sqrt{2.355}} = \frac{7}{1.535} \approx 4.46$

df = 13; t-table at .05, df=13: 2.16

Obtained t = 4.46 >> 2.16 → Reject H₀

Interpretation: Urban and rural boys differ significantly in their attitude toward vocational courses (p < .05). This is a real difference, not due to chance.

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4. Situation 3: Two Large Correlated Samples (N > 30)

When the same group is tested twice (pre-post), or matched pairs are used, the two sets of scores are correlated. The standard formula must be adjusted using the correlation coefficient (r₁₂):

$t = \frac{M_1 \sim M_2}{\sqrt{\sigma_{M_1}^2 + \sigma_{M_2}^2 - 2r_{12}\sigma_{M_1}\sigma_{M_2}}}$

Where: $\sigma_{M_1} = \frac{\sigma_1}{\sqrt{N}}, \quad \sigma_{M_2} = \frac{\sigma_2}{\sqrt{N}}$

df = N – 1 (since it's the same group)

Why Include r₁₂?

In correlated designs, individual differences are controlled — a high scorer at pre-test tends to score high at post-test too. The r₁₂ term removes this shared variance, giving a more sensitive (powerful) test.

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Worked Example: Training Achievement (Large Correlated)

64 students tested before and after vocational training:

	M	σ
Before training	52.50	7.25
After training	58.70	5.30
r₁₂ = 0.50, N = 100 (implied)

Step 1: Calculate σM values $\sigma_{M_1} = \frac{7.25}{\sqrt{100}} = 0.725, \quad \sigma_{M_2} = \frac{5.30}{\sqrt{100}} = 0.530$

Step 2: Apply correlated formula $t = \frac{58.70 - 52.50}{\sqrt{(0.725)^2 + (0.530)^2 - 2(0.50)(0.725)(0.530)}}$ $= \frac{6.2}{\sqrt{0.5256 + 0.2809 - 0.3848}} = \frac{6.2}{\sqrt{0.4217}} = \frac{6.2}{0.6494} \approx 9.54$

df = 99; t at .01 = 2.36; Obtained t = 9.54 >> 2.36 → Reject H₀

Interpretation (One-tailed): The gain after training is highly significant. Vocational training produced real improvement (p < .01).

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5. Situation 4: Two Small Correlated Samples (Difference Method)

When N ≤ 30 and you have paired/correlated data, use the difference method — simpler and does not require computing r₁₂:

Steps:

1. Compute D = (Post – Pre) for each subject
2. Calculate MD = ΣD / N
3. Compute SD = √[Σd²/(N–1)] where d = D – MD
4. Calculate SE_DM = SD / √N
5. Compute t = MD / SE_DM
6. df = N – 1

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Worked Example: Pre-test / Post-test (N = 12)

12 subjects; Pre-test and Post-test scores given.

Subject	Post	Pre	D = Post-Pre	d = D–MD	d²
1	40	42	-2	-10	100
2	62	50	12	+4	16
3	61	51	10	+2	4
4	35	26	9	+1	1
5	30	35	-5	-13	169
6	52	42	10	+2	4
7	68	60	8	0	0
8	51	41	10	+2	4
9	84	70	14	+6	36
10	50	38	12	+4	16
11	72	62	10	+2	4
12	63	55	8	0	0
Total			ΣD=96		Σd²=354

$MD = \frac{96}{12} = 8$ $SD = \sqrt{\frac{354}{11}} = \sqrt{32.18} = 5.67$ $SE_{DM} = \frac{5.67}{\sqrt{12}} = \frac{5.67}{3.46} = 1.64$ $t = \frac{8}{1.64} = 4.88, \quad df = 11$

t-table at .01, df=11: 3.11 (approximately); Obtained t = 4.88 → Reject H₀

Interpretation: The gain on post-test is real and statistically significant in 99 out of 100 cases.

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6. Step-by-Step Process for Significance Testing

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7. Type I and Type II Errors

Every decision in hypothesis testing risks one of two errors:

Type I Error (α Error)

• Definition: Rejecting H₀ when it is actually TRUE
• Also called: False positive / Alpha error
• Probability: = Level of significance (p = .05 or .01)
• Example: Concluding training improves performance when it actually doesn't

Type II Error (β Error)

• Definition: Retaining H₀ when it is actually FALSE
• Also called: False negative / Beta error
• Probability: Higher when significance threshold is very strict (e.g., .001)
• Example: Failing to detect a real training effect because your test was too conservative

The Trade-Off

• Stricter significance level (.01 vs .05) → reduces Type I error BUT increases Type II error
• More lenient level (.05) → reduces Type II error BUT increases Type I error
• Larger N → reduces both errors simultaneously

	H₀ True	H₀ False
Reject H₀	Type I Error (α)	✅ Correct (Power = 1-β)
Retain H₀	✅ Correct	Type II Error (β)

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8. Application in Indian Psychological Research

IGNOU Research Context

IGNOU's own research evaluations often compare student performance across different instructional conditions (print-based vs. multimedia). When sample sizes are small (as in experimental sub-groups), the small-sample t-test with pooled SD is essential.

Clinical Psychology (NIMHANS)

Pre-post comparisons are the backbone of treatment efficacy research in Indian hospitals. A psychologist testing CBT effectiveness on 20 patients would use the difference method (small correlated samples) with df = 19.

Educational Psychology

Studies comparing rural vs. urban students on achievement tests in India routinely use two large independent sample t-tests, with the assumption of approximate normality (valid for N > 30 per group).

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🌐 External Resources

Wikipedia

• Welch's t-test — Extension for unequal variances
• Type I and type II errors — Classic summary
• Paired difference test — Correlated samples

Educational Videos

• 🎥 t-Test for Two Independent Samples — Khan Academy
• 🎥 Paired t-Test — StatQuest

Research Articles

• Funder, D.C. & Ozer, D.J. (2019). Evaluating effect size in psychological research. Advances in Methods and Practices in Psychological Science.
• Open Science Collaboration (2020). Estimating the reproducibility of psychological science. Science.

Additional Resources

• Scribbr — t-Test Guide
• Statistics How To — Pooled Standard Deviation
• Khan Academy — Type I & II Errors
• Real Statistics — Dependent Samples t-test

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🧠 Memory Aids

SIMPLE mnemonic for Hypothesis Testing Steps

State hypothesis → Identify level → Match formula → Plug in values → Look up table → Evaluate result

Independent vs Correlated: "SAME or DIFFERENT?"

Same group tested twice → Correlated (use r₁₂ or difference method)
Different groups compared → Independent (use pooled SD)

Type Errors Poem

"Type ONE is fun — rejecting true null (false alarm!)
Type TWO is blue — missing a real effect (missed the norm!)"

Formula Navigator

Is N large (>30)?
    YES → Use CR/Z formulas
    NO  → Use t formulas with df = N₁+N₂-2 (independent) or N-1 (correlated)

Are groups independent?
    YES → Use S.E.DM with σ₁²/N₁ + σ₂²/N₂
    NO  → Use r₁₂ correction OR difference method

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✅ Self-Assessment Questions

Recall Level

1. What are the four situations in which two means are compared? Give an example for each.
2. State the formula for t when testing two small independent samples.
3. What is Type I error? Give an example in the context of clinical psychology.

Application Level

4. A music interest test is given to 30 girls (M=40.39, σ=8.69) and 25 boys (M=35.81, σ=8.33). Calculate t and interpret results at .05 level (df = 53, t-table = 2.01).
5. 10 subjects are tested before and after a memory training program (pre-test scores: 5, 15, 9, 11, 4, 9, 8, 13, 6, 16; post-test: 7, 9, 4, 15, 6, 13, 9, 5, 6, 12). Using the difference method, test if the gain is significant.

Analysis Level

6. A clinical psychologist sets the significance level at .001 to be "extra sure." What are the consequences for Type I and Type II errors? When might this strict level be justified?
7. A researcher claims to use a one-tailed test because "training can only improve performance." Is this reasoning sound? What happens if the training actually harms performance — what error is the researcher vulnerable to?