Hypothesis Testing (Pearson r & Paired t-Test)
Calculate Pearson r correlation and paired samples t-test with automatic null hypothesis (H₀) accept/reject decisions, degrees of freedom, and copyable APA Chapter 4 statistical narratives.
How to Run Hypothesis Testing in 3 Simple Steps
Select your statistical test: 'Pearson r Correlation' (for relationship between X and Y) or 'Paired Samples t-Test' (for Pre-Test vs. Post-Test differences).
Enter or paste your paired data in the two columns below (or click a sample preset like '🎲 Study Hours vs. Exam Scores').
Instantly view the computed statistic (r or t), degrees of freedom (df), critical value, explicit Null Hypothesis (H₀) Decision badge, and copy the APA Chapter 4 results paragraph.
Statistical Test Selection
Switch between correlational relationship testing and experimental pre-test/post-test difference testing.
Enter paired scores per subject, or paste two columns from Excel/Google Sheets.
Column 1 = · Column 2 =
| # | X × Y | Difference (D = Post - Pre) | |||
|---|---|---|---|---|---|
| Mean | ∑XY: | Mean Diff: |
Current Observations: paired data points.
Degrees of Freedom (df)
Critical Value (t_crit)
Decision Rule: Since computed || = is critical threshold ,
Step-by-Step Mathematical Formula Substitution
Pearson Formula: r = [ n∑XY - (∑X)(∑Y) ] / √[ (n∑X² - (∑X)²) × (n∑Y² - (∑Y)²) ]
1. Numerator: [ ( × ) - ( × ) ] =
2. Denominator: √[ ( × - ²) × ( × - ²) ] =
3. Pearson r: / = (Reported as )
4. Significance t-test: t = [ r√df ] / √(1 - r²) = [ × √ ] / √(1 - ) =
Paired t-Test Formula: t = D̄ / SE_D , where SE_D = S_D / √n
1. Mean Difference (D̄): ∑D / n = / =
2. Standard Deviation of Differences (S_D): √[ ∑(D - D̄)² / (n - 1) ] =
3. Standard Error of Difference (SE_D): / √ =
4. Computed t-Value: / =
5. Cohen's d: |D̄| / S_D = || / = ()
Chapter 4 Statistical Narrative (APA 7th Edition)
Copy and paste directly into your thesis Chapter 4 results and discussion section.
When to Use Pearson r vs. Paired t-Test in Undergraduate and Action Research
A practical guide to hypothesis testing, statistical decision-making, and Chapter 4 APA reporting.
1. Defining Your Research Design: Correlational vs. Experimental
One of the most frequent questions posed by thesis advisers during proposal defense is: "Did you choose the right statistical tool for your statement of the problem?" Choosing between Pearson r and a Paired Samples t-test depends entirely on your research design and research questions:
- Use Pearson Product-Moment Correlation (r) when conducting Descriptive-Correlational Research. Your goal is to determine if two continuous variables covary—whether changes in Variable X are associated with changes in Variable Y. You are not manipulating any variable or providing an intervention. Examples include: "Is there a significant relationship between social media screen time and general weighted average (GWA)?" or "Does math anxiety correlate with achievement test scores?"
- Use Paired Samples t-Test (Dependent t-test) when conducting Experimental, Pre-Experimental (One-Group Pretest-Posttest), or Action Research. In this setup, the same group of participants is measured at two different points in time (Time 1 vs. Time 2)—specifically before and after an educational intervention, mobile app, training seminar, or instructional module. An example is: "Is there a significant difference in student proficiency before and after the implementation of gamified learning worksheets?"
2. Formulating Null (H₀) and Alternative (H₁) Hypotheses in Chapter 1
In Chapter 1, quantitative studies must state hypotheses in null form:
Correlational Hypothesis (Pearson r)
H₀: There is no significant relationship between [Variable X] and [Variable Y] among the respondents.
H₁: There is a significant relationship between [Variable X] and [Variable Y].
Intervention Hypothesis (Paired t-Test)
H₀: There is no significant difference between the pre-test and post-test scores of the participants.
H₁: There is a significant difference between the pre-test and post-test scores of the participants.
3. How to Make the Decision: The Critical Value Rule (α = 0.05)
To test hypotheses objectively, researchers set an alpha risk level—almost universally α = 0.05 in educational and social science research. This signifies a 5% risk of concluding that a difference or relationship exists when it actually does not (Type I error).
The decision rule follows a clean mathematical comparison:
➔ REJECT H₀ · The result is statistically significant.
If |t_computed| < t_critical (or p-value ≥ 0.05):
➔ ACCEPT / FAIL TO REJECT H₀ · The result is not statistically significant.
4. Interpreting Pearson r Strength and Cohen's d Effect Size
Finding statistical significance is only half of your Chapter 4 analysis. You must also interpret the practical magnitude of the effect:
- Pearson r Guidelines: ±0.90 to ±1.00 (Very High), ±0.70 to ±0.89 (High), ±0.50 to ±0.69 (Moderate), ±0.30 to ±0.49 (Low), 0.00 to ±0.29 (Negligible). Furthermore,
r²(the coefficient of determination) indicates the exact percentage of variation in Variable Y explained by Variable X. - Cohen's d Guidelines (Paired t-Test):
d = 0.20indicates a small effect;d = 0.50indicates a moderate effect; andd = 0.80or higher indicates a large educational effect. Action research projects that achieve a Cohen's d ≥ 0.80 provide compelling evidence that an instructional intervention dramatically accelerated student competencies.
Frequently Asked Questions
What is the difference between Pearson r correlation and a Paired Samples t-test?
Pearson r is used in correlational research to measure the strength and direction of a linear relationship between two continuous variables (e.g., Study Hours and Exam Scores). A Paired Samples t-test is used in experimental or action research to compare the means of the same group tested twice (e.g., Pre-Test vs. Post-Test) to evaluate whether an intervention produced a statistically significant difference.
How is the decision to accept or reject the Null Hypothesis (H₀) determined?
At a significance level of α = 0.05, you compare the absolute value of the computed test statistic (|t| or |r|) against the critical value from the t-distribution table for your degrees of freedom (df). If |t_computed| ≥ t_critical (or p < 0.05), you REJECT the null hypothesis (H₀), indicating a statistically significant relationship or difference. If |t_computed| < t_critical (p ≥ 0.05), you FAIL TO REJECT / ACCEPT H₀.
What are the degrees of freedom (df) for Pearson r and Paired t-test?
For Pearson r correlation, degrees of freedom are df = n - 2, where n is the number of paired observations (X, Y). For a paired-samples t-test, degrees of freedom are df = n - 1, where n is the number of paired participants with both pre-test and post-test scores.
What does Cohen's d effect size mean in a paired t-test?
Cohen's d measures the standardized magnitude of the difference between pre-test and post-test scores: d = |Mean Difference| / Standard Deviation of Differences. Benchmarks defined by Cohen (1988) are: d = 0.20 (Small effect), d = 0.50 (Medium effect), and d = 0.80 or greater (Large effect). While p-value indicates statistical significance, Cohen's d proves practical educational significance.
How do I report Pearson r or paired t-test results in APA 7th edition format?
For Pearson r: "A Pearson correlation revealed a significant relationship between Variable X and Variable Y, r(28) = .842, p < .05." For paired t-test: "Scores significantly increased from Pre-Test (M = 14.50, SD = 3.20) to Post-Test (M = 24.80, SD = 3.10), t(24) = 11.23, p < .05, d = 1.15." Our tool automatically generates this exact text for your Chapter 4.