Mean Differences · Parametric Test
Independent Samples t-test Calculator
Determine if there is a statistically significant difference between the continuous means of two independent, unrelated groups.
Configure data
n = 0
n = 0
Descriptive Statistics
Independent Samples Test Results
Group Means & Distribution
Left: Group means with error bars representing ±1 Standard Deviation. Right: Individual data points jittered for visibility.
Effect Size (Cohen's d)
APA 7th edition reporting statement
Copy-paste template
Calculation Details
Interpretation Report
Step-by-step analysis & plain-language conclusions
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Hypotheses
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Assumptions check (Levene's Test)
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Computed statistical values
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Statistical decision
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Effect size & practical significance
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Conclusion & reporting
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All interpretations are generated from your data. Review before use in academic submissions.
Reference · Statistical Theory
How the Independent t-test Works
Theoretical foundation, assumptions, pooled vs unpooled variance, effect size interpretation, and primary references for the Independent Samples t-test.
What is the Independent t-test?
Originally developed by William Sealy Gosset under the pseudonym "Student" (1908), the independent-samples t-test compares the means of two independent groups to determine whether there is statistical evidence that the associated population means are significantly different.
Student's t-test (Equal Variances Assumed):
t = (M₁ - M₂) / √(s²_p / n₁ + s²_p / n₂)
df = n₁ + n₂ - 2
where s²_p (Pooled Variance) = ((n₁ - 1)s₁² + (n₂ - 1)s₂²) / (n₁ + n₂ - 2)
Welch's t-test (Equal Variances NOT Assumed):
t = (M₁ - M₂) / √(s₁²/n₁ + s₂²/n₂)
df is approximated using the Welch-Satterthwaite equation.
Assumptions
- Independence — Observations between and within the two groups must be independent.
- Continuous dependent variable — The outcome variable must be measured at the interval or ratio level.
- Normality — The dependent variable should be approximately normally distributed for each group. The t-test is robust to this if group sizes are relatively large ($n \ge 30$).
- Homogeneity of Variances — The variances of the dependent variable should be equal in both populations. Evaluated using Levene's Test. If violated, Welch's t-test is used, which adjusts the degrees of freedom to account for unequal variances.
Effect size (Cohen's d)
- Cohen's d measures the standardized difference between two means. It is calculated by dividing the mean difference by the pooled standard deviation.
- Conventions (Cohen, 1988):
|d| = 0.20 (Small effect)
|d| = 0.50 (Medium effect)
|d| = 0.80 (Large effect)
APA 7th edition reporting format
Template: "An independent-samples t-test was conducted to compare [Dependent Variable] between [Group 1] and [Group 2]. There was a significant difference in scores for [Group 1] (M = XX.X, SD = X.X) and [Group 2] (M = XX.X, SD = X.X); t(df) = X.XX, p = .XXX, d = X.XX."
Primary references
Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Lawrence Erlbaum Associates.
Field, A. (2018). Discovering statistics using IBM SPSS statistics (5th ed.). SAGE Publications.
Student [W. S. Gosset]. (1908). The probable error of a mean. Biometrika, 6(1), 1-25.
Welch, B. L. (1947). The generalization of ‘Student's’ problem when several different population variances are involved. Biometrika, 34(1/2), 28-35.