T-Test Calculator | Best Powerful Statistical Tool

T-Test Calculator

Our t-test calculator performs statistical hypothesis testing showing t-statistic, p-value, and significance conclusions instantly. This comprehensive t-test calculator performs complete statistical analysis including one-sample t-test comparing sample mean to known population mean, paired t-test calculator for dependent samples measuring before/after differences or matched pairs, two-sample t-test calculator for independent groups comparing means between separate populations, and complete output showing t-statistic measuring effect size in standard error units, p-value indicating probability results occurred by chance under null hypothesis, degrees of freedom affecting critical value thresholds, confidence intervals estimating population parameters, and statistical significance interpretation helping determine whether to reject null hypothesis. Whether you’re analyzing experimental data, conducting research studies, performing quality control testing, or learning statistics fundamentals, our t-test calculator provides accurate results using standard statistical formulas with clear interpretations supporting data-driven decisions, hypothesis testing, and scientific conclusions in academic research, business analytics, and scientific applications requiring rigorous statistical analysis and evidence-based reasoning.

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📊 Statistical Interpretation

Published: January 19, 2025 | Updated: January 19, 2025

How to Use the T-Test Calculator

Using our t-test calculator is straightforward providing accurate statistical analysis for hypothesis testing. Start by selecting the appropriate test type from the dropdown menu based on your experimental design. Choose one-sample t-test when comparing your sample mean to a known population mean (like testing if average test scores differ from national average of 100). Select paired t-test calculator option for dependent samples where same subjects measured twice (before/after treatment) or naturally paired data (twins, matched controls). Choose two-sample t-test calculator for independent groups comparing means between separate populations (males vs females, treatment vs control with different subjects).

Enter your data values in the sample data field separating values by commas, spaces, or line breaks. The t-test calculator accepts any sample size though recommends n≥30 for robust results when distribution uncertain. For one-sample test, input population mean (μ₀) you’re testing against – this is your null hypothesis value. For paired t-test calculator, enter first measurements in Sample 1 field and second measurements in Sample 2 field ensuring equal number of paired observations. For two-sample t-test calculator, enter Group 1 data in first field and Group 2 data in second field with sample sizes that can differ between groups.

Set significance level (α) representing acceptable Type I error rate – probability of rejecting true null hypothesis. Standard α=0.05 means 5% chance concluding difference exists when actually doesn’t, corresponding to 95% confidence level. More conservative α=0.01 requires stronger evidence reducing false positive risk. Less conservative α=0.10 accepts higher false positive risk useful in exploratory research. The paired t-test calculator and two-sample t-test calculator use selected α determining statistical significance threshold for p-value comparison.

Click “Calculate T-Test” to see comprehensive results including t-statistic measuring how many standard errors sample mean(s) deviate from expected value under null hypothesis, p-value showing probability of observing results this extreme if null hypothesis true, degrees of freedom affecting critical value from t-distribution, and clear interpretation stating whether to reject or fail to reject null hypothesis. Results explain statistical significance in practical terms helping you understand whether observed differences represent real effects or random chance variation in your t-test calculator analysis.

Understanding T-Test Calculator Formulas

The t-test calculator uses different formulas depending on test type selected, all based on fundamental principle of comparing observed differences to expected variation under null hypothesis. T-statistic represents signal-to-noise ratio where signal is observed effect (difference between means) and noise is standard error measuring sampling variability. Larger absolute t-values indicate stronger evidence against null hypothesis that no difference exists between populations or that sample comes from specified population.

ONE-SAMPLE T-TEST:
t = (x̄ – μ₀) / (s / √n)

where:
x̄ = sample mean
μ₀ = hypothesized population mean
s = sample standard deviation
n = sample size
df = n – 1

PAIRED T-TEST:
t = d̄ / (s_d / √n)

where:
d̄ = mean of paired differences
s_d = standard deviation of differences
n = number of pairs
df = n – 1

TWO-SAMPLE T-TEST (Equal Variances):
t = (x̄₁ – x̄₂) / √[s_p² (1/n₁ + 1/n₂)]

where:
x̄₁, x̄₂ = sample means
s_p² = pooled variance
n₁, n₂ = sample sizes
df = n₁ + n₂ – 2

P-VALUE:
P(|T| > |t_observed|) using t-distribution with df

These formulas power our t-test calculator for all hypothesis testing computations. One-sample formula compares sample mean to hypothesized population mean accounting for sampling variability through standard error (s/√n). Paired t-test calculator formula treats paired differences as single sample testing whether mean difference significantly differs from zero, effectively one-sample test on difference scores reducing individual variability increasing statistical power. Two-sample t-test calculator formula compares means between independent groups using pooled variance estimate assuming equal population variances, or Welch’s adjustment for unequal variances.

Degrees of freedom (df) affect t-distribution shape determining critical values for significance. Smaller df produce wider t-distributions requiring larger t-statistics for significance. One-sample and paired tests use df = n-1 because estimating one population parameter (mean). Two-sample test uses df = n₁+n₂-2 because estimating two group means. The t-test calculator automatically computes appropriate df based on test type and sample sizes ensuring accurate p-value calculation from correct t-distribution.

P-value represents probability of obtaining t-statistic as extreme or more extreme than observed, assuming null hypothesis true. The paired t-test calculator and two-sample t-test calculator compute p-values from t-statistic and df using cumulative t-distribution function. Two-tailed tests (default) consider extremes in both directions testing for any difference; one-tailed tests consider only one direction testing for specific increase or decrease. Compare p-value to α: if p<α, reject null hypothesis concluding statistically significant difference; if p>α, fail to reject null hypothesis insufficient evidence for difference.

T-Test Calculator Examples and Applications

Example 1: One-Sample T-Test

Scenario: Quality control testing whether average product weight differs from target 500 grams. Sample of 15 products weighed: 498, 502, 495, 501, 499, 503, 497, 500, 498, 502, 496, 501, 499, 500, 498 grams.

Using the t-test calculator:

Test type: One-Sample T-Test
Sample data: 498, 502, 495, 501, 499, 503, 497, 500, 498, 502, 496, 501, 499, 500, 498
Population mean (μ₀): 500
Sample size: n = 15
Sample mean: x̄ = 499.27
Sample SD: s = 2.19
t-statistic: t = (499.27 – 500) / (2.19 / √15) = -1.29
Degrees of freedom: df = 14
P-value: p = 0.218

Interpretation: With p=0.218 > α=0.05, fail to reject null hypothesis. Insufficient evidence to conclude average product weight differs from target 500 grams. While sample mean (499.27g) is slightly below target, difference is not statistically significant and could reasonably occur by chance in random sampling. Production process meets quality standards based on this sample using t-test calculator analysis.

Example 2: Paired T-Test Calculator

Scenario: Testing whether new training program improves test scores. Ten students tested before and after training:

Before: 75, 68, 82, 71, 79, 73, 77, 69, 80, 76

After: 82, 75, 88, 78, 85, 79, 83, 74, 86, 82

Using the paired t-test calculator:

Test type: Paired T-Test
Sample 1 (before): 75, 68, 82, 71, 79, 73, 77, 69, 80, 76
Sample 2 (after): 82, 75, 88, 78, 85, 79, 83, 74, 86, 82
Paired differences: 7, 7, 6, 7, 6, 6, 6, 5, 6, 6
Mean difference: d̄ = 6.2 points
SD of differences: s_d = 0.63
t-statistic: t = 6.2 / (0.63 / √10) = 31.1
Degrees of freedom: df = 9
P-value: p < 0.001

Interpretation: With p<0.001 << α=0.05, strongly reject null hypothesis. Training program produces statistically significant improvement in test scores averaging 6.2 points increase. Extremely large t-statistic (31.1) indicates very strong effect relative to variability. The paired t-test calculator shows training is highly effective with practical significance beyond just statistical significance warranting program continuation and potential expansion.

Example 3: Two-Sample T-Test Calculator

Scenario: Comparing effectiveness of two teaching methods. Method A students (n=12): 78, 82, 75, 88, 79, 85, 76, 83, 80, 84, 77, 81. Method B students (n=10): 85, 88, 82, 91, 87, 89, 83, 86, 84, 90.

Using the two-sample t-test calculator:

Test type: Two-Sample T-Test
Group 1 (Method A): Mean = 80.67, SD = 3.77, n = 12
Group 2 (Method B): Mean = 86.50, SD = 2.84, n = 10
Pooled SD: s_p = 3.40
t-statistic: t = (80.67 – 86.50) / √[3.40²(1/12 + 1/10)] = -3.98
Degrees of freedom: df = 20
P-value: p = 0.001

Interpretation: With p=0.001 < α=0.05, reject null hypothesis. Method B produces statistically significantly higher scores than Method A, with mean difference of 5.83 points. Negative t-statistic indicates Group 1 (Method A) scored lower than Group 2 (Method B). The two-sample t-test calculator demonstrates Method B superior teaching approach, though should consider practical significance, implementation costs, and replication before policy changes.

Choosing the Right T-Test Calculator Type

Selecting appropriate test type is critical for valid statistical conclusions using the t-test calculator. One-sample t-test applies when comparing single sample mean to known or hypothesized population value. Use when you have specific comparison value from theory, published standards, regulatory requirements, or previous research. Examples include testing if average IQ differs from 100, if product meets weight specifications, if treatment effect differs from zero in absolute terms. One-sample tests require knowing population mean but not population standard deviation (estimated from sample). The t-test calculator provides one-sample option for these research designs.

Paired t-test calculator option applies to dependent samples where observations naturally linked. Common designs include before/after measurements on same subjects (pre-test/post-test designs, treatment effects within subjects), matched pairs (twins, siblings, matched controls on relevant characteristics), or repeated measures (multiple trials by same subjects). Pairing reduces variability from individual differences because each subject serves as own control, increasing statistical power detecting smaller effects with fewer subjects. Requirements: equal number of pairs, independence between pairs (dependence within pairs expected), approximately normal distribution of difference scores. Use paired t-test when observations connected, not when groups independent.

Two-sample t-test calculator applies to independent groups comparing means between separate populations. Subjects randomly assigned to groups or naturally fall into distinct categories. Examples include treatment vs control with different subjects, males vs females, Brand A vs Brand B with independent samples. This design requires independence between all observations – measurements from one group don’t influence other group. Sample sizes can differ between groups. The two-sample t-test calculator assumes equal population variances (pooled variance approach) though Welch’s adjustment handles unequal variances. Use two-sample test when groups completely independent with no natural pairing or matching between observations across groups.

Common mistakes in test selection include using two-sample when should use paired (ignoring natural pairing loses statistical power and violates independence assumption), using paired when should use two-sample (incorrect pairing inflates Type I error), or using one-sample when comparing groups (one-sample tests only one mean against constant). The paired t-test calculator and two-sample t-test calculator produce different results from same data because account for design differently. Always match test to experimental design – let data structure determine test type not desired outcome. If uncertain, diagram your study design showing measurement structure clarifying appropriate test for the t-test calculator analysis.

Interpreting T-Test Calculator P-Values

P-value interpretation is crucial for valid conclusions from t-test calculator results. The p-value represents probability of obtaining test results at least as extreme as observed, assuming null hypothesis is true. Not probability null hypothesis is true (common misconception) – rather, probability of observing data this extreme if null were true. Small p-values (p<α) indicate observed data unlikely under null hypothesis providing evidence to reject null concluding alternative hypothesis more plausible. Large p-values (p>α) indicate observed data reasonably likely under null hypothesis insufficient evidence to reject null.

Standard significance threshold α=0.05 means accepting 5% chance of Type I error (false positive – rejecting true null hypothesis). When p<0.05, results are "statistically significant" - difference unlikely due to random chance alone at 5% error rate. The paired t-test calculator and two-sample t-test calculator compare p-value to α: reject null if p<α, fail to reject if p>α. “Fail to reject” doesn’t prove null true – merely insufficient evidence to conclude it false. Like jury verdict “not guilty” doesn’t mean innocent, just insufficient evidence for conviction. Never accept null hypothesis, only fail to reject.

Common p-value interpretations: p<0.001 indicates very strong evidence against null ("highly significant"), p<0.01 indicates strong evidence, p<0.05 indicates moderate evidence (conventional threshold), p<0.10 indicates weak evidence (marginally significant), p>0.10 indicates insufficient evidence. However, arbitrary thresholds criticized – p=0.049 and p=0.051 produce identical evidence despite different significance conclusions. The t-test calculator reports exact p-value allowing nuanced interpretation beyond binary significant/not significant dichotomy. Consider effect size and practical importance alongside statistical significance.

P-value limitations: statistical significance doesn’t guarantee practical importance (large samples detect trivial differences as significant), p-value doesn’t measure effect size (only whether effect differs from zero), multiple testing inflates Type I error (testing 20 hypotheses at α=0.05 produces one false positive on average), and publication bias toward significant results distorts literature. Use the t-test calculator p-value as one piece of evidence alongside confidence intervals showing effect magnitude, effect size measures indicating practical importance, replication in independent samples, and theoretical plausibility. Evidence-based reasoning requires considering multiple factors beyond p-value alone for robust scientific conclusions and data-driven decisions.

T-Test Assumptions and Limitations

Valid t-test calculator results require meeting specific statistical assumptions. First assumption: independence of observations within samples. Each data point must be independent measurement unaffected by other measurements. Violations occur with serial correlation (time series data where observations correlated), spatial correlation (geographic clusters), or hierarchical structure (students within classrooms). Independence assumption most critical – violations invalidate t-test requiring specialized methods like mixed models or time series analysis. The paired t-test calculator requires independence between pairs (not within pairs where dependence expected), while two-sample t-test calculator requires independence between all observations.

Second assumption: approximate normal distribution of data or sampling distribution of means. For one-sample and two-sample tests, data should be approximately normal especially with small samples (n<30). Large samples (n≥30) relax normality requirement through Central Limit Theorem - sampling distribution of mean approaches normal regardless of population distribution. The paired t-test calculator requires normality of difference scores, not individual measurements. Assess normality through histograms, Q-Q plots, or Shapiro-Wilk test. Moderate violations acceptable with larger samples; severe violations (highly skewed, heavy tails, extreme outliers) warrant non-parametric alternatives like Wilcoxon tests or data transformation.

Third assumption for two-sample t-test calculator: equal population variances (homogeneity of variance). Standard t-test assumes σ₁²=σ₂² using pooled variance estimate. Assess with F-test or Levene’s test, or informally by comparing sample standard deviations (s₁ and s₂ shouldn’t differ by more than factor of 2-3). Violations handled by Welch’s t-test using separate variance estimates and adjusted degrees of freedom (provided by many statistical packages). The paired t-test calculator doesn’t require equal variances because analyzes difference scores. One-sample test has no variance assumption as compares to population mean not another sample.

Additional considerations: outliers distort means and increase variability reducing statistical power. Examine data for anomalous values investigating measurement errors or special causes. Small samples (n<10) require very careful attention to assumptions and may lack power detecting moderate effects. Very large samples (n>1000) detect trivial differences as statistically significant requiring emphasis on effect size and confidence intervals over p-values. The t-test calculator provides accurate results when assumptions reasonably met but cannot guarantee validity with severe violations. Always examine data distributions, check assumptions, and consider alternative analyses when assumptions questionable for robust statistical conclusions.

Common T-Test Calculator Mistakes

Don’t confuse statistical significance with practical importance when using the t-test calculator. A result can be statistically significant (p<0.05) yet practically trivial if effect size small. Large samples detect tiny differences as significant - example: two teaching methods differing by 0.5 test points (on 100-point scale) might show p<0.001 with n=1000 per group, but 0.5 point difference has no educational importance. Conversely, moderate effect might miss significance with small sample having insufficient power. The paired t-test calculator and two-sample t-test calculator report p-values indicating statistical significance, but examine mean differences, confidence intervals, and effect sizes (Cohen's d) assessing practical significance for real-world decisions.

Avoid selecting wrong test type for experimental design. Using two-sample t-test calculator for paired data loses statistical power and violates independence assumption because treats paired observations as independent when actually correlated. Example: testing weight loss program with before/after measurements on 20 subjects – using two-sample test (treating as 40 independent observations) instead of paired test (treating as 20 differences) inflates standard error reducing power and producing incorrect p-value. Using paired t-test calculator for independent groups is equally wrong – artificially pairs unrelated observations violating assumptions. Match test to study design, not desired outcome or convenience.

Don’t make directional errors interpreting negative t-statistics. The t-test calculator reports signed t-values where sign indicates direction of difference. Negative t-statistic means sample mean(s) lower than comparison value (one-sample) or first group mean lower than second group mean (two-sample). Positive t-statistic means opposite. When interpreting: “Group A scored significantly higher than Group B” requires positive t-statistic if A listed first, negative if B listed first. Check which group coded as 1 vs 2, and interpret direction accordingly. Magnitude matters for evidence strength (larger absolute value = stronger evidence), but sign indicates direction critical for substantive conclusions.

Avoid dichotomous thinking treating p=0.049 differently than p=0.051 despite nearly identical evidence. The t-test calculator provides exact p-values enabling nuanced interpretation beyond arbitrary threshold. Report exact p-values allowing readers to judge evidence strength. Don’t manipulate α level, test type, or outlier handling seeking significant results (p-hacking inflates false positive rate). Pre-specify analysis plan before seeing data. If exploring data, acknowledge exploratory nature requiring replication. Don’t claim causation from observational data – the paired t-test calculator and two-sample t-test calculator show associations requiring experimental design (random assignment) for causal inference. Use results appropriately within methodological limitations for valid scientific conclusions.

Frequently Asked Questions

How to do t test on calculator?

To perform t-test using our t-test calculator: (1) Select test type (one-sample, two-sample independent, or paired), (2) Enter your data values separated by commas or line breaks, (3) Set significance level (typically α=0.05), (4) For one-sample test enter population mean, (5) Click Calculate to see t-statistic, p-value, degrees of freedom, and significance interpretation. The paired t-test calculator handles dependent samples, two-sample t-test calculator compares independent groups. Results show whether to reject or fail to reject null hypothesis based on p-value compared to alpha level. Our t-test calculator performs all calculations automatically using standard statistical formulas for accurate hypothesis testing in research and data analysis applications.

How to calculate t test on graphing calculator?

Calculate t-test on graphing calculator by entering data and selecting t-test function, but our online t-test calculator provides easier workflow: (1) Input all data values in simple format, (2) Choose appropriate test type for your experimental design, (3) Specify parameters like population mean or second sample, (4) View comprehensive results including t-statistic, p-value, confidence intervals, and effect size. Graphing calculators require menu navigation and separate data entry; our two-sample t-test calculator and paired t-test calculator handle complex calculations instantly with clear output formatting. The t-test calculator shows step-by-step interpretation helping understand statistical significance without manual probability table lookup required by traditional graphing calculators for hypothesis testing applications.

How to do a t test on calculator?

Perform t-test using our calculator by following simple steps: (1) Collect your sample data values, (2) Select appropriate test – use one-sample t-test comparing sample mean to known population mean, use paired t-test calculator for before/after measurements on same subjects, or use two-sample t-test calculator for comparing two independent groups, (3) Enter data separated by commas, (4) Set significance level (default 0.05 for 95% confidence), (5) Review results showing t-statistic measuring difference in standard error units, p-value indicating probability results occurred by chance, and significance conclusion. The t-test calculator handles all mathematical complexity providing immediate accurate results for statistical analysis without requiring manual formula calculations or probability distribution tables.

How to do matched pairs t test on calculator?

Perform matched pairs t-test (also called paired t-test) using our paired t-test calculator: (1) Select ‘Paired T-Test’ option for dependent samples, (2) Enter first set of measurements (before values, treatment A, or first condition), (3) Enter second set of measurements (after values, treatment B, or second condition), (4) Ensure equal number of pairs – each subject measured twice, (5) Calculate to see paired differences, t-statistic, p-value, and significance. The paired t-test calculator computes differences between paired observations, tests if mean difference significantly differs from zero. Use matched pairs when same subjects measured under two conditions, repeated measures designs, or naturally paired data. Our t-test calculator automatically handles paired analysis showing whether intervention or condition caused statistically significant change in dependent samples.

How to do one sample t test on calculator?

Conduct one-sample t-test using our t-test calculator: (1) Select ‘One-Sample T-Test’ to compare sample mean against known population mean, (2) Enter your sample data values, (3) Input hypothesized population mean (μ₀) you’re testing against, (4) Choose significance level (typically 0.05), (5) Calculate to see if sample mean significantly differs from population mean. The one-sample t-test calculator computes t-statistic = (x̄ – μ₀) / (s / √n) where x̄ is sample mean, s is sample standard deviation, n is sample size. Results show p-value indicating probability of observing your data if null hypothesis true. Use one-sample test when comparing sample to theoretical value, published standard, or regulatory requirement. Our t-test calculator provides complete output including confidence interval for population mean estimation.

What is a t-test used for?

A t-test determines whether differences between group means are statistically significant or likely due to random chance. Use the t-test calculator for: (1) One-sample t-test comparing sample mean to known value (does average test score differ from 100?), (2) Paired t-test calculator for before/after comparisons (did treatment reduce blood pressure?), (3) Two-sample t-test calculator comparing two independent groups (do males and females differ in height?). The t-test assumes approximately normal data distribution, independent observations, and similar variances for two-sample tests. Results from our t-test calculator show t-statistic measuring effect size in standard error units and p-value indicating statistical significance. T-tests are fundamental in research, quality control, A/B testing, and scientific experiments for hypothesis testing and data-driven decision making.

How do I interpret t-test results?

Interpret t-test calculator results by examining: (1) P-value – if p < 0.05 (α), reject null hypothesis concluding significant difference exists; if p > 0.05, fail to reject null hypothesis insufficient evidence for difference, (2) T-statistic – larger absolute values indicate stronger effects (±2 typically significant for moderate samples), (3) Degrees of freedom affecting critical t-values, (4) Confidence interval – if doesn’t contain zero (paired/two-sample) or hypothesized mean (one-sample), difference is significant. The paired t-test calculator and two-sample t-test calculator show whether groups truly differ beyond random variation. Statistical significance (p<0.05) doesn't guarantee practical importance - also consider effect size. Our t-test calculator provides complete interpretation noting whether results are statistically significant helping make informed research conclusions and data-driven decisions.

What’s the difference between paired and unpaired t-test?

Paired t-test (dependent samples) analyzes related measurements on same subjects – before/after treatment, matched pairs, repeated measures. The paired t-test calculator tests whether mean difference between paired observations differs from zero. Unpaired t-test (independent samples) compares means of two separate groups – males vs females, treatment vs control with different subjects. The two-sample t-test calculator tests whether group means differ significantly. Key differences: paired tests analyze difference scores reducing variability from individual differences increasing statistical power; unpaired tests treat groups independently requiring larger sample sizes for equivalent power. Use paired t-test calculator when observations naturally linked; use two-sample t-test calculator for independent groups. Our t-test calculator handles both designs showing appropriate statistics and interpretations for your specific experimental design and research questions.

What sample size do I need for a t-test?

Minimum sample size for t-test is technically n≥2, but practical recommendations vary: (1) Aim for n≥30 per group for two-sample t-test calculator applications ensuring Central Limit Theorem applies and normality assumption less critical, (2) Smaller samples (n=10-30) acceptable if data clearly normally distributed, (3) Very small samples (n<10) require caution - verify normality and consider non-parametric alternatives if assumptions violated. The paired t-test calculator needs fewer pairs than two-sample test for equivalent power because reduces individual difference variability. Larger samples increase statistical power detecting smaller effects as significant. Our t-test calculator works with any sample size but may warn about small samples. Consider power analysis beforehand determining adequate sample size for detecting expected effect at desired significance level and power (typically 0.80) in research planning.

What does p-value mean in t-test?

P-value in t-test represents probability of obtaining test results at least as extreme as observed, assuming null hypothesis is true. The t-test calculator computes p-value from t-statistic and degrees of freedom using t-distribution. Interpretation: p<0.05 (typical α) indicates <5% chance results occurred by random chance if no true difference exists, providing evidence to reject null hypothesis concluding statistically significant difference; p>0.05 means insufficient evidence to reject null hypothesis cannot conclude significant difference exists. Smaller p-values indicate stronger evidence against null hypothesis. The paired t-test calculator and two-sample t-test calculator both report p-values for significance testing. P-value does not indicate effect size or practical importance – significant results with large samples may have trivial effects. Our t-test calculator shows p-value alongside effect size for complete interpretation.

Sources and References

This t-test calculator uses standard statistical formulas from authoritative sources to ensure accuracy and validity. The following references were consulted in developing this statistical calculator:

UC Berkeley Statistics Department – Academic institution providing statistical education and t-test methodology
Penn State Statistics Online – Educational resource offering comprehensive t-test instruction
NIST/SEMATECH e-Handbook of Statistical Methods – Federal statistical methods reference for t-tests
American Statistical Association – Professional organization providing statistical standards and guidance
The R Project for Statistical Computing – Statistical software documentation for t-test implementations

Our t-test calculator follows standard statistical methods for hypothesis testing. This tool is for educational and research purposes. Always verify assumptions, consider effect sizes alongside p-values, and consult statisticians for complex experimental designs or critical decisions requiring rigorous statistical analysis.