T-Test Calculator: Analyze Statistical Significance with Ease

How to use the tool

1. Select a test
   
   • One-sample – compare one sample mean to a known value.
   • Independent two-sample – compare two unrelated groups.
   • Paired sample – compare matched observations (e.g., before-after).
2. Choose input method
   Summary statistics or Raw data. Example inputs:
   • Summary: 7.4, 1.9, 18
   • Raw: 9.1, 8.7, 7.5, 10.2, 9.8
3. Fill Sample 1
   Two fresh examples:
   • Raw list: 12.2, 11.7, 10.9, 12.6
   • Summary: 11.9, 0.8, 12
4. Fill Sample 2 (only for two-sample tests)
   
   • Raw list: 13.5, 12.9, 13.1, 13.8
   • Summary: 13.3, 0.6, 12
5. Optional parameters
   
   • μ₀ (e.g., 11.0) for one-sample tests.
   • α significance (e.g., 0.01).
   • Tail: two-, left-, or right-tailed.
6. Press “Calculate” and read t-value, degrees of freedom, p-value, 95 % CI, and decision.

Core formulas

One-sample:

$$ t = \frac{\bar{x}-\mu_0}{s/rac{\sqrt{n}}} ,\qquad df = n-1 $$

Independent two-sample (equal variances):

$$ s_p = \sqrt{rac{(n_1-1)s_1^2 + (n_2-1)s_2^2}{n_1+n_2-2}},\quad t = \frac{\bar{x}_1-\bar{x}_2}{s_p\sqrt{rac{1}{n_1}+rac{1}{n_2}}},\quad df = n_1+n_2-2 $$

Paired sample:

$$ t = \frac{\bar{d}}{s_d/rac{\sqrt{n}}},\qquad df = n-1 $$

95 % confidence interval:

$$ (\Delta \pm t_{0.025,df}\times SE) $$

Worked examples

Example 1 – One-sample

• Input: mean = 10.2, s = 2.3, n = 15, μ₀ = 9.0
• t = (10.2 – 9.0)/(2.3/√15) ≈ 2.02.
• df = 14; two-tailed p ≈ 0.061.
• Decision: fail to reject H₀ at α = 0.05.

Example 2 – Independent two-sample

• Group A: 5.1, 1.2, 20 Group B: 4.7, 1.4, 22
• s_p ≈ 1.30; SE ≈ 0.40; t ≈ 1.00.
• df = 40; p ≈ 0.32; fail to reject H₀.

Quick-Facts

• Two-tailed critical t for df = 20 at α = 0.05 is 2.086 (NIST Handbook, https://itl.nist.gov/div898/handbook).
• T-tests assume normality or n > 30 for robustness (Central Limit Theorem, Ross 2010).
• Effect size for t-tests is Cohen’s d where 0.2 = small, 0.5 = medium, 0.8 = large (Cohen, 1988).
• “Use pooled variance only when Levene’s test is non-significant” (ISO 13528:2022).

FAQ

What does the t-value tell me?

The t-value shows how many standard-error units separate sample and hypothesised means; larger |t| signals stronger evidence against H₀ (Student, 1908).

How do I interpret the p-value?

P-value is the probability of observing that |t| or greater if H₀ is true; p < α means statistically significant (NIST Handbook, URL).

When should I pick a one-tailed test?

Choose one-tailed only with a directional hypothesis set before data collection; otherwise remain two-tailed to avoid inflated Type I error (Rosenthal, 1994).

Can I use raw data with unequal sample sizes?

Yes. Enter each list; the calculator adjusts degrees of freedom accordingly using pooled or Welch’s method where coded.

What if my data are non-normal?

For n ≥ 30 per group, the t-test stays reliable due to the Central Limit Theorem; else use a non-parametric test like Wilcoxon (Conover, 1999).

How is the 95 % confidence interval built?

The tool multiplies the standard error by critical t_0.025 and adds/subtracts around the mean difference for lower and upper limits.

Why do I need degrees of freedom?

Degrees of freedom index sample information; they locate the correct t-distribution curve for critical values and p-values (ISO 3534-1:2006).

Does the calculator handle unequal variances?

The current release applies equal-variance formulas; future update will include Welch’s adjustment for heteroscedastic samples.

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