T-Score Calculator 2026

📊 Interactive T-Test Calculator

📈 One-Sample T-Test

Compare a sample mean to a known population mean

Sample Mean (x̄):

Population Mean (μ₀):

Sample Standard Deviation (s):

Sample Size (n):

Significance Level (α):

Test Type:

📊 Two-Sample T-Test

Compare means of two independent samples

Sample 1:

Mean (x̄₁):

Standard Deviation (s₁):

Sample Size (n₁):

Sample 2:

Mean (x̄₂):

Standard Deviation (s₂):

Sample Size (n₂):

Significance Level (α):

🔗 Paired T-Test

Compare paired observations (before/after, matched pairs)

Mean of Differences (d̄):

Standard Deviation of Differences (sᵈ):

Number of Pairs (n):

Hypothesized Difference (μ₀):

Significance Level (α):

1. What is a T-Score?

A t-score (or t-statistic) is a ratio used in hypothesis testing to determine whether there is a significant difference between sample means or between a sample mean and a population mean.

The t-score follows a Student's t-distribution, which is used when:

Sample size is small (n < 30)
Population standard deviation is unknown
Data is approximately normally distributed

2. T-Score Formulas

One-Sample T-Test

\[t = \frac{\bar{x} - \mu_0}{\frac{s}{\sqrt{n}}}\]

Where:

\(t\) = t-statistic
\(\bar{x}\) = Sample mean
\(\mu_0\) = Hypothesized population mean
\(s\) = Sample standard deviation
\(n\) = Sample size

Two-Sample T-Test (Independent)

\[t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{s_p^2\left(\frac{1}{n_1} + \frac{1}{n_2}\right)}}\]

Pooled Variance:

\[s_p^2 = \frac{(n_1-1)s_1^2 + (n_2-1)s_2^2}{n_1 + n_2 - 2}\]

Degrees of Freedom:

\[df = n_1 + n_2 - 2\]

Paired T-Test

\[t = \frac{\bar{d} - \mu_0}{\frac{s_d}{\sqrt{n}}}\]

Where:

\(\bar{d}\) = Mean of differences
\(\mu_0\) = Hypothesized mean difference (usually 0)
\(s_d\) = Standard deviation of differences
\(n\) = Number of pairs

3. T-Distribution Critical Values

df	α = 0.10 (two-tailed)	α = 0.05 (two-tailed)	α = 0.01 (two-tailed)
1	6.314	12.706	63.657
2	2.920	4.303	9.925
5	2.015	2.571	4.032
10	1.812	2.228	3.169
20	1.725	2.086	2.845
30	1.697	2.042	2.750
∞	1.645	1.960	2.576

4. When to Use Each T-Test

Test Type	Use When	Example
One-Sample	Comparing sample mean to known population mean	Testing if average IQ of class differs from 100
Two-Sample (Independent)	Comparing means of two independent groups	Comparing test scores of two different classes
Paired	Comparing before/after or matched pairs	Comparing weight before and after diet program

5. Hypothesis Testing Steps

State Hypotheses:
- Null hypothesis (H₀): No difference exists
- Alternative hypothesis (H₁): Difference exists
Choose Significance Level: Typically α = 0.05 (5%)
Calculate T-Score: Use appropriate formula
Find Critical Value: From t-distribution table
Make Decision:
- If |t| > critical value: Reject H₀
- If |t| ≤ critical value: Fail to reject H₀

6. Interpretation of Results

P-Value Interpretation

P-Value	Interpretation	Decision
p < 0.001	Very strong evidence against H₀	Reject H₀
0.001 ≤ p < 0.01	Strong evidence against H₀	Reject H₀
0.01 ≤ p < 0.05	Moderate evidence against H₀	Reject H₀ (at α=0.05)
0.05 ≤ p < 0.10	Weak evidence against H₀	Borderline
p ≥ 0.10	Little/no evidence against H₀	Fail to reject H₀

7. Assumptions of T-Tests

Key Assumptions:

Independence: Observations are independent of each other
Normality: Data is approximately normally distributed (especially important for small samples)
Random Sampling: Data comes from a random sample
For two-sample t-test: Equal variances (homogeneity of variance)

8. T-Distribution vs Normal Distribution

Feature	Normal Distribution	T-Distribution
Shape	Bell-shaped, specific curve	Bell-shaped, varies with df
Tails	Lighter tails	Heavier tails (more variability)
Parameters	μ and σ known	Depends on degrees of freedom
Sample Size	Large samples (n ≥ 30)	Small samples (n < 30)
As n increases	-	Approaches normal distribution

9. Degrees of Freedom (df)

Calculation of Degrees of Freedom

One-Sample T-Test: \(df = n - 1\)
Two-Sample T-Test: \(df = n_1 + n_2 - 2\)
Paired T-Test: \(df = n - 1\) (where n = number of pairs)

Degrees of freedom represent the number of independent pieces of information available to estimate a parameter.

10. Common Mistakes to Avoid

⚠️ Typical Errors:

Using t-test when assumptions are violated
Confusing one-tailed and two-tailed tests
Using paired t-test for independent samples (or vice versa)
Forgetting to check for outliers
Misinterpreting p-values as probability of hypothesis being true
Not considering practical significance vs statistical significance
Using t-test for categorical or ordinal data

11. 2026 Statistical Software Integration

Modern T-Test Tools

✅ R: t.test() function
✅ Python: scipy.stats.ttest_ind(), ttest_1samp(), ttest_rel()
✅ SPSS: Analyze > Compare Means > T-Test
✅ Excel: T.TEST() function
✅ GraphPad Prism: Built-in t-test analyses
✅ STATA: ttest command

💡 Pro Tip for 2026

Always visualize your data before conducting a t-test! Box plots, histograms, and Q-Q plots help verify assumptions. Modern statistical software makes this easy and can save you from invalid conclusions.

📚 Note on Statistical Significance

Remember: Statistical significance (p < 0.05) does not always mean practical significance. Consider the effect size and real-world implications of your findings. A statistically significant result with a tiny effect size may not be meaningful in practice.