Question 1

What is a t-score and how does it differ from a z-score?

Accepted Answer

Both standardise a value by dividing by the standard error. A z-score uses the known population standard deviation σ. A t-score uses the estimated sample standard deviation s, introducing additional uncertainty. The t-distribution has heavier tails than the normal to account for this extra uncertainty — especially for small samples. As df increases, the t-distribution converges to normal. For df > 120, t and z values are virtually identical.

Question 2

How do I find the p-value from a t-score?

Accepted Answer

Given t-statistic and df: two-tailed p = 2 × P(T > |t|); one-tailed p = P(T > t) or P(T < t) depending on direction. This calculator uses the regularised incomplete beta function for exact computation. Example: t = 2.45, df = 18, two-tailed → p ≈ 0.025. Since p < 0.05, reject H₀ at the 5% level.

Question 3

What is a critical t-value?

Accepted Answer

The critical t-value t_crit is the threshold beyond which you reject H₀. If |t| > t_crit, reject H₀. t_crit depends on α and df: for α=0.05, two-tailed — df=∞ → t_crit=1.960; df=30 → 2.042; df=20 → 2.086; df=10 → 2.228; df=3 → 3.182. Small samples need a much larger t to achieve significance because the t-distribution has heavier tails.

Question 4

What does degrees of freedom mean in a t-test?

Accepted Answer

Degrees of freedom represent the number of independent pieces of information available to estimate variance. For a one-sample t-test: df = n − 1 (one degree of freedom is used to estimate the mean). For two-sample Welch t-test: df is approximated via the Welch–Satterthwaite formula using both sample sizes and variances. Higher df gives a t-distribution closer to normal.

Question 5

When should I use a one-tailed vs two-tailed t-test?

Accepted Answer

Two-tailed: tests whether the parameter differs in either direction from H₀. Conservative; standard in most research. One-tailed: tests a specific direction (greater than or less than). Has more power but requires committing to a direction before data collection. Selecting one-tailed after seeing data inflates Type I error. Default: two-tailed, unless you had a clear directional hypothesis before collecting data.

Question 6

What t-score is considered statistically significant?

Accepted Answer

At α=0.05, two-tailed: |t| > 1.96 is significant for large df; |t| > 2.086 for df=20; |t| > 4.303 for df=3. At α=0.01: |t| > 2.576 (large df) or |t| > 2.845 (df=20). The threshold increases as df decreases — small samples demand stronger evidence. Always report the exact p-value rather than just comparing to a threshold.

df	α=0.10	α=0.05	α=0.025	α=0.01	α=0.001
1	6.314	12.706	25.452	63.657	636.619
3	2.353	3.182	4.177	5.841	12.924
5	2.015	2.571	3.163	4.032	6.869
10	1.812	2.228	2.634	3.169	4.587
20	1.725	2.086	2.423	2.845	3.850
30	1.697	2.042	2.360	2.750	3.646
60	1.671	2.000	2.299	2.660	3.460
120	1.658	1.980	2.270	2.617	3.373
∞ (z)	1.645	1.960	2.241	2.576	3.291

Condition	Use t	Use z
Population σ known?	No (estimated from sample s)	Yes (known exactly)
Sample size	Any n (essential for small n)	Large n ≥ 30 with CLT
Distribution shape	Normal, or n ≥ 30 for CLT	Normal or large n
Result precision	Exact for normally distributed data	Approximate (CLT)
df needed?	Yes — n−1 or Welch formula	No
Converges to z when?	df > 120 → virtually identical	Always

T-Score Calculator

Formula

Critical t-Values — Two-Tailed Reference Table

t-Test vs z-Test — Which Statistic to Use?

Case Study: Evaluating a New Pain Relief Drug

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