SigmaCalc

T-Score Calculator

The small-sample cousin of the z-score, used when you don't know the true population standard deviation.

Sample mean

Population mean

Sample std dev

Sample size (n)

t-score

1.826

t = (x̄ − μ) ÷ (s ÷ √n)

What you entered

Sample mean: 105Population mean: 100Sample std dev: 15Sample size (n): 30

1
Standard error of the mean
15 ÷ √30= 2.7386
2
t-score
(105 − 100) ÷ 2.7386= 1.8257

Result

t-score: 1.826

A t-score of 1.826 tells you how many standard errors your sample mean is from the population mean — compare it to a t-table critical value for your degrees of freedom to test significance.

T-scores against a population mean of 100

The t-score is the difference between the sample mean and the population mean, divided by the standard error - the standard deviation divided by the square root of n. These use an SD of 15.

Sample mean	Sample size	T-score
105	25	1.667
110	25	3.333
95	30	-1.826
103	50	1.414
108	40	3.373
97	16	-0.800

Sample size matters as much as the difference itself. Rows one and four show a 5-point gap at n=25 giving t=1.667, while a smaller 3-point gap at n=50 gives a comparable 1.414 - because a larger sample shrinks the standard error and makes a smaller difference detectable. That is the entire logic of statistical power. The t-distribution is used rather than z when the population standard deviation is unknown and estimated from the sample; with large samples the two converge. Compare the result against a critical value for your degrees of freedom, which is n minus 1.

t-score vs. z-score

Both measure how far a sample statistic is from an expected value in standard-error units. Use a t-score when your sample is small or you're estimating the population standard deviation from the sample itself, rather than knowing it exactly.

What to do with the result

Compare your t-score's absolute value against a critical value from a t-distribution table for your degrees of freedom (n − 1) and chosen significance level — if it exceeds the critical value, the difference is statistically significant.

Frequently asked questions

A new drug trial has n=25, sample mean=120, population mean=115, SD=10. Is the effect real?

t = (120−115) / (10/√25) = 5/2 = 2.5. With df=24, the critical value at p=0.05 (two-tailed) is 2.064. Since 2.5 > 2.064, the difference IS statistically significant — the drug appears to have a real effect.

When should I use a t-test instead of a z-test?

Use a t-test when sample size is small (typically n < 30) or when you're estimating the population SD from your sample. Use a z-test when you know the true population SD and n is large. In practice, the t-test is almost always appropriate — it converges to the z-test as n grows.

What are degrees of freedom?

For a one-sample t-test: df = n − 1. It reflects how much 'freedom' the data has to vary. With more degrees of freedom, the t-distribution looks more like a normal distribution. At df > 30, they're nearly identical.