Example Data Table
| Case |
Test Type |
Mean 1 |
SD 1 |
N 1 |
Mean 2 |
SD 2 |
N 2 |
Null Value |
| A |
One sample |
72.4 |
8.5 |
28 |
Not used |
Not used |
Not used |
70 |
| B |
Independent Welch |
72.4 |
8.5 |
28 |
68.1 |
7.9 |
30 |
0 |
| C |
Paired summary |
4.3 |
6.2 |
24 |
Not used |
Not used |
Not used |
0 |
Formula Used
One Sample or Paired Summary Test
t = (x̄ - μ0) / (s / √n)
df = n - 1
Independent Two Sample Welch Test
t = ((x̄1 - x̄2) - Δ0) / √(s1² / n1 + s2² / n2)
df = (a + b)² / ((a² / (n1 - 1)) + (b² / (n2 - 1)))
a = s1² / n1 and b = s2² / n2
Independent Pooled Test
sp² = ((n1 - 1)s1² + (n2 - 1)s2²) / (n1 + n2 - 2)
t = ((x̄1 - x̄2) - Δ0) / √(sp²(1 / n1 + 1 / n2))
Confidence Interval
Estimate ± t critical × standard error
How to Use This Calculator
Select the test type first. Use one sample when one mean is compared with a target value. Use independent two sample when two separate groups are compared. Use paired summary when the entered mean is already the mean of paired differences.
Enter each mean, standard deviation, and sample size. For paired tests, enter the mean difference, standard deviation of differences, and number of pairs in the first sample fields. Then enter the hypothesized mean or hypothesized difference.
Choose the tail direction. Use two-tailed when any difference matters. Use right-tailed when the estimate should be greater. Use left-tailed when it should be smaller. Press calculate. Review the result, p-value, degrees of freedom, confidence interval, and export buttons.
Understanding the t-test from mean and SD
A t-test helps compare a sample mean with a target value. It can also compare two sample means. The method is useful when population standard deviation is unknown. It uses sample standard deviation, sample size, and observed mean. This calculator focuses on summary statistics. You do not need raw data.
Why summary inputs matter
Many reports share mean, standard deviation, and count only. These values are enough for common t-tests. A one sample test checks whether a mean differs from a hypothesized mean. An independent test compares two groups. A paired test works when each observation has a matched partner. In that case, enter the mean difference and its standard deviation.
What the result means
The t statistic measures distance from the null value. It divides the observed difference by standard error. A larger absolute t value gives stronger evidence against the null hypothesis. Degrees of freedom control the curve shape. Smaller samples have heavier tails. That makes the test more conservative.
P-value and confidence interval
The p-value estimates how unusual the observed result is, assuming the null hypothesis is true. A two-tailed test checks for any difference. A right-tailed test checks whether the mean is greater. A left-tailed test checks whether it is smaller. The confidence interval gives a practical range for the true mean or mean difference. When the interval excludes the null value, the result usually matches a significant two-tailed test.
Using results responsibly
A t-test assumes independent observations and roughly normal data. For large samples, the method is often stable. For small samples, strong skew or outliers can distort results. Equal variance is optional for two groups. Welch's method is safer when group spreads differ. Pooled variance can be used when both populations have similar variability.
Practical reporting
Report the test type, t statistic, degrees of freedom, p-value, confidence interval, sample sizes, means, and standard deviations. Also explain the context. Statistical significance is not the same as practical importance. Always compare the estimated difference with real-world needs. Use this tool for teaching, checking reports, and quick statistical review.
It supports careful decisions with compact, repeatable summary inputs.
FAQs
What is a t-test calculator using mean and SD?
It is a calculator that uses summary statistics. You enter mean, standard deviation, sample size, and null value. It returns t statistic, degrees of freedom, p-value, and confidence interval.
Can I use this without raw data?
Yes. This tool is made for summary data. Mean, standard deviation, and sample size are enough for one sample, paired summary, and independent two sample tests.
When should I choose Welch?
Choose Welch when two groups may have different variances. It is often safer than the pooled method. It also allows decimal degrees of freedom.
When should I choose pooled variance?
Choose pooled variance only when both groups have similar spread. It assumes equal population variances. If that assumption is doubtful, use Welch instead.
What does the p-value show?
The p-value shows how unusual the observed result is under the null hypothesis. A smaller p-value gives stronger evidence against the null hypothesis.
What does the confidence interval mean?
The confidence interval gives a likely range for the true mean or mean difference. Wider intervals usually mean more uncertainty.
Why is my degrees of freedom decimal?
Welch tests use an approximation. That approximation can produce decimal degrees of freedom. This is normal and expected.
Can I download the results?
Yes. After calculation, use the CSV button for spreadsheet data. Use the PDF button for a simple report file.