Example Data Table
| Test type |
Alpha |
Degrees of freedom |
Target probability |
Critical value |
Decision boundary |
| Two tailed |
0.05 |
10 |
0.975 |
±2.228 |
Reject outside ±2.228 |
| Right tailed |
0.01 |
30 |
0.990 |
2.457 |
Reject above 2.457 |
| Left tailed |
0.05 |
20 |
0.050 |
-1.725 |
Reject below -1.725 |
Formula Used
Two tailed test: tc = F-1(1 - α / 2)
Right tailed test: tc = F-1(1 - α)
Left tailed test: tc = F-1(α)
Here, F-1 is the inverse cumulative t distribution. Alpha is the significance level. The degrees of freedom control the curve shape.
One sample or paired df: df = n - 1
Pooled independent df: df = n1 + n2 - 2
Welch df:
df = (s1² / n1 + s2² / n2)² / [((s1² / n1)² / (n1 - 1)) + ((s2² / n2)² / (n2 - 1))]
How to Use This Calculator
- Select left tailed, right tailed, or two tailed testing.
- Enter alpha or choose confidence level mode.
- Select the degrees of freedom method.
- Enter sample sizes and standard deviations when needed.
- Choose decimal places for the final output.
- Press calculate to see the result above the form.
- Use CSV or PDF buttons to export the current result.
About the Critical Value T Test Calculator
A t critical value is the boundary used in a t test. It marks the point where results become unlikely under the null hypothesis. This calculator helps you find that boundary with clear inputs and direct output. It supports left tailed, right tailed, and two tailed tests. It also accepts alpha or confidence level.
Why Critical Values Matter
Many Statistics problems compare a calculated t statistic with a critical value. When the statistic falls inside the rejection region, the test result is significant. When it stays outside, the evidence is not strong enough. The boundary depends on degrees of freedom, test direction, and chosen error level.
Degrees of Freedom
Degrees of freedom describe how much information can vary. A one sample t test usually uses n minus one. A paired test uses the number of pairs minus one. A pooled independent test uses n one plus n two minus two. Welch tests use a special formula based on standard deviations and sample sizes. The calculator includes each option for quick study and review.
Tail Selection
Tail selection controls where the rejection area is placed. A left tailed test checks for a smaller mean. A right tailed test checks for a larger mean. A two tailed test checks for any difference. In a two tailed test, alpha is split equally across both tails. That is why the positive and negative critical values appear together.
Practical Use
Use this tool before comparing your test statistic. Enter the sample information or enter degrees of freedom directly. Choose alpha when your problem gives a significance level. Choose confidence when your problem gives a confidence level. Then press calculate. The result shows the cutoff and the rejection rule.
Good Habits
Always match the calculator settings to your hypothesis. Do not use a two tailed value for a one tailed claim. Keep alpha consistent with the assignment or report. Check that sample sizes are valid. Review the formula notes when results look unexpected. Small degrees of freedom give wider critical limits. Large degrees of freedom move t values closer to normal z values. For reports, export the answer and keep the same settings beside your conclusion for later review.
FAQs
What is a t critical value?
It is a cutoff from the t distribution. You compare your calculated t statistic with it. If the statistic falls in the rejection region, the result may be statistically significant.
When should I use a two tailed test?
Use a two tailed test when the alternative hypothesis checks for any difference. It does not assume the result must be only larger or only smaller.
What is alpha in a t test?
Alpha is the chosen significance level. Common values are 0.10, 0.05, and 0.01. A smaller alpha creates a stricter rejection boundary.
Can I enter confidence instead of alpha?
Yes. The calculator converts confidence to alpha. For example, 95 percent confidence gives alpha equal to 0.05.
What degrees of freedom should I use?
Use the formula that matches your test. One sample and paired tests use n minus one. Pooled independent tests use n1 plus n2 minus two.
What is Welch degrees of freedom?
Welch degrees of freedom are used when two samples have unequal variances. The formula uses both sample sizes and standard deviations.
Why are two tailed values shown with plus and minus signs?
A two tailed test has rejection areas on both sides. The negative value marks the left cutoff. The positive value marks the right cutoff.
Can I download the result?
Yes. Use the CSV button for spreadsheet data. Use the PDF button for a simple report summary.