Calculator
Example Data Table
| Test case | Alpha | Degrees of freedom | Tail | Critical value |
|---|---|---|---|---|
| Class quiz score check | 0.05 | 10 | Two tailed | -2.22814 to 2.22814 |
| Right side claim test | 0.01 | 20 | Right tailed | 2.52798 |
| Left side process test | 0.05 | 15 | Left tailed | -1.75305 |
| Wide confidence screen | 0.10 | 30 | Two tailed | -1.69726 to 1.69726 |
Formula Used
For a manual critical value, the calculator solves the inverse Student t distribution.
Two tailed test: critical values are -t(1 - alpha / 2, df) and t(1 - alpha / 2, df).
Right tailed test: critical value is t(1 - alpha, df).
Left tailed test: critical value is t(alpha, df).
One sample statistic: t = (x̄ - μ0) / (s / sqrt(n)).
Welch statistic: t = ((x̄1 - x̄2) - Δ0) / sqrt(s1² / n1 + s2² / n2).
Paired statistic: t = (mean difference - Δ0) / (sd difference / sqrt(n)).
How to Use This Calculator
Select the test mode first. Use manual mode when you only need a critical value.
Enter alpha, degrees of freedom, and tail type. Then press Calculate.
For sample based modes, fill the matching sample fields. Leave other fields unchanged.
Read the critical boundary and compare it with the t statistic.
Use the CSV or PDF buttons to save the current calculation.
Understanding t critical values
A t critical value marks a boundary in a t distribution. It helps decide whether a test statistic is unusual enough to reject a null hypothesis. The value depends on three inputs. These are alpha, degrees of freedom, and tail type. Smaller alpha levels move the boundary farther from zero. Lower degrees of freedom also create wider tails. That is why small samples need careful checks.
Why this calculator helps
This calculator is made for manual study and practical reporting. You can enter degrees of freedom directly. You can also let the tool compute them from sample data. It supports one sample tests, paired tests, pooled two sample tests, and Welch two sample tests. Welch mode is useful when sample spreads are not equal. The calculator also returns standard error, the t statistic, p value, and a confidence interval when enough data is entered.
Reading the results
For a two tailed test, the rejection area sits on both sides. The calculator shows a lower and upper critical value. A test statistic beyond either boundary supports rejection. For a right tailed test, only the upper boundary is used. For a left tailed test, only the lower boundary is used. Always compare the result with your original alternative hypothesis.
Good statistical practice
A t test works best with independent observations and roughly normal data. Large samples are more forgiving. Small samples need stronger attention to outliers. For two sample work, choose pooled mode only when equal variance is reasonable. Use Welch mode when that assumption is doubtful. Report alpha, degrees of freedom, tail type, t statistic, p value, and your conclusion. Clear reporting keeps the decision easy to audit.
Limits and interpretation
A critical value is not a full proof. It is a decision rule built from probability. The p value gives related evidence, but it does not measure practical size. A confidence interval gives more context because it shows likely effect values. Use all three parts together. Check units before entering data. Round only after the final result. Keep the raw values in your notes. This protects the conclusion from small rounding errors and makes later review simpler. State any excluded observations before presenting the final answer.
FAQs
What is a t critical value?
It is the cutoff point on a Student t distribution. It separates common test statistic values from values considered rare under the null hypothesis.
Which alpha should I use?
Many school and research examples use 0.05. Stricter tests may use 0.01. Choose alpha before checking the result.
What are degrees of freedom?
Degrees of freedom describe the amount of independent information in the estimate. For one sample and paired tests, they are usually n minus 1.
When should I use a two tailed test?
Use a two tailed test when the alternative hypothesis allows the result to be either higher or lower than the null value.
When should I use Welch mode?
Use Welch mode for two independent samples when equal variance is doubtful. It adjusts the degrees of freedom with a safer approximation.
Can this calculator find a p value?
Yes. When sample data is entered, it computes the t statistic and p value for the selected tail direction.
Why does a small sample need a larger critical value?
Small samples have fewer degrees of freedom. Their t distributions have heavier tails, so the rejection boundary moves farther from zero.
Can I export the results?
Yes. Use the CSV or PDF button after entering the values. The exported file contains the main calculation summary.