Calculator Inputs
Example Data Table
These examples show common t critical value settings.
| Degrees of Freedom | Alpha | Tail Type | Approximate Critical Value | Rejection Region |
|---|---|---|---|---|
| 10 | 0.05 | Two-tailed | ±2.228 | Reject when |t| ≥ 2.228 |
| 20 | 0.01 | Right-tailed | 2.528 | Reject when t ≥ 2.528 |
| 30 | 0.05 | Left-tailed | -1.697 | Reject when t ≤ -1.697 |
| 50 | 0.10 | Two-tailed | ±1.676 | Reject when |t| ≥ 1.676 |
Formula Used
One sample t statistic:
t = (x̄ - μ₀) / (s / √n)
Two-tailed critical values:
Lower = -t1 - α / 2, df, Upper = t1 - α / 2, df
Right-tailed critical value:
Critical value = t1 - α, df
Left-tailed critical value:
Critical value = tα, df
The calculator estimates t distribution probabilities with the regularized incomplete beta function. It then uses binary search to find inverse t critical values.
How to Use This Calculator
- Enter the degrees of freedom for your t distribution.
- Enter alpha, such as 0.05 for a 5 percent test.
- Select two-tailed, right-tailed, or left-tailed testing.
- Enter an observed t statistic when you already have one.
- Or enter sample mean, hypothesized mean, standard deviation, and sample size.
- Choose automatic degrees of freedom when using one sample data.
- Press calculate to view critical values, p value, and decision.
- Use the CSV or PDF button to save the result.
Critical Value T Statistic Guide
What the Calculator Does
A t critical value marks the boundary of a rejection region. It is used when population standard deviation is unknown. This is common in small sample research. The calculator finds that boundary from alpha, tail type, and degrees of freedom. It also checks an observed t statistic when you provide one.
Why Tail Choice Matters
A two-tailed test checks for a difference in either direction. It splits alpha across both tails. A right-tailed test checks for a value greater than the null claim. A left-tailed test checks for a value lower than the null claim. Choosing the correct tail should happen before viewing the data.
Degrees of Freedom
Degrees of freedom shape the t distribution. Low values create thicker tails. This makes the critical value larger. As sample size grows, the t distribution becomes closer to the normal curve. For a one sample mean test, degrees of freedom are usually n minus one.
Interpreting P Values
The p value measures how extreme the observed statistic is. A small p value gives stronger evidence against the null hypothesis. When the p value is less than or equal to alpha, the result is significant. This does not prove practical importance. It only supports a statistical decision under the chosen model.
Confidence Results
If sample inputs are entered, the calculator can also show standard error. For two-tailed settings, it gives a confidence interval. For one-tailed settings, it gives a one-sided confidence bound. These values help describe uncertainty around the sample mean. They should be reported with context and measurement units.
FAQs
1. What is a t critical value?
A t critical value is the cutoff from a t distribution. It defines where the rejection region begins for a selected alpha, tail type, and degrees of freedom.
2. What does alpha mean?
Alpha is the chosen significance level. It represents the risk of rejecting a true null hypothesis. Common alpha values are 0.10, 0.05, and 0.01.
3. When should I use a two-tailed test?
Use a two-tailed test when the alternative hypothesis allows a difference in either direction. It checks whether the statistic is unusually low or unusually high.
4. When should I use a right-tailed test?
Use a right-tailed test when the alternative hypothesis says the parameter is greater than the null value. The rejection region is on the right side.
5. When should I use a left-tailed test?
Use a left-tailed test when the alternative hypothesis says the parameter is less than the null value. The rejection region is on the left side.
6. How are degrees of freedom calculated?
For a one sample t test, degrees of freedom usually equal sample size minus one. Other t tests, such as Welch tests, may use adjusted values.
7. What does the p value show?
The p value shows the probability of getting a result at least as extreme as the observed statistic, assuming the null hypothesis is true.
8. Can I download the result?
Yes. After calculation, use the CSV button for spreadsheet data or the PDF button for a clean printable report.