T Critical Value Calculator

How to Use the T Critical Value Calculator

This calculator helps you find the one-sided and two-sided t-critical values and shows you how to find these values from the t-distribution table. To use the calculator:

1. Select the type of test you’re conducting.
   • Choose two-tailed if you’re testing whether a value is different from a claim.
   • Choose left-tailed if you want to test the hypothesis that a value is less than the claim.
   • Choose right-tailed if you are testing whether a value is greater than the claim.
2. Enter the significance level (α) and the degrees of freedom (df)
3. Click calculate

The calculator will instantly return the t-critical value for your selected type of test. It also provides a clear, step-by-step solution to help you understand how you can get this value from the t-table.

Tip. You should always choose the correct test type to get the correct critical t value. Additionally, the degrees of freedom vary depending on the type of t-test you’re performing.

What Is a T Critical Value?

A t critical value is a cutoff point on the t distribution, which defines the boundary between the rejection region and the non-rejection region in a t-based hypothesis test. It is also used when constructing confidence intervals.

In hypothesis testing, the t critical value is compared with the calculated t statistic to decide whether to reject or fail to reject the null hypothesis.

For a two-tailed test, use this decision rule:

• Reject the null hypothesis (H₀​) if the absolute value of the calculated t statistic is greater than the absolute value of the t critical value. In symbols, Reject H₀​ if ∣t∣>∣t_critical​∣
• Fail to reject the null hypothesis (H₀​) if the absolute value of the calculated t statistic is less than or equal to the absolute value of the t critical value. In symbols, fail to reject H₀​ if ∣t∣≤∣t_critical​∣

For a right-tailed test, reject H₀​ if the calculated t statistic is greater than the positive t critical value. However, for a left-tailed test, reject H₀​ if the calculated t statistic is less than the negative t critical value.

In confidence intervals, the t critical value is used as the multiplier for the standard error. It helps determine the margin of error around the sample estimate.

T Critical Value Formula

The exact t critical value is found from the inverse t distribution. The formula varies depending on the test type.

• For a right-tailed test, the t critical value formula is: t_α, df
• For a left-tailed test, the t critical value formula is: -t_α, df
• For a two-tailed test, the t critical values formula is: t_α/2, df

where:

• α is the significance level
• df is the degrees of freedom

Tips:

• The t-critical value for a right-tailed t-test is always positive
• The t-critical value for a left-tailed t-test is always negative
• For a two-tailed t-test, you must get two critical values (one negative and one positive). This is why two-tailed t-critical values are often written with a ± symbol.

How to Find T Critical Value

Finding a t critical value manually usually involves using a t distribution table. To use the table, you need to know three things:

1. Whether the test is one-tailed or two-tailed
2. The significance level, α
3. The degrees of freedom, df

Once you know these, you look up the row for df and the column for $\alpha$α. The value where the row and column meet is the t critical value.

Example 1. Two-Tailed Test

A researcher wants to test whether the average exam score of a sample of students is different from the national average. The researcher selects a sample of 16 students and uses a significance level of 0.05. Find the t critical value for this test.

Solution

From the question, we know that:

• The correct test is a one-sample t-test
• Sample size, n = 16
• Significance level, α = 0.05
• This is a two-tailed test because we want to test whether it is different from the national average.

By definition, the degrees of freedom formula for a one-sample t-test is: df = n-1

Thus, df = 16-1

= 15

To find the two-tailed t-critical value for this test using t tables, follow these steps:

1. Find a two-sided t-distribution table
2. Locate the df = 15 row and the α = 0.05 column
3. Identify the value at the intersection

From the two-sided t-table, the value where df = 15 and α = 0.05 meets is 2.131, as shown below. Since this is a two-sided test, we write it as ±2.131.

Alternatively, you can use the critical t-value calculator to get instant results by following these steps:

1. Select two-tailed as the test type
2. Enter α = 0.05 and df = 15
3. Click calculate

The calculator will instantly return the two-sided t-critical value as ±2.1314, similar to the t-table results.

Example 2. One-Tailed Test

A physiotherapist wants to test whether a 6-week exercise program increases patients’ flexibility scores. The same 14 patients were measured before and after the program. The researcher uses a significance level of 0.05. Find the t critical value for this test.

Solution

This is a paired t-test because the same patients are measured twice (before and after the program).

From the question, we know that:

• Number of pairs, n_pairs​=14
• Significance level, α =0.05
• This is a right-tailed test because the researcher wants to test whether flexibility scores increased after the program.

For a paired t-test, the degrees of freedom are: df = n_pairs​−1

Therefore: df =14−1

=13

Therefore, to find the one-sided t-critical value for the test using t-tables, follow these steps:

1. Find a one-sided t-table
2. Locate the df = 13 row and α =0.05 column
3. Identify the value at the intersection

The df = 13 and α =0.05 intersects at 1.771, as shown below. Therefore, the correct critical value for the test is: t_0.05, 13 = 1.771.

However, using the calculator, you only need to follow these steps

1. Select right-tailed as the test type
2. Enter α = 0.05 and df = 13
3. Click calculate

The calculator will instantly return the right-tailed t-critical value as: t = 1.7709 (similar to the table results).

Want to quickly find the degrees of freedom for any t-test? Use the degrees of freedom calculator.

When Should You Use This Calculator?

Use this t critical value calculator when you need a cutoff value for a t-based hypothesis test or confidence interval.

You can use it for:

• One-sample t tests
• Paired-samples t tests
• Independent-samples t tests
• Left-tailed t tests
• Right-tailed t tests
• Two-tailed t tests
• Confidence interval problems that require a t-critical value

How do I know whether to choose left-tailed, right-tailed, or two-tailed?

Choose the test type based on the alternative hypothesis. The table below provides a quick summary of how to choose the correct test type.

For example, use a two-tailed test when the alternative hypothesis uses “not equal to.”

Frequently Asked Questions