T Critical Value Calculator

Find the t critical value for a confidence level or significance level, with degrees of freedom and one-tailed or two-tailed rejection regions.

Enter lookup settings

Uses Student's t distribution with your selected degrees of freedom. Results update as you change the degrees of freedom, tail type, or input value.

\nu

Degrees of freedom

For a one-sample mean or paired differences, use n - 1.

Tail type

Input mode

For a two-tailed lookup, the confidence level is the central area and alpha is split equally across both tails.

Common confidence levels

\mathrm{CL}

Confidence level

Enter a percent greater than 0 and less than 100.

Critical value result

$t_1=t_{\alpha/2}$ Lower critical value

-2.262157

$t_2=t_{1-\alpha/2}$ Upper critical value

2.262157

\alpha

0.05

\alpha/2

0.025

2.5%

1-\alpha

95%

Confidence level

Distribution

t_{9}

Student t

F_t^{-1}(p)

0.975

p=1-\alpha/2

Rejection region

Reject H0 when t is less than or equal to the lower critical value or greater than or equal to the upper critical value.

t curve with shaded alpha

Values are rounded for display; use the full precision in the formulas when comparing borderline test statistics.

Step-by-Step Lookup

Convert the input to alpha

\alpha = 1-\frac{95}{100} = 0.05

Assign alpha to the tail or tails

\frac{\alpha}{2}=\frac{0.05}{2}=0.025

Identify the required t quantile

t^{*}=F_t^{-1}\left(1-\frac{\alpha}{2}\right)=F_t^{-1}\left(0.975\right)=2.26215716

Read the critical value

t_1=\left(-2.26215716\right),\ t_2=2.26215716

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What a t critical value means

A critical value is a cutoff on a probability curve. In hypothesis testing, it defines the rejection region: if the test statistic falls beyond the cutoff, the result is unusual enough under the null hypothesis to reject $H_0$ . In confidence intervals, the critical value sets how many standard errors are needed on each side of the estimate.

For a t critical value, the calculator uses Student's t distribution with your selected degrees of freedom. The t curve has heavier tails than the standard normal curve, especially when the degrees of freedom are small. As the degrees of freedom increase, the t distribution approaches the z distribution.

Degrees of freedom

The degrees of freedom, often written $\nu$ or $df$ , control the shape of the t distribution. For a one-sample t procedure or paired differences, the common rule is:

df = n - 1

For example, a sample size of $n=25$ gives $df=24$ . Smaller degrees of freedom produce larger critical values because more probability sits in the tails.

Alpha versus confidence level

The significance level $\alpha$ is the probability assigned to the rejection region. A $5\%$ significance level means $\alpha=0.05$ .

The confidence level is the central coverage used for confidence intervals, usually written as $1-\alpha$ . A $95\%$ confidence level corresponds to:

\alpha = 1 - 0.95 = 0.05

For two-tailed critical values, that total alpha is split equally:

\alpha_{\text{per tail}} = \frac{\alpha}{2}

For one-tailed critical values, the full alpha stays in the selected tail.

One-tailed and two-tailed critical values

In a right-tailed test, the rejection region is in the high end of the curve, so the calculator finds:

t_{\alpha,df} = F_t^{-1}(1-\alpha)

In a left-tailed test, the rejection region is in the low end of the curve:

t_{\alpha,df} = F_t^{-1}(\alpha)

In a two-tailed test, the rejection region is split between both ends:

\pm t^{*} = \pm F_t^{-1}\left(1-\frac{\alpha}{2}\right)

For example, with $df=24$ and a $95\%$ two-tailed lookup, the critical values are about $\pm 2.064$ . With the same degrees of freedom and a right-tailed $\alpha=0.05$ lookup, the critical value is about $1.711$ .

How to use this calculator

Enter the degrees of freedom.
Choose the tail type: two-tailed, right-tailed, or left-tailed.
Choose whether your input is a confidence level or significance level $\alpha$ .
Use a preset confidence level or enter a custom value.
Read the critical value and rejection-region statement.
Compare your test statistic with the critical value or bounds.

How to read the shaded alpha region

The shaded part of the curve is the rejection area. In a two-tailed test, the calculator shades both tails because extreme values in either direction count against the null hypothesis. In a right-tailed test, only the right tail is shaded. In a left-tailed test, only the left tail is shaded.

The vertical marker is the boundary between the non-rejection area and the rejection area. A test statistic beyond the marker is in the shaded alpha region.

Critical values in confidence intervals

For a two-sided t confidence interval for a mean, the same two-tailed lookup is used. A $95\%$ interval leaves $\alpha=0.05$ outside the interval, with $0.025$ in each tail, so the critical value for $df=24$ is $t^* \approx 2.064$ . The margin of error is then:

\text{margin of error} = t^* \times \text{standard error}