Hypothesis Tests
T Critical Value Calculator
A t critical value is the Student cutoff for your degrees of freedom, your alpha, and the tail you chose before looking at the data.
Worksheet
Press Calculate to see the result.
A t critical value is the Student cutoff for your degrees of freedom, your alpha, and the tail you chose before looking at the data.
What this calculator answers
The t critical value is the Student-curve cousin of a z cutoff, and the degrees of freedom decide how heavy the tails are. A small df pushes the cutoff farther from zero than the normal line at the same alpha, which is why a sample of a handful of measurements needs a stricter bar. This page solves the cutoff for the df, alpha, and tail you type. It does not compute a p-value from a data set. That direction of the problem lives on the p-value and t-test pages.
The headline is the positive t cutoff. The rejection region line carries the sign that matches the tail you chose. Supporting counts and intermediate terms sit under the headline so a reader can see what was divided by what. When a checkbox or a tail setting changes the divisor or the rejection region, that choice appears in the steps rather than vanishing into a silent default.
The formula, written out
Student t critical value uses this relationship.
Find t so the tail area beyond it equals alpha, or alpha/2 when the test is two-sided, on df degrees of freedom.
- df. Degrees of freedom of the t curve.
- alpha. Probability in the rejection region.
- tail. Two-sided, left, or right.
- t critical. The positive cutoff on that curve.
The t family converges to the standard normal as df grows, so the cutoff slides toward the z cutoff of the same tail. For a finite df the page searches the t distribution function until the upper probability matches the target tail area. The algebra is the named textbook identity for this worksheet. Intermediate products and quotients appear in order so a calculator and a pencil can agree to the displayed rounding. If a neighbouring page uses a different divisor or a continuity correction, that page says so on its own ticket.
A full worked example
Example loaded in the worksheet
The loaded lookup is df equal to 20, alpha equal to 0.05, and a two-sided tail.
Degrees of freedom are 20, alpha is 0.05, and the tail is two-sided. The t critical value on that curve is 2.086. The rejection region is beyond 2.086 and beyond −2.086.
The matching z cutoff at the same alpha and tail would be 1.96. The t cutoff accounts for estimating a spread on 20 degrees of freedom. Each tail holds alpha over 2, which is 0.025. A one-sample mean uses df equal to n minus 1. This page trusts the df you typed, which is 20.
As df grows, this cutoff moves toward the z figure 1.96. The result is a cutoff, not a p-value. Bring an observed t to the p-value from t page if you already have the statistic.
| T critical value | 2.086 |
|---|---|
| Degrees of freedom | 20 |
| Rejection region | beyond 2.086 and beyond −2.086 |
A second situation
Five degrees of freedom and a right-tailed alpha of 0.05
Degrees of freedom are 5, alpha is 0.05, and the tail is right-tailed. The t critical value on that curve is 2.015. The rejection region is above 2.015.
The matching z cutoff at the same alpha and tail would be 1.6449. The t cutoff accounts for estimating a spread on 5 degrees of freedom. The single tail holds the full alpha of 0.05. A one-sample mean uses df equal to n minus 1. This page trusts the df you typed, which is 5.
As df grows, this cutoff moves toward the z figure 1.6449. The result is a cutoff, not a p-value. Bring an observed t to the p-value from t page if you already have the statistic.
| T critical value | 2.015 |
|---|---|
| Degrees of freedom | 5 |
| Rejection region | above 2.015 |
Dropping from many degrees of freedom to a handful moves the cutoff out, away from the z line. A right-tailed request also changes the rejection region from two sides to the positive side only. The second example changes the feature that this formula is sensitive to, and the ticket shows the new figure beside the same definition. That contrast is the fastest way to see which input owns the movement.
What moves when the input moves
Degrees of freedom fell from 20 to 5. Alpha and the two-sided tail stayed put. The headline result went from 2.086 to 2.5706.
The nudged inputs
Degrees of freedom are 5, alpha is 0.05, and the tail is two-sided. The t critical value on that curve is 2.5706. The rejection region is beyond 2.5706 and beyond −2.5706.
The matching z cutoff at the same alpha and tail would be 1.96. The t cutoff accounts for estimating a spread on 5 degrees of freedom. Each tail holds alpha over 2, which is 0.025. A one-sample mean uses df equal to n minus 1. This page trusts the df you typed, which is 5.
As df grows, this cutoff moves toward the z figure 1.96. The result is a cutoff, not a p-value. Bring an observed t to the p-value from t page if you already have the statistic.
| T critical value | 2.5706 |
|---|---|
| Degrees of freedom | 5 |
| Rejection region | beyond 2.5706 and beyond −2.5706 |
How to read the result
Compare an observed t with this cutoff only when the observed statistic uses the same df. A left-tailed alternative rejects on the negative side of zero, using the negative of the printed cutoff. Say the unit, the sample size or contract terms, and the formula name in the same sentence as the figure. A number without those three pieces invites a later reader to invent a definition that this page never used.
Write t critical, df, alpha, and the tail in the same sentence. A cutoff of 2.086 without df 20 is not checkable. Write the result with its unit, the inputs that produced it, and the name of the formula. If the ticket shows a decision sentence, quote the alpha and the tail with it. If the ticket shows a money or dose figure, quote the rate and the schedule that produced it.
Assumptions that have to stay true
- The null statistic really follows a t distribution with this df.
- Df is positive. A zero-df curve is not defined.
- Alpha is a probability strictly between 0 and 1.
- The tail was chosen with the alternative, not after peeking at the sign of t.
If an assumption above is false, do not average the discomfort away. Change the worksheet or change the question. A correct arithmetic step on the wrong model is still the wrong answer for the decision you are trying to make.
Mistakes that change the answer
- Looking up a z row when df is 5. The normal line is too small.
- Using n as df for a one-sample mean. That test spends one df and keeps n minus 1.
- Reading a two-sided cutoff as if it were one-sided at the same alpha.
- Mixing the df of a pooled test with the fractional df of a Welch test.
Where a different calculator is the honest choice
Use the z critical value calculator when sigma is known and the normal curve is the right model. Use the one-sample t-test calculator when you have data and want the statistic and the p-value, not only the cutoff. The degrees of freedom calculator tells you which df belongs to a design. Related links under the essay open pages that keep their own formulas instead of hiding a mode switch inside this ticket.
When the job changes, move to a page built for that job: Z Critical Value Calculator, P-Value from T Calculator, One Sample T-Test Calculator.
Check yourself
What is the two-sided t critical value at alpha 0.05 with 20 degrees of freedom?
The t critical value is 2.086.
Using the result in a sentence
Mean tests with an estimated standard deviation, regression slopes, and correlation tests all look up a t cutoff with their own df. The number is useless until that df is the one the statistic actually has. Keep the count, the unit, and the assumption list beside any copy of the number that leaves this browser tab.
The cutoff assumes the t model. Strong skew in a tiny sample makes the nominal alpha drift. The page also does not know whether your df formula matched the design. Pair the headline with the steps before you staff a route, set a grade boundary, or compare two contracts from a single number.
Questions people ask next
01What definition does the T Critical Value Calculator commit to?
This page uses Student t critical value: Find t so the tail area beyond it equals alpha, or alpha/2 when the test is two-sided, on df degrees of freedom.. The t family converges to the standard normal as df grows, so the cutoff slides toward the z cutoff of the same tail. For a finite df the page searches the t distribution function until the upper probability matches the target tail area. If a textbook uses a different divisor, a different continuity correction, or a different rounding rule, the last digits can move, so the definition is printed above the result. The worksheet asks for Degrees of freedom, Alpha, Tail, and it refuses a blank that the formula cannot interpret. Keep that printed definition in the sentence you write next to the figure so a later reader can rebuild the arithmetic without guessing which convention you chose.
02How should the worked example be read out loud?
In the loaded example the t critical value is 2.086. Compare an observed t with this cutoff only when the observed statistic uses the same df. A left-tailed alternative rejects on the negative side of zero, using the negative of the printed cutoff. Say the unit and the count before you say whether the number is large or small. Read the steps in order rather than jumping to the headline, because the intermediate terms are where a swapped percent and proportion usually hide. If the ticket shows both a primary figure and a comparison figure, name which one you are quoting.
03What is the fastest way to get a misleading result?
Looking up a z row when df is 5. The normal line is too small. The cutoff assumes the t model. Strong skew in a tiny sample makes the nominal alpha drift. The page also does not know whether your df formula matched the design. A second common failure is leaving a checkbox or tail setting in the wrong state and then treating the number as if the default had matched your plan. Reset the example, change one input, and confirm the ticket moves in the direction the formula predicts before you trust a live run.
04When should I leave this page for a different calculator?
Use the z critical value calculator when sigma is known and the normal curve is the right model. Use the one-sample t-test calculator when you have data and want the statistic and the p-value, not only the cutoff. The degrees of freedom calculator tells you which df belongs to a design. The related links under the essay go to those worksheets, which keep their own formulas instead of hiding them in a mode switch. If the question changed from a summary to a test, from a payment to a refinance, or from a raw score to a percentile, switch pages rather than forcing this ticket to answer a job it was not built for.
05What belongs in the sentence next to the number?
The t critical value is 2.086. Write t critical, df, alpha, and the tail in the same sentence. A cutoff of 2.086 without df 20 is not checkable. A bare number without those three pieces cannot be audited when a score, a rate, or a lab value is challenged later. Store the inputs beside the result whenever the figure will travel into a report.