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P-Value from T-Score Calculator

Approximate the p-value for a t-statistic and its degrees of freedom.

Result

Approximate P-Value
0.02168

This uses a normal-transform approximation of the t-distribution, accurate to within a few thousandths for degrees of freedom of 10 or more - for small samples, a t-distribution table gives an exact result.

About the P-Value from T-Score

This calculator converts a t-statistic and its degrees of freedom into an approximate p-value, for either a one-tailed or two-tailed hypothesis test. It is useful when you already have a computed t-statistic, from a t-test or a regression coefficient, and need the associated significance level without consulting a t-distribution table.

How It Works

You enter the t-score, the degrees of freedom associated with it, and whether you want a one-tailed or two-tailed result. The calculator transforms the t-statistic into an equivalent standard normal z-value using a correction formula, then runs that adjusted value through the normal cumulative distribution function to produce a two-tailed p-value, halving it if you asked for a one-tailed test.

z_approx = (|t| * (1 - 1/(4*df))) / sqrt(1 + t^2 / (2*df)); two-tailed p = 2 * (1 - normalCdf(z_approx)); one-tailed p = two-tailed p / 2

Formula & Methodology

Because the exact t-distribution CDF requires an incomplete beta function that is impractical to evaluate with a closed-form expression, this calculator instead uses a transform that maps the t-statistic onto an approximately equivalent z-score, accounting for the extra spread the t-distribution has at low degrees of freedom. That z-score is then passed through the same normal CDF approximation used elsewhere on the site. The result is accurate to within a few thousandths of the exact value once degrees of freedom reach about 10 or higher; below that, the approximation is still usable but slightly less precise, and an exact t-table is preferable for very small samples.

Examples

Regression Coefficient Check

A t-statistic of 2.5 with 20 degrees of freedom, tested two-tailed, produces an approximate p-value a little above 0.02, comfortably under the common 0.05 significance threshold.

One-Tailed Directional Test

The same t-statistic of 2.5 with 20 degrees of freedom, tested one-tailed instead, produces half that two-tailed value, roughly 0.01, since a one-tailed test only considers deviation in a single predetermined direction.

Advantages

  • Produces both one-tailed and two-tailed p-values from a single t-statistic and degrees-of-freedom input, without needing a t-table.
  • Uses a transform accurate enough for typical study sizes (df of 10 or more) to make quick significance calls.
  • Notes its own approximation error explicitly, so you know when a result of this kind is precise versus approximate.

Common Mistakes

  • Selecting a one-tailed test after already seeing whether the result is significant two-tailed, which inflates the chance of a false positive.
  • Applying this normal-transform approximation to very small samples (single-digit degrees of freedom) where an exact t-distribution table would give a meaningfully different answer.
  • Confusing degrees of freedom with raw sample size, when degrees of freedom typically equals sample size minus the number of estimated parameters.

Edge Cases to Watch For

  • Degrees of freedom must be greater than zero; the calculator returns an error otherwise, since the transform is undefined for zero or negative df.
  • The final p-value is capped at 1, so an unusually small t-statistic near zero cannot return a probability above 100%.
  • For degrees of freedom below about 10, the approximation drifts further from the exact t-distribution value, so results for very small samples should be treated as directional rather than precise.

Common Use Cases

  • Researchers who have a t-statistic from statistical software output and want a quick approximate significance check.
  • Students verifying hand-calculated t-test results against an independent approximation.
  • Analysts reviewing regression or comparison output where only the t-statistic and degrees of freedom were reported.
Written & fact-checked by the Calculateus TeamLast updated August 5, 2026How we verify our formulas

Frequently asked questions

Why is this p-value only an approximation?

The exact t-distribution CDF requires evaluating an incomplete beta function, which isn't practical to compute with a simple formula - this calculator instead transforms the t-statistic into an equivalent normal z-score using a standard correction, which is very close to exact for larger degrees of freedom but slightly less precise for very small samples.

Related calculators

Confidence Interval CalculatorCalculate the confidence interval for a sample mean.Sample Size CalculatorCalculate the minimum sample size needed for a survey or study at a given confidence level and margin of error.Margin of Error CalculatorCalculate the margin of error for a survey result given sample size and confidence level.Correlation Coefficient CalculatorCalculate the Pearson correlation coefficient between two paired sets of numbers.

Conclusion

This calculator turns a bare t-statistic and degrees of freedom into an approximate significance level in seconds, which is convenient for degrees of freedom of 10 or more. For very small samples, cross-checking against an exact t-distribution table remains the more reliable option.