Critical Value Calculator: t, z, F, and Chi-Square

What a critical value is

A critical value is the fixed threshold your test statistic must cross to reject the null hypothesis at a chosen significance level α. Look up a t test critical value, a z critical value, an F critical value, or a chi-square critical value here, set α and the degrees of freedom, and read the cutoff plus the exact rejection region. This is a threshold lookup. It takes no data and computes no p-value; it answers the single question "how large does my statistic have to be?"

The t critical value

For a two-tailed t test at α = 0.05 with ν = 10 degrees of freedom, the critical value is ±2.2281: reject the null when |t| > 2.2281. The tail probability fed to the inverse t CDF is α/2 = 0.025 per tail, so the cutoff is the 97.5th percentile of the t distribution with 10 degrees of freedom. Switch to one-tailed and the same α = 0.05 goes entirely into one tail, giving the smaller cutoff 1.8125. Fractional degrees of freedom are accepted, so a Welch-Satterthwaite ν such as 12.4 works directly.

The z critical value

The z critical value uses the standard normal and needs no degrees of freedom. The two-tailed 5% cutoff is ±1.9600, the 97.5th percentile of the standard normal; the one-tailed 5% cutoff is 1.6449, the 95th percentile. For 99% confidence the two-tailed z critical value is ±2.5758.

The F critical value

The F critical value takes a numerator and a denominator degrees of freedom. For α = 0.05 with df1 = 3 and df2 = 12 the upper critical value is 3.4903: reject when F > 3.4903. F is right-tailed by construction in ANOVA, so the upper tail leads; a labeled two-tailed variance mode returns both bounds when you need them.

The chi-square critical value

The chi-square critical value takes one degrees-of-freedom count. For α = 0.05 with ν = 1 the upper critical value is 3.8415, which is exactly 1.9600² because a chi-square with one degree of freedom is a squared standard normal. Goodness of fit and independence tests are right-tailed, so chi-square also leads with the upper tail.

How one-tailed and two-tailed differ

A two-tailed test splits α into two equal pieces, α/2 in each tail, because the alternative allows a departure in either direction. A one-tailed test commits to a direction and spends the whole α in that single tail, which is why its cutoff sits closer to zero for the same α. This tool signs one-tailed values: a left-tailed selection returns a negative number, so a left-tailed z at α = 0.05 reads Z < -1.6449, not an unsigned 1.6449. The confidence row is labeled Confidence level (1 - α); for a one-tailed selection that is a one-sided level, and a 95% one-sided level pairs with a 90% two-sided interval.

Why chi-square and F default to the upper tail

Goodness-of-fit, test-of-independence, and ANOVA statistics are inherently right-tailed: evidence against the null shows up as a large statistic, never a small one. So the default here is the upper-tail critical value, matching the convention practitioners expect. The separately labeled two-tailed variance mode exists for the chi-square and F variance tests that genuinely inspect both tails; it returns a lower and an upper bound, for example χ² < 3.2470 or χ² > 20.4832 at α = 0.05 with ν = 10.

Which quantile backs each answer

Each answer is a single inverse-CDF call: inverse normal for z, inverse Student's t for t, inverse chi-square for χ², and inverse F for F. The reference tables are the NIST/SEMATECH e-Handbook critical-value tables listed under Sources. The tool re-implements no distribution; it maps your inputs to the correct left-tail probability and sign, then reads the quantile.