Critical Value Calculator

Common questions

What is the critical value for a 95% confidence level?

On the normal distribution it is 1.959964, the two-tailed cutoff at alpha 0.05: 0.025 of the curve sits beyond it at each end. Rounded to 1.96 it is the figure most tables print. When the standard deviation was estimated from the sample rather than known, t is the distribution and the cutoff depends on the degrees of freedom: 2.228139 on 10 of them, 2.063899 on 24, and 1.959988 on 100,000, which is as close to the normal figure as this page goes.

What is the difference between a one-tailed and a two-tailed critical value?

Where alpha goes. Two-tailed splits it, so alpha 0.05 leaves 0.025 at each end and the normal cutoff is 1.959964. One-tailed spends all of it on a single side, so the same alpha leaves 0.05 in one tail and the cutoff is 1.644854. Those are different numbers for the same alpha, which is why the tail count is printed next to the cutoff here and why the table underneath runs both columns. At alpha 0.01 the same pair is 2.575829 and 2.326348.

How do I find the critical value for a t test?

Choose t, type the degrees of freedom and the significance level, then pick the tail count. For 10 degrees of freedom at alpha 0.05 that is 2.228139 either side of zero, or 1.812461 in one tail. The degrees of freedom do not have to be whole: Welch's correction for two samples with unequal variances produces a fractional one, and the cutoff is computed for it directly rather than rounded to the nearest printed table row.

What is the chi-square critical value at 0.05?

It depends on the degrees of freedom and, unlike the normal and t, on which end you mean, because the chi-square curve is not symmetric and its two cutoffs are different distances from the middle. On 3 degrees of freedom the upper-tail cutoff at alpha 0.05 is 7.814728. On 10 degrees of freedom, splitting the same alpha across both ends gives 3.246973 below and 20.483177 above, and this page prints both rather than a single plus or minus.

How do I read an F critical value?

F takes two degrees of freedom: df1 from the variance on top of the ratio and df2 from the one underneath, and the order changes the answer. For df1 3 and df2 12 at alpha 0.05 the upper-tail cutoff is 3.490295. The lower cutoff is the reciprocal of the upper one with the two degrees of freedom swapped, which is why printed tables list only the upper end. This page computes either, and both at once when alpha is split.

Why is a t cutoff always wider than the matching z cutoff?

Because the t distribution carries more of its weight in the tails, which is what allows for a standard deviation estimated from the same sample. At alpha 0.05 two-tailed the cutoff is 2.228139 on 10 degrees of freedom, 2.063899 on 24 and 1.959988 on 100,000: always further out than the normal's 1.959964, falling toward it as the degrees of freedom rise and never reaching it.

What significance levels does this take?

Anything from 1e-10 to 0.9999999999, typed as a decimal like 0.05 or as a percentage like 5%. An alpha of exactly 0 is refused, because leaving no area in the tail puts the cutoff at infinity, and an alpha of exactly 1 is refused for the mirror reason. A bare whole number is answered with the question it usually raises: type 5 and the page asks whether 5% was meant, rather than guessing.

How accurate are these cutoffs?

Each is found by bisection on the distribution's own tail, and that tail is always evaluated as the small side rather than as one minus the other, which is where precision is normally lost. They were then measured against a 60-digit reference written separately from the page: across 4,800 combinations of distribution, degrees of freedom from 1 to 100,000, side and tail probability from 1e-10 upward, the worst relative error in a cutoff was 1.274e-14, well inside the twelve significant figures the accuracy note claims.

Does this tell me whether my result is significant?

No. It reports a number and the geometry behind it: the cutoff, the distribution, the degrees of freedom and how much of the curve lies beyond it. What that means for a particular study depends on the design, the assumptions behind the test and what was decided before the data was collected, none of which this page knows.