Critical value calculator

Critical values for z, t and chi-square — with the surrounding table, because a threshold on its own tells you nothing about how sharp a line it is. Nothing you type here leaves your browser — it is worked out on your own device, so there is nothing to send and nothing to log.

What a critical value is

The value your test statistic has to exceed for its p-value to fall below a threshold you picked. It is derived entirely from the distribution and the alpha you chose, and it knows nothing whatever about your data.

That is worth stating plainly because critical values are often presented as though crossing one were a discovery. It is a comparison against a line you drew yourself.

Why t is always bigger than z, and by how much

At 95% two-tailed, z is 1.960. The t value is 2.776 at 4 degrees of freedom, 2.228 at 10, 2.042 at 30, 1.984 at 100, and 1.962 at 1000.

It approaches 1.96 from above and never reaches it. The gap is the price of having estimated the standard deviation from the same data rather than knowing it — and at small samples that price is large enough that using 1.96 instead is a real error, not a rounding.

Chi-square has only one tail

The statistic is a sum of squared differences, so it can only be large or small, and only large values are surprising. A small chi-square means your data fit the expectation well, which is not evidence of anything except that.

So there is no two-tailed chi-square critical value, and this page says so rather than printing one.

Where to go next