Critical Value Calculator

Φ(z) = 1 − α/2 T(t, df) = 1 − α/2 P(χ² > x, df) = α P(F > f, df₁, df₂) = α

α

The significance level — the tail area the boundary is drawn at, and the same quantity the dropdown above calls a confidence level. The two vocabularies are one subtraction apart: a 95% confidence level is α = 0.05, a 99% level is α = 0.01. Splitting the difference matters, because the page needs to know whether that 5% sits in one tail or is shared by two, and only the α reading makes that visible

Φ⁻¹

The inverse of the standard normal cumulative function — the same curve the z table is read from, run backwards. The table gives an area for a given z; a critical value is the z for a given area, which is exactly why the answer has to come from an approximation rather than from arithmetic

df

Degrees of freedom — the shape parameter of the t and chi-square distributions. It is what makes the same confidence level give different answers on different pages: at 95% two-tailed the boundary is 12.7062 at one degree of freedom, 2.2281 at ten, and 1.9600 once the sample is large enough that the t and the normal are indistinguishable

df₁, df₂

The two degrees of freedom of the F distribution — the numerator and the denominator. An F test compares two variances or several group means at once, so it needs both, and the second field on the page is only read in this mode. Entering them the other way round is a common slip and gives a different answer

tails

Which side of the distribution the boundary sits on. Two tails splits α in half and puts α/2 at each end, so the 95% two-tailed z is 1.9600 while the 95% one-tailed z is 1.6449 — the same confidence, a lower bar, because the whole 5% is now on one side. For chi-square and F there is no two-tailed reading at all: those distributions are lopsided, so a lower boundary is an entirely separate number rather than the negative of the upper one

Use it when you have a confidence level or a significance level and need the number to compare a test statistic against — to finish a hand calculation, to check what a piece of software used, or to see how far the bar moves when the sample size changes. It is also the page that settles the most common confusion in this area: 1.96 and 1.645 are not two conventions for the same test, they are the two-tailed and one-tailed boundaries at the 5% level, and which one applies is decided by whether the hypothesis named a direction before the data was seen. Read the other way round, the page answers the reverse question too — a reported statistic of 2.3 next to a boundary of 2.2281 is significant at 5% two-tailed, and next to 2.5706 at five degrees of freedom it is not. What the page does not decide is which distribution you should have used. That is a question about the study: a z applies when the standard deviation is a known constant, a t when it was estimated from the same data, chi-square to counts and variances, F to ratios of variances. The dropdown is the user's call, and the FAQ below explains the parts of it that are easy to get wrong.

1. The default: the 95% two-tailed z boundary

   1. A 95% confidence level is α = 0.05
   2. Two tails means 0.025 in each, so the upper boundary is where the standard normal curve leaves 0.975 below it
   3. Φ⁻¹(0.975) = 1.9600
   4. The lower boundary is its mirror image at −1.9600, which is why a two-tailed test compares |z| against this one number

   This is the most quoted number in statistics and the right place to start, because it can be checked against memory before trusting anything else on the page. Note what the two-tailed reading does to the problem: it asks for the z that leaves half of the 5% above it, not all of it, and that halving is the entire difference between 1.9600 and 1.6449. Both degrees-of-freedom fields are on screen and unused in this mode — a z test has no degrees of freedom, and the page says so by not reading them rather than by greying them out.

2. The same 95%, but the standard deviation came from the sample

   1. Same confidence level and same number of tails, so the same α/2 = 0.025
   2. The distribution changes from the normal to the t with 10 degrees of freedom, whose tails are heavier
   3. T⁻¹(0.975, 10) = 2.2281
   4. The same boundary at 5 degrees of freedom would be 2.5706, and at 50 it is 2.0086 — it walks down towards 1.9600 as the sample grows

   The gap between 1.9600 and 2.2281 is the price of not knowing the population standard deviation, and it is the reason the two distributions are kept apart rather than treated as interchangeable. It is not a small correction: a statistic of 2.1 clears the z bar and fails the t bar at ten degrees of freedom. The three t values quoted here — 2.5706, 2.2281, 2.0086 — are the same computation at three degrees of freedom, and reading them in sequence is the clearest way to see that the t is not a different rule but the same rule with a penalty that fades.

3. A chi-square boundary has no two-tailed version

   1. Chi-square is a sum of squares, so it is never negative and its curve is lopsided to the right
   2. A test asks whether the observed deviation is too large, so the boundary is the upper one: where the curve leaves 0.95 below it
   3. P(χ² > x, df = 3) = 0.05 gives x = 7.8147
   4. Choosing two tails here is rejected rather than approximated

   Asking for a two-tailed chi-square is the one input the page refuses, and the refusal is the informative part: on a symmetric distribution the two boundaries are negatives of each other, so one number serves both ends. On a lopsided one they are unrelated values, and the upper 5% point of a chi-square with 3 degrees of freedom is 7.8147 while the lower 5% point is 0.3518. Printing either one as "the two-tailed critical value" would be a silent error, so the page stops instead. The lower boundary is available on request and is the one used by goodness-of-fit tests that reject on too little variation.

4. The lower boundary is not the negative of the upper one

   1. An F statistic is a ratio of variances, so like chi-square it cannot be negative and its curve is lopsided
   2. The upper 5% boundary with 10 and 20 degrees of freedom is 2.3479
   3. The lower 5% boundary is the value that leaves 0.05 below it: 0.3605
   4. The two do not stand in a minus-sign relationship — they are two independent points on the same lopsided curve

   This pair is the reason the page asks for a tail even on the asymmetric distributions. Because F is a ratio, its lower tail belongs to unusually small ratios — which is what an equal-variance test looks at when the second group turns out to be the more variable one. The reciprocal identity F(1−α, df₁, df₂) = 1 / F(α, df₂, df₁) is the way these were once read off a single table printed only for the upper tail, which is exactly the arithmetic the page is doing for you.

The page returns a boundary and stops there. It does not compare anything against it, because it has nothing to compare — the test statistic comes from data the page never sees. That separation is deliberate and it is also the reason the page asks you to name a distribution instead of inferring one: a boundary is only meaningful once you know what quantity is being measured against it, and the arithmetic cannot tell a z from a t. The second thing it does not do is sanity-check the degrees of freedom. A t with one degree of freedom gives a perfectly correct answer of 12.7062, and on a study of five people that answer is also nearly useless — the boundary is wide because the estimate of the standard deviation rests on a single degree of freedom. Nothing in the input stream distinguishes a deliberate stress test from a mistake, so the number is returned either way. Finally, all four distributions assume independent observations and, for the F, that the underlying populations are normal; a critical value computed correctly from the wrong distribution is still the wrong bar to clear.

What is the difference between a critical value and a p-value?

They are the same fact read from opposite ends. A critical value starts from the significance level you chose and returns the boundary; a p-value starts from the statistic you observed and returns the tail area beyond it. Because the two are inverses, comparing a statistic against a boundary and comparing a p-value against α always give the same verdict — which is why software can report either one. The practical difference is when the choice gets made: a critical value is fixed before the data is seen and is the same for every study run at that level, while a p-value is a property of the particular sample and cannot be quoted in advance. This page is the boundary half; the p-value page is the other.

Should I use 1.96 or 1.645?

That is a question about tails, not about conventions. 1.9600 is the two-tailed boundary at the 5% level and 1.6449 is the one-tailed boundary at the same level — the first splits the 5% between the two ends, the second puts all of it above. Use the two-tailed value unless a direction was specified before the data was collected, because a one-tailed test buys a lower bar by giving up the ability to detect a difference in the other direction. The table above shows the pattern directly: each α has both readings, and the one-tailed column at α is the two-tailed column at 2α.

Which distribution should I pick?

It follows from how the standard deviation was obtained and what is being compared. Use z when the standard deviation is a known constant rather than something estimated — including large-sample cases where the estimate is stable enough not to matter. Use t when the standard deviation came from the same sample as the mean, which is the usual situation and the reason t is far more common in practice. Use chi-square for counts and variances: goodness-of-fit, independence in a contingency table, a confidence interval for a variance. Use F when the statistic is a ratio of two variances, or when several group means are compared at once in an analysis of variance. The page cannot infer this from your numbers, which is why the choice is a field rather than an automatic step.

Why can I not ask for a two-tailed chi-square or F critical value?

Because those distributions are lopsided and the two tails are not mirror images. On a normal or t curve the lower boundary is exactly the negative of the upper one, so a single number describes both ends and "two-tailed" is a well-defined thing to ask for. Chi-square and F start at zero and tail away to the right, so the upper and lower boundaries are two unrelated values — for a chi-square with 3 degrees of freedom the upper 5% point is 7.8147 and the lower 5% point is 0.3518. Reporting either one as the two-tailed answer would be a quietly wrong number, so the page rejects the combination and asks which tail you meant.

Why does the critical value change when I change the degrees of freedom?

Because degrees of freedom set the shape of the t, chi-square and F distributions, and a boundary is a position on that shape. Small samples give a wider t: at 95% two-tailed the boundary is 12.7062 at one degree of freedom, 2.2281 at ten, and 2.0086 at fifty, approaching 1.9600 as the degrees of freedom grow. That convergence is not a coincidence — the t is what you get when a normal is divided by an estimated spread, and the estimate stops adding uncertainty once the sample is large. The same mechanism makes the chi-square and F boundaries move with their degrees of freedom, which is why neither can be tabulated the way the normal can.

Is the critical value the same thing as the confidence interval?

No, though the confidence interval is built out of it. The width of an interval is the critical value multiplied by a standard error, so the boundary is one of the two ingredients and the spread of the data is the other. That division of labour is why this page can answer without any data at all: the boundary depends only on the confidence level and the degrees of freedom, while the interval needs the sample too. It also explains why two studies with the same confidence level have different intervals — they share the critical value and differ in the standard error. If you want the interval itself rather than the multiplier, the confidence interval page is the one that has the mean and the spread as inputs.

• 1.3.6.7.1. Cumulative Distribution Function of the Standard Normal Distribution — e-Handbook of Statistical Methods (the cumulative function whose inverse the z boundary is, and the table it is normally read from) — National Institute of Standards and Technology (NIST)
• 6.2 Using the Normal Distribution — Introductory Statistics 2e (how the middle of the curve is cut off at a stated confidence level, and what the two tail areas add up to) — OpenStax (Rice University)
• 1.3.6.1. What is a Probability Distribution — e-Handbook of Statistical Methods (a test statistic as a quantity with its own distribution, which is what makes a boundary on that distribution mean anything) — National Institute of Standards and Technology (NIST)