What a critical value is
A critical value is a cut-off on the z scale. It marks the boundary of the region you have agreed in advance to treat as too extreme to be chance. Everything beyond it is the rejection region; everything inside it is consistent with the null hypothesis at your chosen level.
The calculation runs the normal distribution backwards. The normal distribution calculator takes a cut-off and returns an area; this one takes an area and returns the cut-off. Mathematically that is the inverse of the cumulative function Phi, written Phi-1 and called the probit or, on a graphing calculator, invNorm.
Two quantities describe the same decision from opposite ends. The confidence level is the share of the distribution you keep inside; the significance level alpha is the share you leave outside. They sum to 1, so a 95% confidence level is an alpha of 0.05, and this page lets you enter whichever you have.
The other choice that changes the answer is whether alpha sits in one tail or is split between two. That is not an arithmetic detail: at alpha = 0.05 the two-tailed cut-off is 1.96 and the one-tailed cut-off is 1.645, a difference large enough to flip a conclusion.
One tail or two, and where the halving happens
For a two-tailed test the total alpha is split evenly, so each tail gets alpha/2. You want the z value with alpha/2 above it, which means inverting the cumulative probability 1 - alpha/2. At alpha = 0.05 that is Phi-1(0.975) = 1.959964.
For a one-tailed test the whole of alpha sits in a single tail, so you invert 1 - alpha instead. At alpha = 0.05 that is Phi-1(0.95) = 1.644854. For a left-tailed test the magnitude is identical and the sign flips, because the normal curve is symmetric: the cut-off is -1.644854.
That relationship produces a useful check. A one-tailed test at alpha is always the same number as a two-tailed test at 2 x alpha, so the one-tailed 5% value equals the two-tailed 10% value. The table on this page shows exactly that alignment, and if your one-tailed answer does not match the two-tailed answer one level down, you have halved alpha in the wrong place.
Converting to raw units is a rearrangement of the z-score formula: x = mu + z x sigma. Enter your distribution's mean and standard deviation and the calculator returns the cut-offs in your own units. When you are testing a sample mean rather than an individual value, the relevant spread is the standard error sigma/sqrt(n), so enter that as the standard deviation - and the two cut-offs become the ends of a confidence interval for the mean.
The inverse normal has no closed-form expression. Printed tables were read backwards for decades; software now uses rational approximations that are accurate to far more digits than any application needs.
Worked example: a 95% interval, then a one-tailed test
Start with the interval everyone quotes. You want 95% confidence, two-tailed.
- Convert to alpha. 1 - 0.95 = 0.05.
- Split between the tails. 0.05 / 2 = 0.025 in each tail.
- Find the cumulative probability to invert. 1 - 0.025 = 0.975.
- Invert. Phi-1(0.975) = 1.959964, the familiar 1.96.
Now put it on an IQ scale with mean 100 and standard deviation 15. The cut-offs are 100 - 1.959964 x 15 = 70.60 and 100 + 1.959964 x 15 = 129.40. Exactly 95% of scores fall in that band, with 2.5% below and 2.5% above.
Now a different question. A supplier claims the mean fill weight of a bottling line is 500 ml, and you only care whether it is under that - overfilling costs money but does not fail the specification. That is a left-tailed test. At alpha = 0.05:
- All of alpha goes in the lower tail. perTail = 0.05.
- Invert 1 - 0.05 = 0.95. Phi-1(0.95) = 1.644854, and the left-tailed critical value is -1.644854.
- Convert to millilitres. With a process standard deviation of 4 ml and a sample of 25 bottles, the standard error is 4 / sqrt(25) = 0.8 ml, so the cut-off is 500 - 1.644854 x 0.8 = 498.68 ml.
A sample mean below 498.68 ml rejects the claim at the 5% level. Note that the one-tailed cut-off of 1.645 is closer to the centre than the two-tailed 1.96, so a one-tailed test rejects on weaker evidence in the direction you chose - and has no power at all in the other direction, which is the price of that sensitivity.
Using the cut-off, and what it does not tell you
Compare your test statistic with the critical value. For a two-tailed test, reject the null when the absolute z exceeds the cut-off. For a right-tailed test, reject when z exceeds it; for a left-tailed test, reject when z falls below the negative cut-off. The alternative route is to compute the p-value from the normal distribution calculator and compare it with alpha - the two approaches always agree, because they are the same comparison read on different scales.
Alpha is the rate of false positives you are accepting, not a property of your data. Choosing 0.05 means that if the null hypothesis is true, 5% of studies like yours would reject it anyway. It says nothing about how often the null is actually true, which is a Bayesian question requiring a prior - the Bayes' theorem calculator shows how much difference that makes.
Choose the number of tails before you see the data, and be honest about it. A one-tailed test is legitimate when a result in the other direction would lead to exactly the same action as no result at all. Running a two-tailed test, seeing the direction, and then reporting it as one-tailed roughly doubles the true false-positive rate while quoting the original alpha.
Finally, use z only when the population standard deviation is genuinely known. When sigma is estimated from the sample, the correct cut-off comes from the t distribution with n - 1 degrees of freedom, and it is larger - at 95% two-tailed with n = 10 it is 2.262 rather than 1.960. The two converge as n grows and are within about 2% of each other by n = 60.
Critical z values at the standard levels
| Confidence | Alpha | Two-tailed z | One-tailed z |
|---|---|---|---|
| 80% | 0.20 | 1.281552 | 0.841621 |
| 85% | 0.15 | 1.439531 | 1.036433 |
| 90% | 0.10 | 1.644854 | 1.281552 |
| 95% | 0.05 | 1.959964 | 1.644854 |
| 98% | 0.02 | 2.326348 | 2.053749 |
| 99% | 0.01 | 2.575829 | 2.326348 |
| 99.5% | 0.005 | 2.807034 | 2.575829 |
| 99.9% | 0.001 | 3.290527 | 3.090232 |
Read down the two columns and you will see each one-tailed value repeated as the two-tailed value one row higher up the table: the one-tailed 95% figure 1.644854 is the two-tailed 90% figure, because both leave 0.05 in a single tail.
Mistakes worth avoiding
- Forgetting to halve alpha. A two-tailed 95% test inverts 0.975, not 0.95. Inverting 0.95 gives 1.645 and silently turns your 5% test into a 10% one.
- Choosing the number of tails after seeing the data. This inflates the real false-positive rate to roughly double the alpha you report. Fix the direction in advance and state it.
- Using z when sigma is estimated. With an estimated standard deviation the correct distribution is t, whose critical values are larger for small n. Using z there produces intervals that are too narrow and tests that reject too often.
- Reading alpha as the probability the null is true. Alpha is the false-positive rate conditional on the null being true. Converting it into a probability that the null is true requires a prior probability, which no significance test supplies.
- Using a critical value with the wrong standard deviation. Testing a sample mean requires the standard error sigma/sqrt(n), not sigma. Using sigma makes the interval sqrt(n) times too wide.
- Confusing a confidence level with a coverage guarantee for one interval. The 95% refers to the long-run performance of the procedure. A particular interval either contains the true value or does not; the probability statement is about the method.
Where critical values are used
The most common use is the confidence interval: sample mean plus or minus z times the standard error. With 95% confidence and a standard error of 0.8, the interval is the mean plus or minus 1.568. The width depends on three things - the confidence level through z, the underlying spread through sigma, and the sample size through the square root of n - and only the last is usually under your control.
Critical values also set sample sizes. Requiring a margin of error E at confidence level z gives n = (z sigma / E)2, which is why halving a margin of error costs four times the sample. The same z appears in A/B test sample size formulas alongside a second z for the desired power.
In quality engineering, control chart limits are conventionally placed at three standard deviations, which is a critical value of 3 corresponding to a two-tailed alpha of 0.0027 - about one false alarm in 370 samples. That specific trade-off, chosen by Shewhart, balances false alarms against missed shifts, and it is the reason three sigma rather than 1.96 became the industrial standard.
To go the other way and turn an observed statistic into a probability, use the z-score calculator for the statistic and the normal distribution calculator for the tail area. To read a cut-off straight off empirical data with no normal assumption, the percentile calculator does the same job non-parametrically, which is the better choice whenever the data are visibly skewed.