Critical Value Calculator
At 95% confidence with a two-tailed z-test, the critical value is about ±1.96. This calculator finds the critical value for a z (standard normal) or t (Student's t) distribution at any common confidence level, for both one-tailed and two-tailed tests. Pick your distribution, confidence level, number of tails, and degrees of freedom (for t only), and it returns the exact cutoff value used to build a confidence interval or decide whether to reject a null hypothesis. Critical values are the numbers that mark the boundary of the rejection region in a hypothesis test, or the multiplier used to build a margin of error around a sample estimate. Instead of flipping through a printed z-table or t-table and interpolating between rows, this tool computes the value directly using the same numerical methods statistical software relies on: a rational approximation for the normal inverse CDF and a continued-fraction incomplete beta function for the t-distribution. The t-distribution is used instead of z whenever the population standard deviation is unknown and has to be estimated from a small sample. It has heavier tails than the normal distribution at low degrees of freedom, which makes its critical values larger, and it converges toward the z critical value as the sample size grows.
Quick answer
A critical value marks the boundary of the rejection region in a hypothesis test, or the multiplier applied to standard error when building a confidence interval.
What this tells you
- •A critical value marks the boundary of the rejection region in a hypothesis test, or the multiplier applied to standard error when building a confidence interval.
- •Use the z distribution when the population standard deviation is known or the sample size is large, typically above 30.
- •Use the t distribution when the population standard deviation is unknown and estimated from the sample, especially with small samples.
- •A two-tailed test splits the total alpha (1 minus confidence level) evenly between both tails, so each side uses alpha divided by 2.
- •A one-tailed test puts the entire alpha in a single tail, which produces a smaller critical value than the two-tailed case at the same confidence level.
- •Degrees of freedom for a one-sample t-test equal the sample size minus 1, and larger degrees of freedom pull the t critical value closer to the z value.
- •The calculator rounds displayed values to 4 decimal places but performs the underlying math in full floating-point precision.
How to Use
- 1Choose the distribution: Z for a known population standard deviation or a large sample, or T for a small sample with an unknown population standard deviation.
- 2Select the confidence level you need, such as 95% for a standard hypothesis test or 99% for a stricter threshold.
- 3Select whether your test is two-tailed (testing for any difference) or one-tailed (testing for a difference in one specific direction).
- 4If you selected T, enter the degrees of freedom, which for a one-sample test equals your sample size minus 1. This field is ignored when Z is selected.
- 5Click Calculate to get the critical value, the alpha used in the lookup, and the range you can apply directly to your confidence interval or test statistic.
How It Works
Formula
alpha = 1 - confidence level / 100. Two-tailed: critical value = InverseCDF(1 - alpha/2). One-tailed: critical value = InverseCDF(1 - alpha).Alpha is the total probability of a Type I error you are willing to accept, found by subtracting the confidence level from 1. For a two-tailed test, that alpha is split between the two tails of the distribution, so the lookup uses alpha divided by 2 on each side, and the critical value is reported as plus or minus the same number. For a one-tailed test, the full alpha sits in a single tail, so the lookup uses alpha directly and the result is reported as one positive number, which can be negated for a lower-tail test. The InverseCDF step finds the point on the distribution where the cumulative probability equals 1 minus that per-tail alpha. For Z, this uses a rational approximation of the standard normal inverse cumulative distribution function. For T, it uses the inverse of the regularized incomplete beta function evaluated at the chosen degrees of freedom, found numerically by narrowing in on the value where the t-distribution's CDF matches the target probability.
Calculation note: values are processed in the order shown above, using the current input units.
Worked Examples
Z critical value at 95% confidence, two-tailed
Alpha = 1 - 0.95 = 0.05, split into 0.025 per tail for a two-tailed test. The z value where 97.5% of the standard normal distribution falls below it is about 1.9600, so the rejection region sits beyond plus or minus 1.96 standard errors from the mean.
T critical value at 95% confidence, two-tailed, df = 9
With a sample size of 10, degrees of freedom equal 9. Because the t-distribution has heavier tails than the normal distribution at low degrees of freedom, the two-tailed 95% critical value of about 2.2622 is noticeably larger than the z value of 1.96 for the same confidence level.
T critical value at 90% confidence, one-tailed, df = 20
A one-tailed test at 90% confidence puts the full 0.10 alpha in a single tail rather than splitting it. With 20 degrees of freedom, the resulting critical value is about 1.3253, smaller than a two-tailed critical value would be at the same confidence level because the whole rejection region sits on one side.
Z critical value at 99% confidence, two-tailed
Alpha = 1 - 0.99 = 0.01, split into 0.005 per tail. A stricter 99% confidence level pushes the critical value further from zero than the 95% case, up to about plus or minus 2.5758, widening the confidence interval or rejection region accordingly.
Common Critical Values by Confidence Level
Two-tailed critical values for the z distribution and for the t distribution at a few common degrees of freedom, rounded to 4 decimal places.
| Confidence level | Z (two-tailed) | t, df=9 | t, df=30 | t, df=120 |
|---|---|---|---|---|
| 80% | 1.2816 | 1.3830 | 1.3104 | 1.2893 |
| 90% | 1.6449 | 1.8331 | 1.6973 | 1.6577 |
| 95% | 1.9600 | 2.2622 | 2.0423 | 1.9799 |
| 98% | 2.3263 | 2.8214 | 2.4573 | 2.3578 |
| 99% | 2.5758 | 3.2498 | 2.7500 | 2.6175 |
As degrees of freedom increase, the t critical value shrinks toward the z critical value in the same row, and by df = 120 the two are within about 0.02 of each other at every confidence level shown.
What a critical value means in hypothesis testing
A critical value is the cutoff on a probability distribution that separates the region where you fail to reject the null hypothesis from the region where you reject it. In a hypothesis test, you compare your calculated test statistic (a z-score or t-score from your sample data) against the critical value. If your test statistic falls beyond the critical value, in the tail of the distribution, the result is considered statistically significant at your chosen confidence level and you reject the null hypothesis. The same number also builds confidence intervals: multiply the critical value by the standard error of your estimate, and add and subtract that margin from your sample statistic to get the interval's upper and lower bounds.
Choosing between the z and t distribution comes down to what you know about the population and how much data you have. Use z when the population standard deviation is known outright, which is uncommon outside of controlled settings, or when the sample size is large enough (commonly cited as 30 or more) that the sample standard deviation reliably estimates the population value. Use t whenever the population standard deviation is unknown and has to be estimated from the sample itself, which describes most real-world statistics problems, especially with small samples. The t-distribution accounts for the extra uncertainty that comes from estimating both the mean and the spread from the same limited data, which is why its critical values run larger than z at low degrees of freedom.
As the sample size grows, the t-distribution's shape converges toward the normal distribution because the sample standard deviation becomes a more reliable stand-in for the true population value. By around 100 to 120 degrees of freedom, the t and z critical values for a given confidence level are close enough that the difference rarely changes a real-world decision, which is why some textbooks simplify by using z for any sample above 30.
Common mistakes
- Using the z distribution for a small sample with an unknown population standard deviation. Below about 30 observations, the t-distribution's wider tails give a more honest, slightly larger critical value.
- Forgetting to split alpha in half for a two-tailed test. A two-tailed 95% test uses 0.025 in each tail, not 0.05, which is the one-tailed lookup instead.
- Entering the sample size instead of the degrees of freedom. For a one-sample t-test, degrees of freedom equal sample size minus 1, so a sample of 25 uses df = 24.
- Mixing up one-tailed and two-tailed tests. A one-tailed test only makes sense when the research question specifies a direction in advance, such as testing whether a value is greater than a baseline, not just different from it.
- Comparing a test statistic directly to a confidence level percentage instead of to the critical value itself. The confidence level sets which critical value to look up, but the actual comparison is against that critical value, not the percentage.
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