Critical Value Calculator

A critical value is the number your test statistic has to beat before you reject the null hypothesis. Pick a distribution, a significance level and a tail direction, and this calculator returns the exact cutoff: 1.9600 for a two-tailed z test at α = 0.05, 2.1448 for a t test with 14 degrees of freedom, and 11.0705 for a right-tailed chi-square test with 5 degrees of freedom.

Three sister tools cover the neighboring questions. The Z-Score Calculator standardizes one observation and gives its percentile, so use it when you already have a value and want to know where it sits. The Standard Deviation Calculator turns a list of raw numbers into the mean, variance and standard deviation you feed into a test statistic. The Statistics Calculator gives the full descriptive summary of a data set in one pass. Come back to this page for the last step: the fence that decides the hypothesis test.

What a critical value actually is

Every hypothesis test starts by assuming the null hypothesis is true. Under that assumption the test statistic follows a known distribution: standard normal for a z test, Student's t with some degrees of freedom for a t test, chi-square for a goodness-of-fit or variance test. You then agree in advance how much risk of a false alarm you will accept. That risk is the significance level, written α, and 0.05 is the usual default.

The critical value is the point on that distribution which leaves exactly α of the probability beyond it. Everything past the cutoff is the rejection region: results so extreme that, if the null hypothesis were true, they would happen less than α of the time. Everything on the near side is the do-not-reject region. The whole decision reduces to one comparison, which is why the critical value approach was the standard classroom method long before software could compute a p-value on demand.

How the critical value is calculated

A critical value is a quantile: the inverse of the cumulative distribution function. In symbols, for a right-tailed test:

where F is the cumulative distribution function of the relevant distribution. For a left-tailed test you use F⁻¹(α). For a two-tailed test the risk is split evenly, so each side gets α / 2 and the pair of cutoffs is:

For the symmetric z and t distributions those two numbers are mirror images, which is why results are usually quoted as a single value with a plus-or-minus sign. Chi-square is not symmetric: it starts at zero and stretches to the right, so a two-tailed chi-square test needs two genuinely different numbers.

There is no closed-form algebra for any of these quantiles. This calculator solves them numerically. The normal quantile uses the Wichura AS241 rational approximation, accurate to roughly sixteen digits across the whole range. The t quantile is found by inverting the regularized incomplete beta function, and the chi-square quantile by inverting the regularized lower incomplete gamma function, both to about ten significant digits. That is why the output matches a printed table exactly at the three decimals a table shows, and keeps going for degrees of freedom no table lists.

Worked example: a one-sample t test

A quality lab measures the fill weight of 15 bottles from a line that is supposed to average 50.0 grams. The sample mean comes out at 51.3 grams with a sample standard deviation of 3.2 grams. Is the line off target at the 5% level?

• Hypotheses: the null says the true mean is 50.0; the alternative says it is not 50.0. "Not equal" means a two-tailed test.
• Distribution: the population standard deviation is unknown and estimated from the sample, so this is a t test.
• Degrees of freedom: n − 1 = 15 − 1 = 14.
• Critical value: t with 14 degrees of freedom at α = 0.05 two-tailed is ±2.1448.
• Standard error: 3.2 ÷ √15 = 3.2 ÷ 3.8730 = 0.8262.
• Test statistic: (51.3 − 50.0) ÷ 0.8262 = 1.3 ÷ 0.8262 = 1.5734.
• Decision: 1.5734 is inside the interval from −2.1448 to +2.1448, so it does not reach the rejection region. Do not reject the null hypothesis.

The p-value route gives the same answer: the two-tailed p-value for t = 1.5734 with 14 degrees of freedom is 0.1379, comfortably above 0.05. The same critical value also builds the confidence interval. The margin of error is 2.1448 × 0.8262 = 1.7721, so the 95% interval for the true mean runs from 49.5279 to 53.0721 grams. Because that interval contains 50.0, the test and the interval agree, as they always must when they share a critical value.

Notice how much the tail choice matters. Had the lab only cared about overfilling and written a one-tailed alternative, the critical value would drop to 1.7613, and 1.5734 would still fall short. Had it used α = 0.01 two-tailed instead, the cutoff would climb to 2.9768.

Z critical values by confidence level

The z table below is the one you use when the population standard deviation is known, when the sample is large, or when you are building a confidence interval for a proportion. It is fixed: there are no degrees of freedom, so these numbers never change.

The pattern worth memorizing: the one-tailed value at one confidence level equals the two-tailed value at the next level down the list. The one-tailed 99% value of 2.3263 is exactly the two-tailed 98% value, because both leave 1% in a single tail.

T critical values by degrees of freedom

The t table has one row per degrees of freedom because the shape of the distribution depends on how much data you have. The last row is the limit as degrees of freedom go to infinity, where t becomes the normal distribution.

The convergence is fast at first and then very slow. Measured against the two-tailed z value of 1.9600, the t cutoff sits 41.7% higher at df = 4, 15.4% higher at df = 9, 9.4% higher at df = 14, 4.4% at df = 29, 2.1% at df = 59 and 1.0% at df = 119. That is the arithmetic behind the old rule of thumb that above about 30 observations the two are close enough for rough work, though the calculator removes any need to approximate.

Chi-square critical values (right tail)

Chi-square critical values are almost always right-tailed, because goodness-of-fit and independence tests only get large when the data disagree with the model. Small chi-square values mean the fit is good, so there is nothing to reject on the left.

Second worked example: chi-square goodness of fit

A store tracks which of six checkout lanes 300 customers choose, expecting them to be used equally often. Equal use means 300 ÷ 6 = 50 customers per lane. The observed counts are 52, 48, 61, 39, 55 and 45. Do the lanes differ at the 5% level?

The chi-square statistic sums the squared gap between observed and expected, divided by expected:

Term by term: 0.08, 0.08, 2.42, 2.42, 0.50 and 0.50, which add to 6.00. Degrees of freedom are the number of categories minus one, so 6 − 1 = 5. The right-tailed critical value at α = 0.05 with 5 degrees of freedom is 11.0705. Since 6.00 is well below 11.0705, the counts are not far enough from equal to reject the null hypothesis; the matching p-value is 0.3062. Raising the bar to α = 0.01 would move the cutoff to 15.0863, and relaxing it to α = 0.10 would only bring it down to 9.2364, so the conclusion is stable across every conventional level.

How to use this calculator

1. Choose the distribution. Use z when the population standard deviation is known or the sample is large, t when you estimated the standard deviation from the sample, and chi-square for goodness-of-fit, independence and variance tests.
2. Choose the tail. A "not equal to" alternative is two-tailed. A "greater than" alternative is right-tailed and a "less than" alternative is left-tailed. Decide this from the hypothesis, never from the direction the data happened to go.
3. Set the significance level. Type any value between 0 and 1, or tap one of the presets. The card shows the matching confidence level so you can sanity-check it.
4. Enter degrees of freedom for t and chi-square. The field disappears for z because the normal distribution has none.
5. Read the rejection region. The result card spells out the comparison in words, so you can copy it straight into a write-up, and the table underneath shows the same cutoff at five other significance levels.

Who this calculator is for

Students working through an introductory statistics course use it to replace the back-of-the-book table, especially for degrees of freedom the printed table skips. Lab technicians and quality engineers use it to set acceptance limits for control charts and capability studies. Survey researchers use the z values to build margins of error. Analysts running an A/B test use it to fix the decision threshold before the experiment starts, which is the honest way to run one. Anyone checking a homework answer or a spreadsheet formula can use it as a second opinion, since the numbers are the same ones printed in standard tables.

Key terms

• Significance level (α): the probability of rejecting a true null hypothesis, chosen before the data are seen. Also called the Type I error rate.
• Confidence level: 1 − α, expressed as a percentage. A 95% confidence level is the same choice as α = 0.05.
• Degrees of freedom: the number of independent pieces of information left after estimating parameters. It controls the shape of the t and chi-square curves.
• Rejection region: the set of test-statistic values beyond the critical value, where you reject the null hypothesis.
• Test statistic: the standardized number computed from your data, such as t = (sample mean − hypothesized mean) ÷ standard error.
• Two-tailed test: a test where the alternative allows a difference in either direction, splitting α between both tails.
• Quantile: the value below which a stated proportion of the distribution falls. A critical value is a quantile of the null distribution.

Choosing one tail or two

One-tailed tests are more powerful in the direction you chose, because the whole of α goes into a single tail and the cutoff moves closer to zero. On the normal curve at α = 0.05, the one-tailed value of 1.6449 is 16.1% closer to zero than the two-tailed 1.9600. That extra sensitivity is real, but it is paid for: a one-tailed test cannot detect an effect in the opposite direction no matter how large it is.

The rule that keeps you honest is to fix the direction before collecting data, and only when a result in the other direction would genuinely lead to the same action as no result at all. Testing whether a new drug is better than the placebo may qualify. Testing whether two groups differ almost never does, and switching to one tail after seeing which way the data went roughly doubles the real false-positive rate.

What changes the critical value

• The significance level. A smaller α always pushes the cutoff further out. Two-tailed on the normal curve: 1.6449 at 0.10, 1.9600 at 0.05, 2.5758 at 0.01 and 3.2905 at 0.001.
• The number of tails. Splitting α across two tails leaves half as much in each, so every two-tailed cutoff is further from zero than the matching one-tailed value.
• Degrees of freedom. Only for t and chi-square. More data pulls the t cutoff toward the z value and pushes the chi-square cutoff up, because a chi-square statistic with more terms is naturally larger.
• The distribution itself. Choosing t instead of z with a small sample is not a technicality: at df = 4 the two-tailed cutoff is 2.7764 against 1.9600, a 41.7% difference that can flip a decision.
• Nothing else. The sample mean, the observed spread and the effect size do not move the critical value at all. They move the test statistic. Keeping the two apart is the whole point of the method.

Critical value or p-value?

Both approaches answer the same question and always agree. With the critical value you fix the cutoff first and ask whether the statistic crossed it. With the p-value you compute the tail probability of the observed statistic and ask whether it is below α. In the bottle example, t = 1.5734 fell short of 2.1448 and the p-value of 0.1379 sat above 0.05: the same verdict twice.

The critical value has one practical advantage worth keeping. Because it does not depend on the data, you can write it down before the experiment and hand it to someone else, which makes it much harder to move the goalposts afterwards. It is also the number you need for a confidence interval, where a p-value on its own tells you nothing about the size of the effect.

Critical values in a confidence interval for a variance

Chi-square critical values do double duty in interval estimation. A confidence interval for a population variance uses both tails: the lower bound is (n − 1)s² divided by the upper critical value, and the upper bound is (n − 1)s² divided by the lower critical value. With n = 25 and a sample standard deviation of 12, degrees of freedom are 24 and (n − 1)s² = 3,456. The two-tailed cutoffs at α = 0.05 are 12.4012 and 39.3641, giving a variance interval from 87.796 to 278.684, or 9.370 to 16.694 in standard-deviation units. Notice how lopsided it is around 12: chi-square asymmetry carries straight through to the interval.

Practical tips

• Write the hypotheses in words first. The tail direction falls out of the alternative hypothesis automatically, and most tail mistakes are really hypothesis mistakes.
• Compute degrees of freedom before anything else. A single off-by-one in df changes the cutoff more than most people expect at small sample sizes.
• Quote critical values to at least three decimals in a report. Rounding 1.9600 to 2 inflates the margin of error by about 2%.
• If you plan to run several tests on the same data, adjust α first, then look up the critical value. A Bonferroni correction across five tests turns α = 0.05 into 0.01, which moves the two-tailed z cutoff from 1.9600 to 2.5758.
• Use the same critical value for the test and the confidence interval so the two cannot contradict each other.
• When the printed table stops at df = 120, do not jump to the infinity row automatically. At df = 200 the two-tailed 5% value is still 1.9719, not 1.9600.

Limitations

A critical value is only correct if the test statistic really follows the assumed distribution under the null hypothesis. A t test assumes roughly normal data or a large enough sample for the central limit theorem to work; a chi-square goodness-of-fit test is unreliable when expected counts fall below about five per cell. In those situations the cutoff is still computed correctly, but it is the cutoff for a distribution your statistic does not follow.

This tool also does not cover every distribution. F-distribution critical values, used in ANOVA and variance-ratio tests, are not included, and neither are non-integer degrees of freedom such as those produced by the Welch approximation for unequal variances. Finally, a critical value is a threshold, not a measure of importance: crossing it says the effect is unlikely to be noise, not that it is large enough to matter.

Related calculators

• To standardize one observation and read its percentile, use the Z-Score Calculator.
• To get the mean, variance and standard deviation from a list of numbers, use the Standard Deviation Calculator.
• For a full descriptive summary including median, mode and range, use the Statistics Calculator.
• For the chance of simple and combined events, use the Probability Calculator.
• To count arrangements and selections behind a discrete distribution, use the Permutation & Combination Calculator.

Sources

Every number on this page comes from the exact mathematical definitions of three standard probability distributions, so no external data source is involved and none is cited. Specifically:

• Deterministic mathematics, no source needed: the critical value is the quantile function of the standard normal, Student t and chi-square distributions, each defined by its own density. The quantiles are computed numerically to about ten significant digits, and the worked examples apply only the definitions of the test statistic and the confidence interval.
• Algorithms used: Wichura's AS241 rational approximation for the normal quantile; inversion of the regularized incomplete beta function for the t quantile; inversion of the regularized lower incomplete gamma function for the chi-square quantile. These are the published algorithms behind the same values printed in standard statistical tables.
• Verification: outputs were checked against the classic table entries z = 1.9600 at 95% two-tailed, t = 2.2281 at 10 degrees of freedom two-tailed at 0.05, and chi-square = 11.0705 at 5 degrees of freedom right-tailed at 0.05. All three match to the printed precision.