5
%
2.5
%
4.2207
4.9448
0.7241
5
%
2.5
%
4.2207
4.9448
0.7241
The Critical Value Calculator determines the boundary values used in hypothesis testing and confidence interval construction. A critical value defines the cutoff point on a probability distribution beyond which results are considered statistically significant. This calculator provides both z-critical values (from the standard normal distribution) and t-critical values (from Student's t-distribution) for common confidence levels: 90%, 95%, and 99%.
In a two-tailed test at the 95% confidence level, for example, the z-critical value is $$\pm 1.96$$, meaning that if your test statistic exceeds 1.96 in absolute value, you reject the null hypothesis. The t-critical value depends on degrees of freedom (df) and is always larger than the corresponding z-value, producing wider rejection regions that account for the extra uncertainty in small samples.
Critical values are essential tools in every branch of applied statistics — from medical research determining whether a treatment is effective, to quality control deciding whether a manufacturing process has shifted, to social science testing whether survey differences are meaningful.
The calculator maps your chosen confidence level to exact z-critical values stored for the three standard levels:
Z-critical values (standard normal):
T-critical approximation: For the t-distribution, the calculator uses the Cornish-Fisher expansion, a well-known series approximation: $$t \approx z + \frac{z + z^3/4}{4\,df} + \frac{5z + 16z^3/3 + 3z^5/4}{96\,df^2}$$ This formula provides excellent accuracy for df ≥ 5 and converges to the z-value as df → ∞. For df ≥ 200, the calculator returns the z-value directly since the t-distribution is virtually identical to the normal distribution at that point.
The significance level $$\alpha = 1 - \text{confidence level}$$ represents the total probability of Type I error (rejecting a true null hypothesis). In a two-tailed test, $$\alpha/2$$ falls in each tail.
Compare your test statistic to the critical value. If |test statistic| > critical value, the result is statistically significant at your chosen level. Use z-critical when you know the population standard deviation or n is large (> 30). Use t-critical for small samples with unknown population SD. Higher confidence levels (99% vs 95%) produce larger critical values, making it harder to reject the null hypothesis and reducing Type I error at the cost of increased Type II error. Always choose your significance level before examining the data.
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Z = 1.96 for large samples; t ≈ 2.23 for 10 df — the t-value is larger, reflecting greater uncertainty with small samples.
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At 99% confidence with 25 df, t ≈ 2.79. The exact table value is 2.787, demonstrating the approximation's accuracy.
Z-critical values come from the standard normal distribution and are used when the population standard deviation is known or the sample is large. T-critical values come from Student's t-distribution, which has heavier tails than the normal distribution, accounting for the additional uncertainty when estimating the standard deviation from a small sample. As degrees of freedom increase, t-values converge to z-values.
Degrees of freedom depend on the test: for a one-sample t-test, df = n − 1; for a two-sample t-test (equal variances), df = n₁ + n₂ − 2; for chi-square goodness of fit, df = k − 1 where k is the number of categories. Use the separate Degrees of Freedom Calculator in this suite for detailed guidance.
The significance level $$\alpha$$ is the probability of rejecting the null hypothesis when it is actually true (Type I error). At α = 0.05, there is a 5% chance of a false positive. Lower α values (like 0.01) are more conservative — they require stronger evidence to reject H₀ but increase the chance of missing real effects (Type II error).
Use a two-tailed test when you are testing for any difference (greater or lesser). Use a one-tailed test when you have a specific directional hypothesis (e.g., treatment improves outcomes). For a one-tailed test at 95% confidence, the critical z-value is 1.6449 instead of 1.96. This calculator shows two-tailed values; divide α by 1 instead of 2 for one-tailed use.
The Cornish-Fisher expansion used here is accurate to within 0.01 for df ≥ 5 at common confidence levels. For df < 5, the approximation may deviate slightly from exact table values. For critical applications requiring maximum precision with very small df, consult published t-distribution tables or statistical software.
A higher confidence level means you want to be more certain before rejecting the null hypothesis. To achieve 99% confidence, you must push the critical boundary further into the tails, capturing 99% of the distribution in the middle. This makes the critical value larger (2.576 vs 1.96 for z), requiring a more extreme test statistic to declare significance.
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