Hong Life CalculatorsUnderstanding the critical value calculator — concept and principle
This calculator finds critical values of the standard normal (z), Student's t, chi-square and F distributions for a chosen significance level, with two-tailed, left-tailed or right-tailed tests. It also states the rejection rule and the matching confidence level. Use it instead of a printed statistical table.
How it is calculated
- Right-tailed: the (1 − α) quantile of the distribution.
- Left-tailed: the α quantile.
- Two-tailed: the α/2 and 1 − α/2 quantiles; for z and t these are ± the same value.
- Quantiles are found by inverting the cumulative distribution function numerically.
- Common z values: 1.645 (α = 0.10 two-tailed or 0.05 one-tailed), 1.960 (0.05 two-tailed), 2.576 (0.01 two-tailed).
- χ² and F are not symmetric, so their two tails have different cut-offs; F uses numerator and denominator degrees of freedom.
The principle
A hypothesis test asks whether a statistic is too extreme to be explained by chance if the null hypothesis is true. The critical value marks where "too extreme" begins: the area beyond it under the null distribution equals α, the risk of a false alarm you accept. In a two-tailed test that risk is split between both ends. The t distribution has heavier tails than z for small samples, so its critical values are larger; as degrees of freedom grow, t approaches z.
Worked example
Default inputs: Student's t, α = 0.05, two-tailed, 15 degrees of freedom.
- Each tail gets α/2 = 0.025, so we need the 0.975 quantile of t with 15 df.
- Upper critical value = 2.131; lower = −2.131.
- Rejection rule: reject H₀ when |t| > 2.131.
- Confidence level = 1 − α = 95 %; degrees of freedom used 15.
| Significance level α | Critical value |
|---|---|
| 0.0125 | 2.837 |
| 0.025 | 2.49 |
| 0.0375 | 2.282 |
| 0.05 | 2.131 |
| 0.0625 | 2.012 |
| 0.075 | 1.913 |
| 0.1 | 1.753 |
How to read the result
Compare your test statistic with the critical value: if it falls beyond it, the result is significant at level α. The same number builds a confidence interval: mean ± 2.131 × standard error gives a 95 % interval for n = 16 observations. With z instead of t you would use 1.960, a narrower interval that ignores the extra uncertainty of a small sample.
Common mistakes
- Using a one-tailed value for a two-tailed test.
- Using z when σ is unknown and the sample is small.
- Taking the wrong degrees of freedom, such as n instead of n − 1.
- Using a symmetric ± value for χ² or F.
Frequently asked questions
What is the critical t value for 95 % confidence and 15 df?
About 2.131 for a two-tailed test.
What is the z critical value at 95 %?
1.960 for two-tailed, 1.645 for one-tailed.
Why is the t value larger than z?
The t distribution accounts for estimating σ from the sample, giving heavier tails.
Sources: NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.6.7 Tables for Probability Distributions