Critical Value Calculator (Z and T)
Find the critical value for confidence intervals and hypothesis tests using either the standard normal (Z) or Student's t distribution.
What is a critical value?
A critical value is a cutoff point that separates the rejection region from the non-rejection region in hypothesis testing. It is also used in confidence intervals to determine how far you extend from a sample estimate.
In simple terms: once you pick a confidence level (or significance level), the critical value tells you how extreme a test statistic must be before you reject the null hypothesis.
How this calculator works
Step 1: Convert confidence level to alpha
The calculator converts confidence level to significance level using:
- α = 1 − confidence level (in decimal form)
Step 2: Determine the probability position
- Two-tailed: use p = 1 − α/2
- Right-tailed: use p = 1 − α
- Left-tailed: use p = α
Step 3: Pick the correct distribution
- Z distribution when population standard deviation is known or sample size is large.
- T distribution when population standard deviation is unknown (especially with smaller samples).
Z vs. t critical values
Use z critical values for normal-based procedures. Use t critical values when working with sample standard deviation and finite degrees of freedom.
T critical values are typically larger than z critical values at the same confidence level because they account for extra uncertainty in estimating variability.
Common critical values (quick reference)
- Two-tailed 90% confidence (z): ±1.645
- Two-tailed 95% confidence (z): ±1.960
- Two-tailed 99% confidence (z): ±2.576
- Right-tailed 95% (z): 1.645
- Left-tailed 95% (z): -1.645
Worked examples
Example 1: Two-tailed z test at 95%
Set distribution to Z, tails to two-tailed, and confidence to 95. The calculator returns approximately ±1.95996.
Example 2: Two-tailed t test at 95%, df = 10
Set distribution to T, tails to two-tailed, confidence to 95, and df = 10. The critical value is about ±2.228.
Why critical values matter
Critical values are central to confidence intervals, A/B testing, quality control, medical studies, social science research, and more. A wrong critical value can lead to incorrect conclusions, so selecting the right tail type and distribution is essential.
Tips to avoid mistakes
- Match the tails to your alternative hypothesis before calculating.
- Do not confuse confidence level with alpha.
- For t procedures, verify degrees of freedom carefully.
- Use enough decimal places when doing manual calculations.
Final note
This calculator gives precise numerical critical values for both z and t distributions, making it easy to move from problem setup to final statistical decision. If you are building confidence intervals or running hypothesis tests, it can save time and reduce errors.