Critical Values Calculator
Z-Distribution (Normal)
T-Distribution (Student's)
Chi-Square Distribution
Two-tailed
Right-tailed (Upper)
Left-tailed (Lower)
Critical Value
0.0000
What are Critical Values?
In statistics, a critical value is a point on the scale of the test statistic beyond which we reject the null hypothesis. It defines the boundary of the "rejection region"—the area of the distribution where the observed data is considered statistically significant.
Whether you are performing a Z-test, T-test, or Chi-Square test, the critical value depends on your chosen significance level (α) and the nature of your alternative hypothesis (one-tailed or two-tailed).
Commonly Used Critical Values
For a standard normal distribution (Z-distribution), the following values are frequently used:
| Confidence Level | Alpha (α) | Z-Value (Two-tailed) |
|---|---|---|
| 90% | 0.10 | 1.645 |
| 95% | 0.05 | 1.960 |
| 99% | 0.01 | 2.576 |
How to Calculate Critical Values
1. Z-Distribution: Used when the population standard deviation is known or the sample size is large (n > 30). It follows the standard normal curve.
2. T-Distribution: Used when the population standard deviation is unknown and the sample size is small. It requires Degrees of Freedom (df), usually calculated as n – 1.
3. Chi-Square: Used for tests of independence or goodness-of-fit. These values are always non-negative and typically used for right-tailed tests.
Calculation Example
Suppose you are performing a two-tailed T-test with a 95% confidence level (α = 0.05) and a sample size of 15. Your degrees of freedom would be 14 (15 – 1). Using this calculator, you would select "T-Distribution", enter 0.05 for Alpha, 14 for Degrees of Freedom, and select "Two-tailed". The resulting critical value is ±2.1448.