Critical Value Calculator

The Threshold That Decides Significance

Before a test statistic can be judged significant or not, it needs a boundary to compare against — the critical value. Set a significance level (alpha), pick a distribution, and the critical value marks the edge of the rejection region: any test statistic beyond it leads to rejecting the null hypothesis. This calculator solves for that boundary across four distributions using numerical inversion of each distribution's CDF.

The Formula

Each mode inverts the corresponding cumulative distribution function to find the value x where CDF(x) equals the target probability:

Z: z* = Φ⁻¹(1 − α/2) for two-tailed, Φ⁻¹(1 − α) for one-tailed
T: t* = inverse Student's t CDF at 1 − α/2 (or 1 − α) with given df
Chi-Square: χ²* = inverse χ² CDF at 1 − α (right-tailed) with given df
F: F* = inverse F CDF at 1 − α with df1, df2

Reference Table: Standard Two-Tailed Z Critical Values

Z critical values for common confidence levels (two-tailed)
Confidence Level	Alpha	z*
90%	0.10	±1.645
95%	0.05	±1.960
98%	0.02	±2.326
99%	0.01	±2.576

Reference Table: T Critical Values (Two-Tailed, α = 0.05)

t* narrows toward the z critical value as degrees of freedom grow
df	t*
5	2.571
10	2.228
20	2.086
30	2.042
60	2.000
120	1.980

Values computed by numerically inverting the same t-distribution CDF this calculator uses; they match standard published t-tables.

Where This Calculation Matters

Manual hypothesis testing — comparing a computed test statistic against the critical value is the classic "reject or fail to reject" decision rule taught before p-values became the norm.
Building confidence intervals — the margin of error in most confidence intervals is critical value × standard error, so this number is a direct input to interval construction.
Quality control charts — control limits in process monitoring are frequently set at specific z or t critical values corresponding to a target false-alarm rate.

How to Use This Calculator

Choose the distribution: Z Critical Value, T Critical Value, Chi-Square Critical Value, or F Critical Value.
Enter alpha, the significance level (e.g., 0.05 for 95% confidence).
For t, chi-square, and F, enter the relevant degrees of freedom.
Select the tail type where applicable (two-tailed, left-tailed, or right-tailed).
Select Calculate to get the critical value.

Related Calculations

Once you have a test statistic to compare, get its exact tail probability with the P-Value Calculator, or build a full interval with the Confidence Interval Calculator.