1. What is a Critical Z Value?

The critical Z value is the point on the standard normal distribution that marks the boundary for a given significance level (α). For α = 0.05 (5% significance), the two-tailed critical value is approximately ±1.96.

2. How Does the Calculator Work?

The calculator uses the inverse standard normal distribution function:

Where:

• \( \Phi^{-1} \) — Inverse standard normal cumulative distribution function
• \( \alpha \) — Significance level (typically 0.05, 0.01, etc.)

Explanation: The critical value separates the likely (central region) from the unlikely (tails) under the null hypothesis.

3. Importance of Critical Values

Details: Critical values are essential for hypothesis testing, determining confidence intervals, and establishing statistical significance thresholds.

4. Using the Calculator

Tips: Enter your desired alpha level (default is 0.05) and select whether you need a one-tailed or two-tailed critical value.

5. Frequently Asked Questions (FAQ)

Q1: Why is the two-tailed 0.05 critical value ±1.96?
A: Because 95% of the standard normal distribution lies between -1.96 and 1.96, leaving 2.5% in each tail (total 5%).

Q2: What's the difference between one-tailed and two-tailed?
A: One-tailed tests use all α in one direction, while two-tailed tests split α between both tails.

Q3: When would I use a one-tailed critical value?
A: When you have a directional hypothesis (e.g., testing if a value is specifically greater than expected).

Q4: How accurate is this calculator?
A: It uses a numerical approximation accurate to about 4 decimal places for typical α values.

Q5: What if I need t critical values instead of Z?
A: For small samples (<30) or unknown population variance, use a t-distribution calculator instead.