1. What is the Z Score?

The Z score (standard score) represents how many standard deviations an element is from the mean. In hypothesis testing, it's used to determine critical values based on the significance level (alpha).

2. How Does the Calculator Work?

The calculator uses the inverse normal distribution function:

Where:

• \( \alpha \) — Significance level (typically 0.05 for 95% confidence)
• \( \text{invNorm} \) — Inverse standard normal distribution function

Explanation: For a two-tailed test, we divide alpha by 2 to get the critical value for each tail.

3. Importance of Z Score Calculation

Details: Z scores are essential for hypothesis testing, confidence intervals, and determining statistical significance in normally distributed data.

4. Using the Calculator

Tips: Enter the desired significance level (alpha) between 0 and 1 (e.g., 0.05 for 95% confidence). The calculator returns the two-tailed Z critical value.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between one-tailed and two-tailed Z scores?
A: One-tailed uses the full alpha in one direction, while two-tailed splits alpha between both tails (hence alpha/2).

Q2: What are common alpha values?
A: Common values are 0.10 (90% CI), 0.05 (95% CI), and 0.01 (99% CI).

Q3: How is this related to p-values?
A: The Z score can be converted to a p-value. If your test statistic exceeds the Z critical value, p < alpha.

Q4: When should I use Z scores vs t scores?
A: Use Z when population standard deviation is known (or large sample size), t when estimating from sample data.

Q5: What does a higher Z score mean?
A: A higher absolute Z score means the result is further from the mean, more statistically significant.