1. What is Z-Score Probability?

The Z-score probability represents the cumulative probability from negative infinity to the given Z-score in a standard normal distribution (mean = 0, standard deviation = 1). It tells you what percentage of values fall below a particular Z-score.

2. How Does the Calculator Work?

The calculator uses the standard normal cumulative distribution function:

Where:

• \( P \) — Cumulative probability (0 to 1)
• \( z \) — Standard normal deviate (Z-score)
• \( \Phi \) — Standard normal CDF

Explanation: The function calculates the area under the standard normal curve to the left of the given Z-score.

3. Importance of Z-Score Probability

Details: Z-score probabilities are fundamental in statistics for hypothesis testing, confidence intervals, and determining percentiles in normally distributed data.

4. Using the Calculator

Tips: Enter any Z-score (positive or negative) to get its cumulative probability. For example, a Z-score of 1.96 gives a probability of ~0.975.

5. Frequently Asked Questions (FAQ)

Q1: What does a probability of 0.5 mean?
A: A probability of 0.5 corresponds to a Z-score of 0, which is the mean of the standard normal distribution.

Q2: How do I find the probability for values above a Z-score?
A: Subtract the cumulative probability from 1. For P(Z > z), calculate 1 - Φ(z).

Q3: What's the probability for Z = 1.96?
A: Approximately 0.975, meaning 97.5% of values fall below Z = 1.96 in a standard normal distribution.

Q4: Can I use this for non-normal distributions?
A: The calculator is specifically for standard normal distributions. For other distributions, different methods are needed.

Q5: What's the range of possible probabilities?
A: Probabilities range from 0 to 1 (or 0% to 100%), though extreme Z-scores will approach these limits asymptotically.