1. What is the Normal Distribution Z-Score?

The Z-score represents how many standard deviations an element is from the mean in a standard normal distribution. The cumulative probability P(Z ≤ z) gives the area under the normal curve to the left of a given 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 score (Z-score)
• \( \Phi \) — Standard normal CDF

Explanation: The calculator numerically approximates the integral of the standard normal probability density function.

3. Importance of Z-Score Calculation

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

4. Using the Calculator

Tips: Enter any Z-score value (positive or negative) to get the corresponding cumulative probability. The result represents the probability that a random variable from a standard normal distribution will be less than or equal to the given Z-score.

5. Frequently Asked Questions (FAQ)

Q1: What does a Z-score of 0 mean?
A: A Z-score of 0 corresponds to the mean of the distribution, with a cumulative probability of 0.5 (50%).

Q2: How do I interpret negative Z-scores?
A: Negative Z-scores indicate values below the mean. The cumulative probability for z=-1.96 is approximately 0.025.

Q3: What's the relationship between Z-scores and percentiles?
A: The cumulative probability multiplied by 100 gives the percentile rank. For example, P=0.8413 corresponds to the 84.13th percentile.

Q4: What's the probability for Z=1.96?
A: The cumulative probability for Z=1.96 is approximately 0.975, meaning 97.5% of the area under the curve is to the left of this point.

Q5: Can this be used for non-standard normal distributions?
A: Yes, after standardizing your data: \( z = \frac{x - \mu}{\sigma} \), where μ is the mean and σ is the standard deviation.