1. What is a Z Score?

A Z-score (standard score) represents how many standard deviations an element is from the mean. In a standard normal distribution, the Z-score corresponds to a specific percentile.

2. How Does the Calculator Work?

The calculator uses the inverse normal distribution function:

Where:

• \( z \) — Standard score (Z-score)
• \( percentile \) — The percentile rank (0-100%)
• \( \text{invNorm} \) — Inverse standard normal distribution function

Explanation: The function finds the Z-score that corresponds to a given cumulative probability in a standard normal distribution.

3. Importance of Z Scores

Details: Z-scores are essential in statistics for comparing different normal distributions, determining probabilities, and identifying outliers.

4. Using the Calculator

Tips: Enter a percentile value between 0.01 and 99.99. The calculator will return the corresponding Z-score from the standard normal distribution.

5. Frequently Asked Questions (FAQ)

Q1: What does a Z-score of 0 mean?
A: A Z-score of 0 corresponds to the 50th percentile, meaning the value is exactly at the mean of the distribution.

Q2: What Z-score corresponds to the 95th percentile?
A: Approximately 1.645. This means 95% of values fall below this point in a standard normal distribution.

Q3: Can I convert Z-scores back to percentiles?
A: Yes, using the standard normal cumulative distribution function (CDF).

Q4: Are Z-scores only for normal distributions?
A: While primarily used for normal distributions, Z-scores can be calculated for any distribution, but interpretation differs.

Q5: What's the difference between Z-scores and T-scores?
A: T-scores are a type of standardized score with a mean of 50 and standard deviation of 10, while Z-scores use mean 0 and SD 1.