1. What is the Z-Score?

The Z-score (standard score) indicates how many standard deviations an element is from the mean. It's a dimensionless quantity used to compare data points from different normal distributions.

2. How Does the Calculator Work?

The calculator uses the Z-score formula:

Where:

• \( N \) — Z-score (dimensionless)
• \( x \) — Raw score (value)
• \( \mu \) — Population mean (value)
• \( \sigma \) — Population standard deviation (value)

Explanation: The formula standardizes any normal distribution to a standard normal distribution with mean 0 and standard deviation 1.

3. Importance of Z-Score Calculation

Details: Z-scores are crucial in statistics for probability calculations, comparing different data sets, and identifying outliers (typically values beyond ±3).

4. Using the Calculator

Tips: Enter the raw value, population mean, and population standard deviation. Standard deviation must be positive.

5. Frequently Asked Questions (FAQ)

Q1: What does a Z-score of 0 mean?
A: A Z-score of 0 indicates the value is exactly at the mean of the distribution.

Q2: What is considered an unusual Z-score?
A: Typically, Z-scores beyond ±2 are considered unusual, and beyond ±3 are very unusual.

Q3: Can Z-scores be used with non-normal distributions?
A: While you can calculate them, interpretation is most meaningful for normal distributions.

Q4: How do I interpret a negative Z-score?
A: A negative Z-score means the value is below the mean.

Q5: What's the relationship between Z-scores and percentiles?
A: Z-scores can be converted to percentiles using standard normal distribution tables.