1. What is a Z-Score?

A Z-score (standard score) measures how many standard deviations an element is from the mean. It's used to compare data points from different normal distributions and identify outliers.

2. How Does the Calculator Work?

The calculator uses the Z-score formula:

Where:

• \( z \) — Z-score (dimensionless)
• \( x \) — Value being measured
• \( \mu \) — Mean of the population
• \( \sigma \) — Standard deviation of the population

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, quality control, and standardized testing.

4. Using the Calculator

Tips: Enter the raw value (x), population mean (μ), and population standard deviation (σ). Standard deviation must be greater than 0.

5. Frequently Asked Questions (FAQ)

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

Q2: What is considered a "high" Z-score?
A: Typically, Z-scores beyond ±2 are considered unusual, and beyond ±3 are very rare in a normal distribution.

Q3: Can Z-scores be negative?
A: Yes, negative Z-scores indicate values below the mean, while positive scores are above the mean.

Q4: How is Z-score related to probability?
A: Z-scores can be converted to probabilities using standard normal distribution tables or functions.

Q5: When shouldn't I use Z-scores?
A: Z-scores assume normal distribution and aren't appropriate for non-normal data without transformation.