1. What is a Z-Score?

The Z-score (standard score) measures how many standard deviations an element is from the mean. It allows comparison of scores from different normal distributions by standardizing them.

2. How Does the Calculator Work?

The calculator uses the Z-score formula:

Where:

• \( Z \) — Z-score (dimensionless)
• \( x \) — Raw score/value
• \( \mu \) — Mean of the population
• \( \sigma \) — Standard deviation of the population

Explanation: The formula shows how far a data point is from the mean in terms of standard deviations.

3. Importance of Z-Score

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

4. Using the Calculator

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

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

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 different from T-score?
A: T-scores are adjusted Z-scores with a mean of 50 and standard deviation of 10, often used in bone density measurements.

Q5: Where are Z-scores commonly used?
A: In standardized testing, quality control, finance (Altman Z-score), and medical measurements like bone density.