1. What is Z-Score?

The Z-score measures how many standard deviations an element is from the mean. It's a statistical measurement that describes a value's relationship to the mean of a group of values.

2. How Does the Calculator Work?

The calculator uses the Z-Score formula:

Where:

• \( X \) — The value being measured
• \( \mu \) — The mean of the population
• \( \sigma \) — The standard deviation of the population

Explanation: A Z-score of 0 indicates the value is identical to the mean. Positive Z-scores indicate values above the mean, while negative Z-scores indicate values below the mean.

3. Importance of Z-Score

Details: Z-scores are crucial in statistics for comparing different data points from different normal distributions. They're used in quality control, finance, medicine, and many other fields.

4. Using the Calculator

Tips: Enter the value you want to analyze, the population mean, and the standard deviation. Standard deviation must be greater than zero.

5. Frequently Asked Questions (FAQ)

Q1: What does a Z-score of 1.0 mean?
A: A Z-score of 1.0 means the value is one standard deviation above the mean.

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

Q3: Can Z-scores be used with non-normal distributions?
A: While possible, interpretation is less straightforward as the properties of normal distribution don't apply.

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