1. What is the Z-Score?

The Z-score (standard score) measures 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:

• \( Z \) — Z-score (dimensionless)
• \( x \) — Value being measured
• \( \mu \) — 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. Positive values are above the mean, negative values are below.

3. Importance of Z-Score Calculation

Details: Z-scores are crucial in statistics for comparing values from different normal distributions, identifying outliers, and standardizing data for analysis.

4. Using the Calculator

Tips: Enter the 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 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 possible, interpretation is less straightforward as the properties of normal distributions 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: For normal distributions, Z-scores can be converted to percentiles using standard normal distribution tables.