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: A positive Z-score indicates the value is above the mean, while a negative Z-score indicates it's below the mean.

3. Importance of Z-Score Calculation

Details: Z-scores are crucial in statistics for comparing different data sets, identifying outliers, and standardizing measurements 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 means the value is exactly equal to the mean of the population.

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

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

Q4: What's the difference between Z-score and T-score?
A: T-scores are a type of standardized score with a mean of 50 and standard deviation of 10, commonly used in psychological testing.

Q5: How are Z-scores used in real-world applications?
A: They're used in quality control, finance (risk management), standardized testing, and medical measurements like bone density.