1. What is a Z-score?

A Z-score measures how many standard deviations an element is from the sample mean. It standardizes different data points to allow comparison across different distributions.

2. How Does the Calculator Work?

The calculator uses the Z-score formula:

Where:

• \( x \) — The value to standardize
• \( \text{Sample Mean} \) — Mean of the sample population
• \( \text{Sample SD} \) — Standard deviation of the sample population

Explanation: The Z-score transforms data onto a standard scale where the mean is 0 and standard deviation is 1.

3. Importance of Z-scores

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

4. Using the Calculator

Tips: Enter the raw value, sample mean, and sample standard deviation. 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 sample.

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

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

Q4: What's the difference between population and sample Z-scores?
A: Population Z-scores use population parameters (μ, σ) while sample Z-scores use sample statistics (x̄, s).

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