1. What is a Z-Score?

A 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 (standard score)
• \( x \) — Raw score/value to be standardized
• \( \mu \) — Mean of the population
• \( \sigma \) — Standard deviation of the population

Explanation: The formula shows how many standard deviations a value (x) is above or below the population mean (μ).

3. Importance of Z-Score Calculation

Details: Z-scores are crucial in statistics for comparing different data sets, identifying outliers, and standardizing scores for comparison. They're widely used in research, quality control, and standardized testing.

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 unusual (outliers).

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

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, while Z-scores have mean 0 and SD 1.

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