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 different data points within a normal distribution.

2. How Does the Calculator Work?

The calculator uses the Z-score formula:

Where:

• \( Z \) — Z-score (dimensionless)
• \( x \) — Value to standardize
• \( \mu \) — 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.

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 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 means the value is exactly equal to the mean.

Q2: What is considered a "high" 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 negative?
A: Yes, negative Z-scores indicate values below the mean.

Q4: What's the difference between Z-score and T-score?
A: T-scores are a type of standardized score where the mean is 50 and standard deviation is 10, while Z-scores use mean 0 and SD 1.

Q5: How are Z-scores used in real life?
A: They're used in standardized testing, quality control, medical measurements, and financial risk assessment.