1. What is a Z-Score?

A Z-score (or standard score) represents how many standard deviations an element is from the mean. It's a statistical measurement that describes a value's relationship to the mean of a group of values.

2. How Does the Calculator Work?

The calculator uses the Z-score formula:

Where:

• \( Z \) — Z-score
• \( x \) — Raw score/value to standardize
• \( \mu \) — Mean of the population
• \( \sigma \) — Standard deviation of the population

Explanation: The formula measures the distance between a data point and the population mean in terms of standard deviations.

3. Importance of Z-Score Calculation

Details: Z-scores are crucial in statistics for comparing results to a normal population, identifying outliers, standardizing scores for comparison, and probability calculations.

4. Using the Calculator

Tips: Enter the raw value, population mean, and population standard deviation. 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 that the data point's score is identical to the mean score.

Q2: What is considered a significant Z-score?
A: Typically, Z-scores beyond ±1.96 are considered statistically significant at the 0.05 level, and beyond ±2.58 at the 0.01 level.

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

Q4: How are Z-scores used in real-world applications?
A: Z-scores are used in quality control, finance (Altman Z-score), psychology testing, medical measurements, and many other fields where standardization is needed.

Q5: What's the difference between Z-scores and T-scores?
A: Z-scores are based on the population standard deviation, while T-scores are used when working with sample data and involve degrees of freedom.