1. What is a Standardized Z-Score?

The standardized z-score (or standard score) measures how many standard deviations an element is from the mean. It allows comparison of scores from different normal distributions by standardizing them to a common scale.

2. How Does the Calculator Work?

The calculator uses the z-score formula:

Where:

• \( Z \) — Standardized z-score (dimensionless)
• \( x \) — Raw score/value to be standardized
• \( \mu \) — Mean of the population
• \( \sigma \) — Standard deviation of the population

Explanation: The z-score transforms data to a standard normal distribution with mean 0 and standard deviation 1, allowing comparison across different scales.

3. Importance of Z-Score Calculation

Details: Z-scores are crucial in statistics for probability calculations, identifying outliers, comparing results from different tests, and standardizing measurements in research studies.

4. Using the Calculator

Tips: Enter the raw value, population mean, and population standard deviation. Standard deviation must be greater than zero. All values should be in the same units.

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 equal to the mean of the population.

Q2: How do you interpret positive and negative z-scores?
A: Positive z-scores indicate values above the mean, negative z-scores indicate values below the mean.

Q3: What is considered an unusual z-score?
A: Typically, z-scores beyond ±2 are considered unusual (about 5% of data), and beyond ±3 are very unusual (about 0.3% of data).

Q4: Can z-scores be used for non-normal distributions?
A: While possible, interpretation is less straightforward as the properties of the standard normal distribution won't apply.

Q5: How are z-scores related to percentiles?
A: For normal distributions, z-scores can be converted to percentiles using standard normal distribution tables or functions.