1. What Is The Z-Score?

The Z-score (standard score) measures how many standard deviations an element is from the mean. It allows comparison of data points from different normal distributions by standardizing them.

2. How Does The Calculator Work?

The calculator uses the Z-score formula:

Where:

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

Explanation: The formula shows how far a data point deviates from the mean in terms of standard deviation units.

3. Importance Of Z-Score Calculation

Details: Z-scores are crucial in statistics for probability calculations, identifying outliers, comparing different data sets, and standardizing variables for analysis.

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 the data point is exactly at the mean of the distribution.

Q2: How do you interpret positive and negative Z-scores?
A: Positive Z-scores are above the mean, negative are below. The magnitude shows how many standard deviations away.

Q3: What is considered an unusual Z-score?
A: Typically, Z-scores beyond ±2 are considered unusual, and beyond ±3 are very unusual (outliers).

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

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