1. What is a Z-Score?

A Z-score (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.

2. How Does the Calculator Work?

The calculator uses the Z-score formula:

Where:

• \( Z \) — Z-score (dimensionless)
• \( x \) — Value being standardized
• \( \mu \) — 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 deviations. Positive values are above the mean, negative values are below.

3. Importance of Z-Score Calculation

Details: Z-scores are essential in statistics for probability calculations, comparing different data sets, identifying outliers, 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 means the value is exactly equal to 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 rare in normal distributions.

Q3: Can Z-scores be used with any distribution?
A: While you can calculate Z-scores for any data, they are most meaningful for normally distributed data.

Q4: How do you convert Z-scores back to raw values?
A: Use the formula: \( x = \mu + Z \times \sigma \)

Q5: What's the relationship between Z-scores and percentiles?
A: In normal distributions, Z-scores correspond to specific percentiles (e.g., Z=1.96 ≈ 97.5th percentile).