1. What is the Z-score?

The Z-score (standard score) measures how many standard deviations an element is from the mean. It's a dimensionless quantity used to compare data points from different normal distributions.

2. How Does the Calculator Work?

The calculator uses the Z-score formula:

Where:

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

3. Importance of Z-score Calculation

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

4. Using the Calculator

Tips: Enter the value you want to analyze, the population mean, and the standard deviation (must be > 0). The calculator will compute how many standard deviations your value is from the mean.

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 at 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 them for any distribution, they're most meaningful for normal (bell-shaped) distributions.

Q4: How is Z-score different from T-score?
A: T-scores are adjusted Z-scores with a mean of 50 and standard deviation of 10, often used in standardized testing.

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