1. What is 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 different data points across 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 calculates how far a data point is from the mean in terms of standard deviations. Positive values indicate above the mean, negative values indicate below the mean.

3. Importance of Z-score Calculation

Details: Z-scores are crucial in statistics for comparing different data points across different distributions, identifying outliers, and standardizing scores for comparison.

4. Using the Calculator

Tips: Enter the value (x), population mean (μ), and standard deviation (σ). 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 at the mean of the distribution.

Q2: What is considered an unusual Z-score?
A: Typically, Z-scores beyond ±2 are considered unusual (only about 5% of values in a normal distribution fall outside this range).

Q3: Can Z-scores be used with any distribution?
A: While Z-scores can be calculated for any distribution, they are most meaningful for normal (bell-shaped) distributions.

Q4: How is Z-score different from T-score?
A: T-scores are a type of standardized score where the mean is 50 and standard deviation is 10, while Z-scores have mean 0 and SD 1.

Q5: What are practical applications of Z-scores?
A: Used in quality control, finance (comparing investment returns), education (standardized testing), and medical research (growth charts).