1. What is a Z-Score?

A Z-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 the Calculator Works

The calculator uses the Z-score formula:

Where:

• \( X \) — Raw score
• \( \mu \) — Mean of the population
• \( \sigma \) — Standard deviation of the population

For the probability between two Z-scores:

Where \( \Phi \) is the standard normal cumulative distribution function.

3. Understanding the Results

Interpretation: The probability result shows what percentage of values in a normal distribution fall between your two Z-scores.

4. Practical Applications

Uses: Z-scores are used in statistics, quality control, finance, and research to identify outliers and compare different data sets.

5. Frequently Asked Questions (FAQ)

Q1: What does a negative Z-score mean?
A: A negative Z-score indicates the raw score is below the mean.

Q2: What's the range of possible Z-scores?
A: In theory, Z-scores can range from -∞ to +∞, but in practice most fall between -3 and +3.

Q3: How is this different from a percentile?
A: A Z-score tells you how many SDs from the mean, while a percentile tells you what percentage of scores are below yours.

Q4: When would I need to calculate between two Z-scores?
A: Useful when you want to know the probability of values falling within a specific range in a normal distribution.

Q5: Can I use this for non-normal distributions?
A: Z-scores are most meaningful for normal distributions, though they can be calculated for any distribution.