1. What is Standard Normal Distribution?

The standard normal distribution is a special case of the normal distribution with a mean of 0 and standard deviation of 1. Z-scores represent how many standard deviations an element is from the mean.

2. How Does the Z-Score Calculator Work?

The calculator uses the standard normal distribution formula:

Where:

• \( Z \) — Standard score (z-score)
• \( X \) — Raw score (value to standardize)
• \( \mu \) — Mean of the population
• \( \sigma \) — Standard deviation of the population

Explanation: The z-score tells you how many standard deviations away from the mean your data point is. Positive z-scores indicate values above the mean, negative z-scores indicate values below the mean.

3. Importance of Z-Score Calculation

Details: Z-scores are essential in statistics for comparing different normal distributions, identifying outliers, and calculating probabilities using standard normal distribution tables.

4. Using the Calculator

Tips: Enter the raw value (X), population mean (μ), and population 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: How do I interpret z-scores?
A: Each whole number z-score represents one standard deviation from the mean. For example, z=1.5 means 1.5 standard deviations above the mean.

Q3: What's the range of possible z-scores?
A: In theory, z-scores can range from -∞ to +∞, but in practice, most values fall between -3 and 3 in a normal distribution.

Q4: Can I use this for non-normal distributions?
A: While you can calculate z-scores for any distribution, the standard normal interpretation only applies to normally distributed data.

Q5: How are z-scores used in real-world applications?
A: They're used in standardized testing, quality control, finance (risk assessment), and medical research (growth charts, lab values).