1. What is a Standard Normal Value (Z-score)?

A Z-score measures how many standard deviations an element is from the mean. It standardizes different data points to be comparable on the same scale.

2. How Does the Calculator Work?

The calculator uses the standard normal formula:

Where:

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

Explanation: The formula transforms any normal distribution to the standard normal distribution (μ=0, σ=1).

3. Importance of Z-scores

Details: Z-scores are crucial for comparing values from different normal distributions, calculating probabilities, and performing hypothesis testing in statistics.

4. Using the Calculator

Tips: Enter the raw value (X), population mean (μ), and standard deviation (σ). Standard deviation must be greater than 0.

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 a normal distribution.

Q3: How is this related to TI-84 calculations?
A: This formula matches the standard normal calculations performed by TI-84's normal distribution functions.

Q4: Can Z-scores be negative?
A: Yes, negative Z-scores indicate values below the mean.

Q5: What's the difference between Z-scores and T-scores?
A: T-scores are a transformation of Z-scores to have mean 50 and standard deviation 10 (used in some psychological tests).