1. What is a Z-Score?

A Z-score (or standard score) represents how many standard deviations an element is from the mean. It's a statistical measurement that describes a value's relationship to the mean of a group of values.

2. How Does the Calculator Work?

The calculator uses the Z-score formula:

Where:

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

Explanation: The formula calculates how many standard deviations a data point is above or below the population mean.

3. Importance of Z-Score Calculation

Details: Z-scores are crucial in statistics for comparing results to a normal distribution, identifying outliers, standardizing scores for comparison, and in hypothesis testing.

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 positive/negative z-score mean?
A: A positive z-score indicates the value is above the mean, while a negative z-score indicates it's below the mean.

Q2: What is considered an extreme z-score?
A: Typically, z-scores beyond ±2 are considered unusual, and beyond ±3 are considered extreme outliers.

Q3: Can z-scores be used with any distribution?
A: While z-scores can be calculated for any distribution, they are most meaningful when the underlying distribution is normal or approximately normal.

Q4: How are z-scores related to probability?
A: In a standard normal distribution, z-scores correspond to specific probabilities. For example, about 68% of values fall between z = -1 and z = +1.

Q5: What's the difference between z-scores and t-scores?
A: Z-scores are based on population parameters (known μ and σ), while t-scores are used when working with sample data and estimating population parameters.