1. What is the Z-Score?

The Z-score, also known as the standard score, measures how many standard deviations an element is from the mean. It's a dimensionless quantity that allows comparison of observations from different normal distributions.

2. How Does the Calculator Work?

The calculator uses the Z-score formula:

Where:

• \( z \) — Z-score (dimensionless)
• \( x \) — 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 data points from different normal distributions, identifying outliers, and standardizing scores for comparison across different measures.

4. Using the Calculator

Tips: Enter the value (x), population mean (μ), and population standard deviation (σ). Standard deviation must be greater than zero. All values are dimensionless.

5. Frequently Asked Questions (FAQ)

Q1: What does a positive/negative z-score mean?
A: A positive z-score indicates the data point is above the mean, while a negative z-score indicates it's below the mean.

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

Q3: Can z-scores be used with any distribution?
A: Z-scores are most meaningful with normally distributed data, though they can be calculated for any distribution.

Q4: How is z-score different from t-score?
A: Z-score uses population parameters (μ and σ) while t-score uses sample statistics and is used when population parameters are unknown.

Q5: What are practical applications of z-scores?
A: Z-scores are used in quality control, academic testing, medical measurements, financial analysis, and many other fields where standardization is needed.