1. What is a Z-Score?

A Z-score (standard 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 Does the Calculator Work?

The calculator uses the Z-score formula:

Where:

• \( z \) — Z-score
• \( x \) — Raw data value
• \( \mu \) — 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 essential in statistics for comparing data points from different normal distributions, identifying outliers, and standardizing scores for comparison across different measurements.

4. Using the Calculator

Tips: Enter the raw data value, population mean, and population standard deviation. 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 data point is above the mean, while a negative Z-score indicates it's below the mean.

Q2: What is considered a significant Z-score?
A: Typically, Z-scores beyond ±2 are considered unusual, and beyond ±3 are considered outliers in a normal distribution.

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-scores are based on population parameters (μ and σ), while T-scores are based on sample statistics and are 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.