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
• \( x \) — Individual data point 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 crucial in statistics for comparing data points from different normal distributions, identifying outliers, and standardizing scores for comparison across different datasets.

4. Using the Calculator

Tips: Enter the data point value (x), the population mean (μ), and the 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 data point 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: Z-scores are most meaningful with normally distributed data, but can be calculated for any distribution.

Q4: How are Z-scores related to probability?
A: In a normal distribution, Z-scores can be converted to percentiles using standard normal distribution tables.

Q5: What's the difference between Z-scores and T-scores?
A: T-scores are a type of standardized score with a mean of 50 and standard deviation of 10, while Z-scores have a mean of 0 and standard deviation of 1.