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 of a normal distribution. It provides a way to compare individual data points to the overall distribution.

2. How Does the Calculator Work?

The calculator uses the Z-score formula:

Where:

• \( X \) — The value being measured
• \( \mu \) — The mean of the population
• \( \sigma \) — The standard deviation of the population
• \( Z \) — The resulting Z-score (unitless)

Explanation: The formula calculates how many standard deviations a value is above or below the mean of a normal distribution.

3. Importance of Z-Score Calculation

Details: Z-scores are crucial in statistics for standardizing data, identifying outliers, calculating probabilities, and comparing results from different normal distributions.

4. Using the Calculator

Tips: Enter the 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 Z-score mean?
A: A positive Z-score indicates the value is above the mean of the distribution.

Q2: What does a negative Z-score mean?
A: A negative Z-score indicates the value is below the mean of the distribution.

Q3: What is considered an extreme Z-score?
A: Typically, Z-scores beyond ±2 are considered unusual, and beyond ±3 are considered extreme outliers.

Q4: Can Z-scores be used with non-normal distributions?
A: While Z-scores can be calculated for any distribution, their interpretation is most meaningful for normal distributions.

Q5: How are Z-scores related to probability?
A: Z-scores can be converted to probabilities using the standard normal distribution table or calculator, showing the likelihood of a value occurring.