1. What is the Z-Score?

The Z-score measures how many standard deviations an element is from the mean. A Z-score of 2 means the value is 2 standard deviations above the mean, while -2 means it's 2 standard deviations below the mean.

2. How Does the Calculator Work?

The calculator uses the Z-score equation:

Where:

• \( Z \) — Z-score (dimensionless)
• \( x \) — Value being measured
• \( \mu \) — Mean of the population
• \( \sigma \) — Standard deviation of the population

Explanation: The equation standardizes any normal distribution, allowing comparison across different scales and measurements.

3. Importance of Z-Score Calculation

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

4. Using the Calculator

Tips: Enter the value you want to evaluate, the population mean, and the standard deviation. The calculator will show the Z-score and whether it falls within 2 standard deviations of the mean.

5. Frequently Asked Questions (FAQ)

Q1: What does a Z-score of 2 mean?
A: A Z-score of 2 means the value is 2 standard deviations above the mean. In a normal distribution, about 95% of values fall within ±2 standard deviations.

Q2: What are common uses of Z-scores?
A: Z-scores are used in quality control, finance, medicine, psychology, and any field where standardization of measurements is needed.

Q3: How is this related to standard deviation?
A: The Z-score directly measures distance from the mean in units of standard deviation.

Q4: What's considered an unusual Z-score?
A: Typically, Z-scores beyond ±2 are considered unusual, and beyond ±3 are very unusual in many applications.

Q5: Can Z-scores be negative?
A: Yes, negative Z-scores indicate values below the mean, while positive scores are above the mean.