1. What is the Z-Score Formula?

The Z-Score (standard score) measures how many standard deviations an element is from the mean. It's a dimensionless quantity used to compare data points from different normal distributions.

2. How Does the Calculator Work?

The calculator uses the Z-Score formula:

Where:

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

Explanation: The formula standardizes values, allowing comparison across different datasets by expressing them in terms of standard deviations from the mean.

3. Importance of Z-Score Calculation

Details: Z-Scores are crucial in statistics for probability calculations, identifying outliers, comparing results from different tests, and standardizing measurements.

4. Using the Calculator

Tips: Enter the value, population mean, and standard deviation. Standard deviation must be greater than zero. All values can be any real number.

5. Frequently Asked Questions (FAQ)

Q1: What does a Z-Score of 0 mean?
A: A Z-Score of 0 indicates the value is exactly equal to the mean of the population.

Q2: What is considered a "high" Z-Score?
A: Typically, Z-Scores beyond ±2 are considered unusual, and beyond ±3 are very unusual (outliers).

Q3: Can Z-Scores be negative?
A: Yes, negative Z-Scores indicate values below the mean, while positive scores are above the mean.

Q4: What's the relationship between Z-Scores and percentiles?
A: Z-Scores can be converted to percentiles using the standard normal distribution table (e.g., Z=1.96 ≈ 97.5th percentile).

Q5: When shouldn't Z-Scores be used?
A: Z-Scores assume normal distribution and aren't appropriate for highly skewed distributions or small sample sizes.