1. What is a Z-Score?

The 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 Z-Score Calculator Work?

The calculator uses the Z-score formula:

Where:

• \( Z \) — Z-score (standard score)
• \( x \) — Raw score (value to standardize)
• \( \mu \) — Mean of the population
• \( \sigma \) — Standard deviation of the population

Explanation: The formula calculates how many standard deviations a value is above or below the mean. Positive Z-scores indicate values above the mean, negative scores indicate values below the mean.

3. Importance of Z-Score Calculation

Details: Z-scores are crucial in statistics for comparing different data sets, identifying outliers, and performing hypothesis testing. They're widely used in research, quality control, and standardized testing.

4. Using the Calculator

Tips: Enter the value you want to standardize, the population mean, and the population standard deviation. All values must be valid (standard deviation must be greater than 0).

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 significant Z-score?
A: Typically, Z-scores beyond ±2 are considered unusual, and beyond ±3 are considered highly unusual (outliers).

Q3: Can Z-scores be used with any distribution?
A: While Z-scores can be calculated for any distribution, they are most meaningful for normal (bell-shaped) distributions.

Q4: How do you interpret a Z-score of 1.5?
A: A Z-score of 1.5 means the value is 1.5 standard deviations above the mean.

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