1. What is the Z-Score?

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

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 shows how far a data point is from the mean in terms of standard deviations. Positive values are above the mean, negative values are below.

3. Importance of Z-Score

Details: Z-Scores are crucial in statistics for comparing different data sets, identifying outliers, and standardizing measurements for comparison. They're widely used in research, finance, and quality control.

4. Using the Calculator

Tips: Enter the value you want to analyze, the population mean, and the standard deviation. Standard deviation must be greater than zero.

5. Frequently Asked Questions (FAQ)

Q1: What does a Z-Score of 0 mean?
A: A Z-Score of 0 means the value is exactly at the mean of the distribution.

Q2: What is considered a significant Z-Score?
A: Typically, Z-Scores beyond ±2 are considered unusual, and beyond ±3 are very rare in a normal distribution.

Q3: Can Z-Scores be used for non-normal distributions?
A: While possible, interpretation is less straightforward as the properties of normal distributions don't apply.

Q4: How is Z-Score different from T-Score?
A: T-Scores are adjusted Z-Scores with a mean of 50 and standard deviation of 10, often used in educational testing.

Q5: What are practical applications of Z-Scores?
A: Used in standardized testing, financial risk assessment, quality control processes, and medical measurements.