1. What is a Z-Score?

A Z-score (standard score) measures how many standard deviations an observation is above or below the mean. It allows comparison of data points from different normal distributions by standardizing them.

2. How Does the Calculator Work?

The calculator uses the Z-score formula:

Where:

• \( observation \) — The data point being evaluated
• \( mean \) — The average of the dataset
• \( standard\ deviation \) — The measure of dispersion in the dataset

Explanation: The formula calculates how many standard deviations a particular observation is from the mean of the distribution.

3. Importance of Z-Score Calculation

Details: Z-scores are crucial in statistics for identifying outliers, comparing results from different tests, and standardizing scores for analysis. They're widely used in research, quality control, and various scientific fields.

4. Using the Calculator

Tips: Enter the observation value, the mean of the dataset, and the standard deviation. Standard deviation must be greater than zero for valid calculation.

5. Frequently Asked Questions (FAQ)

Q1: What does a positive/negative Z-score mean?
A: A positive Z-score indicates the observation is above the mean, while a negative Z-score indicates it's below the mean.

Q2: What is considered a significant Z-score?
A: Typically, Z-scores beyond ±2 are considered unusual, and beyond ±3 are considered outliers in a normal distribution.

Q3: Can Z-scores be used with any distribution?
A: While Z-scores can be calculated for any distribution, their interpretation assumes a normal distribution for probability calculations.

Q4: How are Z-scores related to percentiles?
A: In a normal distribution, Z-scores can be converted to percentiles. For example, a Z-score of 0 corresponds to the 50th percentile.

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