1. What is the Z-Score?

The Z-score, also known as the standard score, measures how many standard deviations an element is from the mean of a normal distribution. It provides a way to compare individual data points to the overall distribution.

2. How Does the Calculator Work?

The calculator uses the Z-score formula:

Where:

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

Explanation: The formula calculates how many standard deviations a particular data point is above or below the population mean.

3. Importance of Z-Score Calculation

Details: Z-scores are crucial in statistics for standardizing data, identifying outliers, comparing results from different normal distributions, and calculating probabilities in standard normal distributions.

4. Using the Calculator

Tips: Enter the raw score value, population mean, and population standard deviation. The standard deviation must be greater than zero.

5. Frequently Asked Questions (FAQ)

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

Q2: How is the z-score interpreted?
A: A z-score of 1.0 means the value is one standard deviation above the mean. A z-score of -2.0 means it's two standard deviations below the mean.

Q3: What is considered an unusual z-score?
A: Typically, z-scores beyond ±2.0 are considered unusual, and beyond ±3.0 are considered highly unusual in a normal distribution.

Q4: Can z-scores be used for non-normal distributions?
A: While z-scores can be calculated for any distribution, their interpretation is most meaningful for normal distributions.

Q5: How are z-scores related to probability?
A: In a standard normal distribution, z-scores correspond to specific probabilities or percentiles that can be found using z-tables or statistical software.