Z-Score Calculator
Calculate Z-scores (standard scores) to determine how many standard deviations a value is from the mean. Essential for statistics, probability, and data analysis.
Calculator Input
Results
Normal Distribution Visualization
Step-by-Step Calculation
How to Use
Basic Steps
- Select the calculation type you need
- Enter the required values (raw score, mean, standard deviation)
- Check "Sample Mean" if working with sample data
- View the calculated Z-score and interpretation
- Use the visualization to understand the position
Input Guidelines
- Standard deviation must be positive
- Use sample mean option for sample statistics
- Z-scores typically range from -3 to +3
- Try quick examples for common scenarios
Understanding Z-Scores
What is a Z-Score?
A Z-score (standard score) measures how many standard deviations a value is from the mean of a distribution.
Formula:
For sample mean: Z = (X̄ - μ) / (σ/√n)
Where X is the raw score, μ is the mean, and σ is the standard deviation.
Interpretation
Applications
Standardization
Compare values from different distributions by converting them to a common scale.
Outlier Detection
Identify unusual values that fall far from the mean (typically |Z| > 2 or 3).
Probability
Calculate probabilities and percentiles using the standard normal distribution.
Example Calculations
Example 1: IQ Score
Problem: What's the Z-score for an IQ of 130?
Given: X = 130, μ = 100, σ = 15
Interpretation: The IQ of 130 is 2 standard deviations above average, placing it in the 97.7th percentile.
Example 2: SAT Score
Problem: What raw score corresponds to Z = -1.5?
Given: Z = -1.5, μ = 1000, σ = 200
Interpretation: A Z-score of -1.5 corresponds to an SAT score of 700, which is below average.
Frequently Asked Questions
What does a negative Z-score mean?
A negative Z-score indicates that the value is below the mean. For example, Z = -1.5 means the value is 1.5 standard deviations below the average.
When should I use the sample mean option?
Use the sample mean option when you're working with a sample mean rather than individual values. This adjusts the standard error by dividing the population standard deviation by the square root of the sample size.
What's considered an extreme Z-score?
Generally, Z-scores beyond ±2 are considered unusual (occurring in about 5% of cases), and Z-scores beyond ±3 are considered very rare (occurring in about 0.3% of cases).