Z-Score Calculator
Calculate z-scores to standardize your data and determine how many standard deviations each value is from the mean.
Understanding Z-Scores
Z-scores (also called standard scores) measure how many standard deviations a data point is from the mean of a dataset. They are useful for standardizing data and identifying outliers.
What is a Z-Score?
A z-score represents the number of standard deviations a data point is away from the mean. Positive z-scores indicate values above the mean, while negative z-scores indicate values below the mean.
Formula: Z = (X - μ) / σ
Where:
- Z = Z-score
- X = Individual data point value
- μ (mu) = Mean of the population
- σ (sigma) = Standard deviation of the population
Interpreting Z-Scores
Z-scores allow you to understand where a value falls within a distribution:
- Z = 0: The value equals the mean
- Z = 1: The value is 1 standard deviation above the mean
- Z = -1: The value is 1 standard deviation below the mean
- Z = 2: The value is 2 standard deviations above the mean
- Z = -2: The value is 2 standard deviations below the mean
In a normal distribution, approximately 68% of values fall within 1 standard deviation of the mean (|Z| < 1), 95% fall within 2 standard deviations (|Z| < 2), and 99.7% fall within 3 standard deviations (|Z| < 3).
Population vs. Sample Standard Deviation
When calculating z-scores, you can use either population or sample standard deviation:
Population Standard Deviation (σ):
Use when your data represents the entire population.
Formula uses N as the divisor.
Sample Standard Deviation (s):
Use when your data is a sample from a larger population.
Formula uses N-1 as the divisor (Bessel's correction).
Uses of Z-Scores
- Standardizing Data: Convert different datasets to a common scale for comparison
- Identifying Outliers: Points with |Z| > 3 are often considered outliers
- Calculating Probabilities: In normal distributions, z-scores can be used to find probabilities
- Creating Percentiles: Convert z-scores to percentiles using standard normal tables
- Comparing Across Distributions: Compare values from different datasets with different units
When to Use Z-Scores
- When comparing values from different datasets
- When identifying unusual or extreme values in your data
- When determining the relative position of a value within a distribution
- When calculating probabilities in a normal distribution
- When creating standardized test scores
How to Use This Calculator
- Enter your dataset in the text box, separating values with commas, spaces, or new lines
- Select your preferred decimal precision
- Choose whether to calculate using population (N) or sample (N-1) standard deviation
- Optionally, enter a specific value to calculate its z-score
- Click "Calculate" to see the results
- Analyze the z-scores to understand how each value relates to the mean and standard deviation