Z-Score Calculator
Calculate z-scores (standard scores) to determine how many standard deviations a value is from the mean. Convert between z-scores, percentiles, and probabilities with detailed interpretations.
Z-Score Calculation
Value → Z-Score
Z-Score → Percentile
Calculate the z-score for a given value using the population mean and standard deviation.
The data point you want to analyze
Average of all values in the population
Must be positive (> 0)
Results
0.00
50th percentile
Z-Score
Interpretation
Enter values to see the interpretation
0%
Area Below
0%
Area Above
0%
Percentile
Z-Score Interpretation Ranges
Very Low
z < -2
Below Average
-2 ≤ z < -1
Slightly Below
-1 ≤ z < 0
Slightly Above
0 < z ≤ 1
Above Average
1 < z ≤ 2
Very High
z > 2
How to Use the Z-Score Calculator
A z-score (standard score) measures how many standard deviations a particular value is away from the mean of a distribution. Z-scores standardize values from different normal distributions, allowing meaningful comparisons and probability calculations. Our calculator offers two modes: converting raw values to z-scores using the formula z = (X - μ) / σ, and converting z-scores to percentiles. This standardization is fundamental in statistics, quality control, academic assessment, and research, helping identify outliers and determine the relative position of any data point within its distribution.
Z-Score Formula:
z = (X - μ) / σ
Where:
X = Raw value (data point)
μ = Population mean
σ = Population standard deviation
z = (X - μ) / σ
Where:
X = Raw value (data point)
μ = Population mean
σ = Population standard deviation
🎓 SAT Score Example:
Scenario: Analyzing SAT Math performance
Population Parameters: μ = 500, σ = 100
Student Score: X = 650 points
Z-Score Calculation: z = (650 - 500) / 100 = 1.5
Interpretation: The student scored 1.5 standard deviations above average
Percentile: ~93rd percentile (better than 93% of test-takers)
Significance: This is an excellent score, well above the typical range
Z-scores are invaluable for standardizing data across different scales and distributions. The empirical rule (68-95-99.7 rule) states that approximately 68% of values fall within z = ±1, 95% within z = ±2, and 99.7% within z = ±3. Values with |z| > 2 are often considered outliers worthy of investigation. In quality control, z-scores help identify defective products; in finance, they assess investment risk; in research, they validate experimental results. Understanding z-scores enables you to make probabilistic statements about data and compare measurements from entirely different contexts on a common scale.