Z-Score Probability Calculator
A Z-Score Probability Calculator helps determine the probability of a value occurring in a normal distribution. It's essential for statistics, quality control, and research analysis. By converting raw scores to standardized values, users can assess how unusual a result is compared to a population mean, enabling better decision-making in fields like psychology, finance, and scientific research.
Calculator
Formula
Z = (X - μ)/σ
Probability is calculated using the cumulative distribution function (CDF) of the standard normal distribution:
Φ(z) = ½[1 + erf(z/√2)]
How to Use
Enter your z-score in the input field. Click "Calculate" to see the probability of values being below (left-tailed) this z-score. The result shows both the percentage and decimal format. Use "Clear" to reset. For two-tailed probabilities, double the result for extreme values in both directions.
Derivation Process
The z-score formula standardizes normal distributions by measuring how many standard deviations a value is from the mean. The probability calculation comes from integrating the standard normal distribution's probability density function. This integration is approximated using the error function (erf), which is implemented numerically in the calculator.
FAQs
What is a z-score?
A z-score measures how many standard deviations a data point is from the population mean. It standardizes different normal distributions to allow comparison and probability calculations.
How accurate is this calculator?
The calculator uses the error function approximation with precision up to 4 decimal places, making it suitable for most statistical applications but not for extreme precision scientific calculations.
What does the probability value mean?
The result shows the cumulative probability from negative infinity to your z-score. For example, 0.975 means 97.5% of data falls below this z-score in a normal distribution.
Can I calculate right-tailed probabilities?
Yes, subtract the left-tailed probability from 1. For z=1.96, left-tailed is 0.975, so right-tailed is 0.025 (1 - 0.975).
What's the difference between one-tailed and two-tailed?
One-tailed considers probabilities in one direction, two-tailed combines both extremes. For two-tailed, calculate both sides and sum the probabilities if symmetric.
What z-score gives 95% confidence?
±1.96 z-scores mark 95% confidence interval in two-tailed tests. This corresponds to 2.5% in each tail of the normal distribution.
Can I use negative z-scores?
Yes, negative z-scores indicate values below the mean. The calculator handles negative values automatically, showing probabilities less than 50%.
Why is my probability over 1?
Probabilities always range 0-1. If you see values over 1, check your input. The calculator displays percentages (0-100%) and decimals (0-1) separately.
What's the range of valid z-scores?
While any number is accepted, z-scores beyond ±4 are extremely rare (probability >99.99% or <0.01%). Most applications use ±3.
How is this different from t-scores?
Z-scores assume known population variance and large samples, while t-scores use sample variance and small samples. Use t-distribution calculators for small sample sizes.