Standard Normal Distribution Probability Calculator
Standard Normal Distribution Probability Calculator helps determine the probability that a statistic is below, above, or between standard normal Z-scores. Essential for hypothesis testing, quality control, and risk assessment in statistics.
Formula
P(Z ≤ z) = Φ(z) = (1/√(2π)) ∫-∞z e-t²/2dt
How to Use
1. Enter Z-score
2. Click Calculate
3. Get left-tail probability
4. Use Clear to reset
Supports values between -3.99 and 3.99
FAQs
1. What does the result represent?
The probability value shows the likelihood of data points occurring below the entered Z-score in a standard normal distribution (mean=0, SD=1). For example, Z=1.96 gives P=0.9750.
2. Can I use negative Z-scores?
Yes, negative Z-scores calculate left-tail probabilities. The calculator automatically handles values below the mean. For Z=-1, P=0.1587 indicates 15.87% probability below -1 SD.
3. How accurate are the results?
Results are precise to 4 decimal places using error function approximation. Maximum error margin ±0.0002 compared to standard normal tables.
4. When is this calculator useful?
Ideal for statistical analysis, Six Sigma projects, academic research, and any scenario requiring conversion between Z-scores and probabilities.
5. Can I calculate right-tail probabilities?
Subtract left-tail probability from 1. For Z=1.65: Right-tail P = 1 - 0.9505 = 0.0495 (4.95%).
6. What about non-standard distributions?
Convert to Z-scores first using: Z = (X - μ)/σ. This calculator only handles standardized values.
7. How are probabilities calculated?
Using the error function approximation: Φ(z) ≈ ½[1 + erf(z/√2)]. This method provides quick computational results without full integration.
8. Why am I getting an error?
Ensure Z-scores are between -3.99 and 3.99. Values beyond this range have probabilities approaching 0/1 (0.00003 for Z=-4, 0.99997 for Z=4).
9. Is this calculator suitable for education?
Yes, it's perfect for verifying textbook problems and learning normal distribution concepts. Always understand the theory behind the calculations.
10. Can I find Z-scores from probabilities?
This version calculates probabilities from Z-scores. Inverse calculations require separate functionality (Z = Φ⁻¹(p)).