🎓 Statistics & Data • Probability & Distributions
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Normal Distribution Calculator (Gaussian Bell Curve)
Evaluates Gaussian bell curve cumulative probabilities P(X ≤ x), Z-scores, and upper tail probabilities.
Gaussian Normal Distribution
Normal Distribution & Z-Table Area
Calculate cumulative area under the standard normal curve for single and two-tailed probabilities with an interactive bell curve.
Presets:
Cumulative Probability P(X ≤ 115)
84.13%
Decimal Area = 0.8413
Z-Score+1 σ
P(X ≤ 115) (Left)84.13%
P(X > 115) (Right)15.87%
Gaussian Normal Bell Curve & Shaded Probability Density
Standardization & Probability Derivation
1. Standardized Score: z = (x - μ) / σ = (115 - 100) / 15 = 1
2. Standard Normal CDF: Φ(1) = 0.8413 (84.13%)
3. Complement Upper Tail: 1 - Φ(1) = 0.1587 (15.87%)
What is the Normal Distribution?
The normal distribution (Gaussian distribution) is a symmetric continuous probability distribution characterized by its bell-shaped probability density curve.
Formula & Step-by-Step Calculation
f(x) = (1 / [σ√(2π)]) · e^[ -(x - μ)² / (2σ²) ]
Gaussian probability density function with mean μ and standard deviation σ.
Worked Step-by-Step Examples
Example 1
Test with mean 100, SD 15. What % of students score ≤ 115?
Solution: P(X ≤ 115) = 84.13% (z = +1.00 σ)
• z = (115 - 100) / 15 = 1.00; Normal CDF(1.00) ≈ 0.8413
Common Real-World & Academic Use Cases
- ✓ Quality control Six Sigma processes
- ✓ IQ and cognitive benchmark testing
- ✓ Biological biometric measurements (human height, weight)
How to Use the Normal Distribution Calculator (Gaussian Bell Curve)
1
Input Mean & Std Dev
Enter μ and σ.
2
Input Observation x
Enter target value.
3
Inspect Cumulative Probability
Review area under curve P(X ≤ x).
Frequently Asked Questions
Q: What is the empirical rule (68-95-99.7)?
68% of data falls within 1σ, 95% within 2σ, and 99.7% within 3σ of the mean.