Interactive Calculator
Calculate normal distribution probabilities, density, and z-scores quickly.
Tip: For a standard normal distribution, use μ = 0 and σ = 1.
What Is a Normal Distribution?
The normal distribution (also called a Gaussian distribution) is one of the most important ideas in statistics. It describes data that clusters around a central value, with fewer observations as you move away from the center. Many real-world measurements approximately follow this shape: test scores, heights, measurement errors, blood pressure, and more.
Key characteristics
- Symmetric bell shape: Left and right sides mirror each other.
- Mean (μ): The center of the distribution.
- Standard deviation (σ): The spread or variability around the mean.
- Total area under the curve = 1: This represents 100% probability.
How to Use This Normal Distribution Calculator
This calculator lets you work with the most common normal distribution questions without manually using a z-table.
1) Set your distribution
Enter the mean (μ) and standard deviation (σ). For standard normal calculations, use μ = 0 and σ = 1.
2) Pick a calculation type
- P(X ≤ x): Probability of observing a value at or below x.
- P(X ≥ x): Probability of observing a value at or above x.
- P(a ≤ X ≤ b): Probability between two bounds.
- f(x): The probability density (height of the curve) at x.
- Z-score: How many standard deviations x is from the mean.
3) Enter x (or a and b), then calculate
Click Calculate to instantly see the result and interpretation.
Formulas Behind the Calculator
Z-score
z = (x - μ) / σ
Probability density function (PDF)
f(x) = (1 / (σ√(2π))) × exp(-0.5 × ((x - μ) / σ)2)
Cumulative distribution function (CDF)
P(X ≤ x) is computed using a numerical approximation of the error function (erf), which is standard for normal CDF calculations.
Practical Examples
Exam scores
Suppose scores are normally distributed with mean 75 and standard deviation 10. To find the chance a student scores at least 90, set μ=75, σ=10, pick P(X ≥ x), and enter x=90.
Quality control
If part diameters are normally distributed, use P(a ≤ X ≤ b) to estimate what percentage of products meet tolerance limits.
Standardization
Need to compare values from different scales? Use the z-score mode to normalize observations and compare them consistently.
68-95-99.7 Rule (Quick Reference)
- About 68% of values fall within μ ± 1σ
- About 95% of values fall within μ ± 2σ
- About 99.7% of values fall within μ ± 3σ
Common Mistakes to Avoid
- Using a non-positive standard deviation (must be > 0).
- Confusing PDF value with direct probability.
- Swapping upper and lower bounds when computing interval probability.
- Rounding too early when high precision is needed.
Final Thoughts
A normal distribution calculator is a simple but powerful tool for data analysis, forecasting, hypothesis testing, and decision-making. Whether you're a student, analyst, or researcher, fast access to CDF, PDF, and z-score calculations can save time and reduce errors.