Interactive Normal Distribution Calculator
Enter your parameters to compute z-score, PDF, CDF, tail probabilities, and area between two values.
Tip: For a standard normal distribution, use μ = 0 and σ = 1.
Distribution Curve (shaded area = P(lower ≤ X ≤ upper))
What this normal distribution curve calculator does
This tool helps you work with a Gaussian (normal) distribution quickly. It computes key values used in statistics, data science, quality control, psychometrics, and finance. Instead of flipping through a z-table, you can enter your numbers and instantly get probabilities and curve-based interpretation.
- Z-score: How many standard deviations a value is from the mean.
- PDF: Relative density at a specific x-value (the curve height).
- CDF: Probability that a random variable is less than or equal to x.
- Interval probability: Area under the curve between a lower and upper bound.
How to use it
1) Enter distribution parameters
Set the mean (μ) and standard deviation (σ). The standard deviation must be positive. If you are working with z-scores directly, keep μ = 0 and σ = 1.
2) Enter the point and interval
Provide an x-value for point probabilities and cumulative probability at that point. Then set lower and upper limits to find the probability in a range.
3) Click calculate
The result panel shows all main probabilities, and the chart visualizes your interval as the shaded region under the bell curve.
Core formulas behind the calculator
For a normal random variable with mean μ and standard deviation σ, the probability density function is:
f(x) = (1 / (σ√(2π))) × exp(-0.5 × ((x - μ)/σ)2)
The z-score transformation is:
z = (x - μ) / σ
The cumulative distribution function (CDF) uses the error function approximation to estimate: P(X ≤ x).
Worked example
Suppose test scores are normally distributed with mean 70 and standard deviation 10. You want to know:
- Probability a score is less than or equal to 85
- Probability a score lies between 60 and 80
Enter μ = 70, σ = 10, x = 85, lower = 60, upper = 80. The calculator gives the cumulative probability at 85 and the interval probability for 60 to 80. This is exactly what z-tables are used for, but faster and less error-prone.
Interpretation tips
- The PDF value is not a direct probability by itself; area under the curve gives probability.
- If your interval is very wide (for example, μ ± 3σ), probability approaches 1.
- For continuous distributions, exact single-point probability P(X = x) is 0.
- Always check that your data are approximately normal before relying on normal-model probabilities.
Quick rule of thumb: 68-95-99.7 rule
In any normal distribution:
- About 68% of values fall within μ ± 1σ
- About 95% fall within μ ± 2σ
- About 99.7% fall within μ ± 3σ
The result panel also computes these for your selected μ and σ so you can compare your interval against common benchmarks.