Normal Distribution Calculator
Calculate probabilities, percentiles, and areas under the normal distribution curve (bell curve)
Normal Distribution Parameters
Center of the distribution
Spread of the distribution (must be positive)
The value for which you want to calculate probabilities
Probability Results
Empirical Rule (68-95-99.7 Rule)
Example: Height Distribution
Adult Male Height in US
Mean height: 175.7 cm (μ = 175.7)
Standard deviation: 10 cm (σ = 10)
Question: What's the probability of being taller than 185 cm?
Solution
1. Calculate Z-score: Z = (185 - 175.7) / 10 = 0.93
2. Find P(X > 185) = 1 - P(X < 185) = 1 - 0.8238 = 0.1762
Answer: 17.62% probability of being taller than 185 cm
Distribution Properties
Bell-shaped and symmetric about the mean
Mean = Median = Mode
Total area under curve = 1
68% within 1 standard deviation
95% within 2 standard deviations
99.7% within 3 standard deviations
Key Formulas
Probability Density Function (PDF)
Z-Score
Standard Normal
Understanding Normal Distribution
What is Normal Distribution?
The normal distribution, also known as the Gaussian distribution, is a continuous probability distribution that is symmetric and bell-shaped. It's one of the most important distributions in statistics because many natural phenomena follow this pattern.
Key Characteristics
- •Symmetric about the mean
- •Mean, median, and mode are equal
- •Asymptotic to x-axis
- •Total area under curve equals 1
Parameters
Mean (μ)
Determines the center of the distribution
Standard Deviation (σ)
Controls the spread or width of the distribution
Real-World Examples
- • Heights and weights of people
- • Test scores and IQ measurements
- • Blood pressure readings
- • Measurement errors in experiments
- • Stock price changes