Normal Distribution Calculator
Calculate probabilities, z-scores, and percentiles for normal distributions
Bell Curve Visualization
Enter values to see the normal distribution curve
Understanding Normal Distribution
The normal distribution, also known as the Gaussian distribution or bell curve, is one of the most important probability distributions in statistics. It appears naturally in many real-world phenomena.
Key Concepts
- Mean (μ): The center of the distribution, representing the average value.
- Standard Deviation (σ): Measures how spread out values are from the mean. A larger σ means more spread.
- Z-Score: The number of standard deviations a value is from the mean. Formula: z = (x - μ) / σ
- Percentile: The percentage of values that fall below a given point.
The 68-95-99.7 Rule
- ~68% of values fall within 1 standard deviation of the mean (μ ± σ)
- ~95% of values fall within 2 standard deviations (μ ± 2σ)
- ~99.7% of values fall within 3 standard deviations (μ ± 3σ)
Real-World Examples
IQ Scores
Mean = 100, SD = 15
IQ scores are designed to follow a normal distribution, making it easy to compare individual scores.
Heights
Adult heights in populations
Most people cluster around average height, with fewer very tall or very short individuals.
Test Scores
Standardized exams (SAT, GRE)
Large groups taking tests typically produce normally distributed scores.
Pro Tip: Try experimenting with different values! Set mean=100, SD=15, and try x-values of 85, 100, and 115 to see how they correspond to different percentiles.