Normal Distribution Explorer
Interactively explore the normal distribution by adjusting mean (μ) and standard deviation (σ) with sliders. Visualize the bell curve, calculate probabilities for shaded regions, and understand the empirical rule in real-time.
Interactive Controls
-10010
0.12.55.0
Distribution Statistics
Mean
μ = 0.00
Standard Deviation
σ = 1.00
Variance
σ² = 1.00
Interactive Bell Curve
Empirical Rule (68-95-99.7)
68% of data falls within ±1σ
[-1.00, 1.00]
95% of data falls within ±2σ
[-2.00, 2.00]
99.7% of data falls within ±3σ
[-3.00, 3.00]
Theory & Formula
What is the Normal Distribution?
The normal distribution, also known as the Gaussian distribution or bell curve, is a continuous probability distribution that is symmetric about the mean. It is one of the most important distributions in statistics.
Probability Density Function
The normal distribution is defined by its probability density function (PDF):
\(f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}\)Where: μ = mean (center), σ = standard deviation (spread)
Key Properties
- Symmetric about the mean μ
- Mean = median = mode
- Total area under the curve equals 1
- Asymptotic to x-axis (tails never touch zero)