Normal Distribution Explorer
Interactive exploration of normal distribution properties and probabilities
Explore the bell curve
Drag the sliders to see how the mean shifts the curve and how the standard deviation makes it tighter or wider.
Quick presets
0.00
-10.00200.00
1.00
0.1030.00
Try this
Set the mean to 100 and the standard deviation to 15. The curve now models classic IQ test scores.
Predict what will happen
If you double the standard deviation, what happens to the peak of the curve?
The total area under the curve always stays equal to 1.
Why it works
Increasing σ spreads probability over a wider range of x, so the peak height must decrease to keep the total area equal to 1.
Distribution Statistics
Mean
μ = 0.00
Standard Deviation
σ = 1.00
Variance
σ² = 1.00
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)