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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.

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)