Normal Distribution
Probability density of the bell curve.
Formula
Variables
Example
About 68% of a normal distribution lies within one σ of the mean, 95% within two.
How it works
This formula gives the height of the normal distribution curve at a particular value x, so it is a probability density rather than a probability by itself. The mean μ sets the centre of the bell curve, while the standard deviation σ controls how spread out the data are; the exponential term makes values far from the mean less likely in a smooth, symmetric way. The constant 1/(σ0 √2 a0 π) is chosen so the total area under the curve is exactly 1, which is what any valid probability distribution must satisfy.
When to use it
Use this when modelling a continuous variable that clusters around an average with symmetric variation, such as measurement error, heights, exam scores, or manufacturing tolerances. It is also the starting point for computing probabilities via z-scores and for many inferential methods that assume normality.
Worked example, step by step
Suppose exam scores are normally distributed with mean μ = 70 and standard deviation σ = 8. Find the density at x = 78. Substitute into the formula: f(78) = 1/(8\u221a(2\u03c0)) \u00d7 e^{-((78-70)^2)/(2\u00d78^2)} Compute the exponent: (78-70)^2 = 8^2 = 64, and 2\u00d78^2 = 2\u00d764 = 128, so the exponent is -64/128 = -0.5 So f(78) = 1/(8\u221a(2\u03c0)) \u00d7 e^{-0.5} Approximate the constants: \u221a(2\u03c0) \u2248 2.5066, so 8\u221a(2\u03c0) \u2248 20.053 Then 1/20.053 \u2248 0.0499 and e^{-0.5} \u2248 0.6065 Multiply: f(78) \u2248 0.0499 \u00d7 0.6065 \u2248 0.0303 Final answer: the density at 78 is about 0.030 per score point.
Common mistakes to avoid
- Treating f(x) as a probability instead of a density
- Forgetting that \u03c3 must be squared inside the exponent
- Using the formula directly to find interval probabilities without integrating or standardising first
Did You Know?
The bell curve is so universal that Francis Galton called it a "wild form of cosmic order".
Share this formula
More in Statistics & Probability
View allArithmetic Mean
BasicThe average of a data set.
Median
BasicThe middle value separating the higher and lower halves.
Variance
IntermediateAverage squared deviation from the mean.
Standard Deviation
IntermediateSquare root of the variance; typical spread.
Z-Score
IntermediateNumber of standard deviations a value is from the mean.
Bayes' Theorem
IntermediateUpdates a probability given new evidence.