Normal Distribution PDF
Bell-curve probability density function.
Formula
Variables
Example
Symmetric about the mean \mu
How it works
This formula gives the height of the normal distribution at a particular value x, so it is a density rather than a direct probability. The curve is centered at the mean \(\mu\), spreads out according to the standard deviation \(\sigma\), and the exponential term makes values farther from the mean drop off rapidly. The constant \(1/(\sigma\sqrt{2\pi})\) is chosen so that the total area under the curve is exactly 1, which is what makes it a valid probability model.
When to use it
Use this when modeling continuous data that cluster around an average with most values near the center and fewer extreme values, such as measurement error, test scores, or manufacturing tolerances. It is also the starting point for calculating probabilities over intervals in any normal-distribution problem.
Worked example, step by step
Find the density at \(x=70\) for a normal distribution with \(\mu=65\) and \(\sigma=4\). Step 1: Write the formula. \(f(x)=\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}\) Step 2: Substitute the values. \(f(70)=\frac{1}{4\sqrt{2\pi}}e^{-\frac{(70-65)^2}{2(4^2)}}\) Step 3: Simplify the exponent. \((70-65)^2=25\) \(2(4^2)=2(16)=32\) So, \(f(70)=\frac{1}{4\sqrt{2\pi}}e^{-25/32}\) Step 4: Evaluate the constants. \(4\sqrt{2\pi} \approx 4(2.5066)=10.0264\) \(e^{-25/32}=e^{-0.78125} \approx 0.4580\) Step 5: Multiply. \(f(70) \approx \frac{0.4580}{10.0264} \approx 0.0457\) Final answer: \(f(70) \approx 0.0457\) This means the height of the normal curve at \(x=70\) is about 0.0457, not that the probability of exactly 70 is 0.0457.
Common mistakes to avoid
- Treating f(x) as a probability instead of a density
- Forgetting to square the standard deviation in the denominator of the exponent
- Using the formula to find probabilities over an interval without integrating or using a CDF
Did You Know?
The normal distribution arises naturally via the Central Limit Theorem.
Share this formula
More in Probability & Combinatorics
View allPermutations
BasicNumber of ordered arrangements of r items from n.
Combinations
BasicNumber of unordered selections of r items from n.
Factorial
BasicProduct of all positive integers up to n.
Permutations with Repetition
BasicOrdered arrangements of r items from n when repetition is allowed.
Permutations of a Multiset
IntermediateArrangements of n items where some are identical.
Classical Probability
BasicProbability of an event with equally likely outcomes.