Normal density is one of 1 distributions formula in the statistics section of this library, and it is used at ap · university level.
Why normal density works
The exponent is minus half a squared z-score, so height drops quickly as you move away from the centre, and the same distance in either direction gives the same height. The fraction in front is the exact constant needed to make the total area under the curve come out to 1.
What each symbol means
$\mu$ is mean and $\sigma$ standard deviation.
Normal density: when it holds
$\sigma>0$; probabilities are areas under the density, not density heights alone.
When it stops applying
The curve stretches over every real number, so it assigns some probability to impossible values whenever the quantity is bounded. Modelling a waiting time with a mean of 5 minutes and a spread of 4 puts about 10.6 percent of the probability below zero minutes, which no data can ever fill.
Normal density: a worked example
The standard normal uses $\mu=0$ and $\sigma=1$.
The mistake to avoid
What people do: Reading the height of the curve as a probability.
Why it goes wrong: For the standard normal, the peak height is about 0.3989, and that is not a 39.89 percent chance of anything. Continuous variables assign probability to intervals, and the chance of any single exact value is zero.
Do this instead: Find the area between two values instead, using a table, a calculator, or software, and reserve the density value for drawing the curve.
Normal density: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $\mu$ is mean and $\sigma$ standard deviation.
- Check the conditions before substituting. $\sigma>0$; probabilities are areas under the density, not density heights alone.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most statistics slips.
Where this formula fits
- Subject
- Statistics formulas — 21 entries in this library
- Topic
- Distributions
- Level
- AP · University
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where normal density comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Probability Foundations — the lesson behind this formula: model outcomes, complements, and conditional information.
- Statistics Calculator — check your substitution and the value it produces.
- Study statistics — the subject guide that explains the ideas these formulas compress.
- Statistics Practice — questions that make you retrieve the formula instead of recognising it.
- All 21 statistics formulas — the full grouped reference, or the complete formula library.
Questions about normal density
Why does pi appear in a formula about data?
It comes from the integral that makes the total area equal 1. The constant has no interpretation on its own; it is the scaling that turns the bell shape into a genuine probability distribution.
What percentage lies within one, two and three standard deviations?
About 68.27, 95.45 and 99.73 percent respectively. The rough version taught as 68, 95, 99.7 is accurate enough for reading a graph but not for exact work.
Will I ever plug numbers into this formula directly?
Almost never. In practice you standardise and look up areas, and this expression is shown mainly to explain where the shape and the tables come from.
Is the area under the whole curve really exactly 1?
Yes, that is what the constant in front guarantees. The tails never actually reach the axis, but the area they add beyond four standard deviations is smaller than one part in ten thousand.