In probability theory, a normalizing constant or normalizing factor is used to reduce any probability function to a probability density function with total probability of one.

For example, a Gaussian function can be normalized into a probability density function, which gives the standard normal distribution. In Bayes' theorem, a normalizing constant is used to ensure that the sum of all possible hypotheses equals 1. Other uses of normalizing constants include making the value of a Legendre polynomial at 1 and in the orthogonality of orthonormal functions.

A similar concept has been used in areas other than probability, such as for polynomials.

In probability theory, a normalizing constant is a constant by which an everywhere non-negative function must be multiplied so the area under its graph is 1, e.g., to make it a probability density function or a probability mass function.

If we start from the simple Gaussian function $${\displaystyle p(x)=e^{-x^{2}/2},\quad x\in (-\infty ,\infty )}$$ we have the corresponding Gaussian integral $${\displaystyle \int _{-\infty }^{\infty }p(x)\,dx=\int _{-\infty }^{\infty }e^{-x^{2}/2}\,dx={\sqrt {2\pi \,}},}$$

Now if we use the latter's reciprocal value as a normalizing constant for the former, defining a function ${\displaystyle \varphi (x)}$ as $${\displaystyle \varphi (x)={\frac {1}{\sqrt {2\pi \,}}}p(x)={\frac {1}{\sqrt {2\pi \,}}}e^{-x^{2}/2}}$$ so that its integral is unit $${\displaystyle \int _{-\infty }^{\infty }\varphi (x)\,dx=\int _{-\infty }^{\infty }{\frac {1}{\sqrt {2\pi \,}}}e^{-x^{2}/2}\,dx=1}$$ then the function ${\displaystyle \varphi (x)}$ is a probability density function. This is the density of the standard normal distribution. (Standard, in this case, means the expected value is 0 and the variance is 1.)

And constant ${\textstyle {\frac {1}{\sqrt {2\pi }}}}$ is the normalizing constant of function ${\displaystyle p(x)}$.

Similarly, $${\displaystyle \sum _{n=0}^{\infty }{\frac {\lambda ^{n}}{n!}}=e^{\lambda },}$$ and consequently $${\displaystyle f(n)={\frac {\lambda ^{n}e^{-\lambda }}{n!}}}$$ is a probability mass function on the set of all nonnegative integers. This is the probability mass function of the Poisson distribution with expected value λ.