White noise is created by generating a random signal made up of serially uncorrelated random variables that has the same intensity at many different frequencies. To be precise, the variable is not entirely random as the bandwidth is bounded by the actual physical mechanism that is used to generate the noise. The range of audible sound frequencies stretches from 20 to 20,000 hertz. A randomly generated sample of white noise in this bounded area sounds like a hissing noise.
We tend to hear white noise as having more high-frequency content than low, even though the actual generation process involves randomly generating noise at all frequencies. This is because each octave has twice as many frequencies as the one below it. From 100 hertz to 200 hertz there are 100 discrete frequencies. The next octave, from 200 hertz to 400 hertz, contains twice as many frequencies as that. The octave after that contains 400 discrete frequencies. The octave after that contains 800 frequencies and so on. In order to combat this issue, you can instead generate ‘pink noise’, in which some of the higher frequencies are damped down in order to generate a noise that seems more consistent across the full range of frequencies.
It is important to understand how the random variables that define the frequencies on which noise is generated are derived. It is sometimes incorrectly stated that white noise is the same thing as Gaussian noise. Gaussian noise is made up of a random signal, in much the same way as white noise is. However, Gaussian noise follows the normal statistical distribution (also known as Gaussian distribution or the bell curve), as suggested by the name. Gaussian noise sounds similar to white noise but the two are not necessarily identical. The random generation of the frequencies in standard white noise need not follow the normal distribution.
Any given random vector, meaning a process that produces vectors of real numbers following a process that is not fully determinate, can be described as white noise if it has a probability distribution with no mean, a finite variance and statistically uncorrelated components. For the components to be statistically uncorrelated it is necessary that they have a covariance of zero. (Covariance is a statistical measure of how correlated the variance of two variables are, in other words how the variance of the variables varies. The more correlated the variance of the variables, the more positive the covariance value will be. Two variables with negative covariance are variables that tend to vary in opposite directions.) Neither a positive nor negative covariance should be shown by the variables that make up true white noise. The variance of the variables should not vary, and they should have a covariance of zero.
If you want Gaussian white noise, rather than other types of white noise, you would also need each of the variables to have a normal distribution, as well as a zero mean. You would also need each of the variables to have the exact same variance. Note that it is also possible to generate white noise with different types of statistical distribution. For instance, you can use a Poisson distribution or a Cauchy distribution.
Some people use white noise in order to help them sleep. Science suggests that it is not actual noises in the night that wake you up, but changes in frequency. Since white noise consist of continuous noise at every frequency, an additional noise on top of white noise will not consist of a particular change in frequency. So a consistent background of white noise will damp down any extraneous noises and allow for more consistent levels of sleep, with less variance and variability in the consistency.