Random number generator

How the numbers are drawn

For whole numbers from a to b, there are b − a + 1 possible values, and each draw picks one with equal probability:

P(any one value) = 1 ÷ (b − a + 1) value = a + ⌊u × (b − a + 1)⌋, u uniform on [0, 1)

Decimals work the same way on a finer grid. With d decimal places the step is 10⁻ᵈ, and the generator picks a random whole number of steps above the lowest value. Because every value on the grid is equally likely, the ends of the range are as likely as the middle. The random u comes from 53 bits of crypto.getRandomValues, so even very wide ranges show no bias toward particular values.

A worked example

The default draws 10 whole numbers from 1 to 100 with repeats allowed. There are 100 possible values, so each draw has a 1 in 100 chance of any given number. The results will look lumpy: a few close together, some gaps. That is normal.

With repeats allowed, how likely is at least one repeated number among the 10? It is the birthday problem with 100 “days”:

P(repeat) = 1 − (100 × 99 × … × 91) ÷ 100¹⁰ = 0.372

So more than a third of the time a list of 10 contains a duplicate. If that is not what you want, choose “No repeats”, which draws the way you would pull numbered tickets out of a hat.

The mean of the 10 numbers is shown against the expected mean, (1 + 100) ÷ 2 = 50.5. The standard deviation of one draw is √((100² − 1) ÷ 12) = 28.9, so the mean of 10 draws typically lands within 28.9 ÷ √10 = 9.1 of 50.5.

Using the numbers

For a random sample from a numbered list, number the items 1 to N, draw without repeats, and take those items. For random assignment to two groups, draw N unique numbers from 1 to N; the first half go to group A. Sorting the output makes it easier to check against a list, and does not change which numbers were drawn.

Repeats or no repeats?

Allowing repeats models independent draws with replacement, like rolling a die or spinning a wheel several times. Forbidding repeats models draws without replacement, like dealing cards or picking raffle tickets. The choice matters for statistics: a simple random sample from a population is drawn without replacement, while a bootstrap resample is drawn with it.

Common mistakes

• Asking for more unique numbers than the range holds. Ten unique whole numbers from 1 to 5 is impossible, and the generator will say so.
• Reading patterns into a short list. Clusters and gaps are what randomness looks like.
• Rounding random decimals yourself. That can make the end values half as likely as the others; set the decimal places here instead.