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I am new to both stats and scientific Python, so apologies if this question breaches any guidelines.

I want to know what x = np.random.randn(500) actually does, in terms I can understand. Why exactly does plotting x give values in the range shown in this image:

enter image description here

?

It looks like the values range roughly from -3 to 3, yet I can find no mention of the range in documentation for numpy.random.randn. Therefore I'm assuming this property must somehow be intrinsic to a standard normal distribution, but I don't have enough knowledge of this entity to understand why.

Is it possible, although wildly improbable, that values with a much greater magnitude could be generated? Is it a feature of the specific implementation that the values range from -3 to +3 ish, rather than, say, -10 to +10 ish?

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    $\begingroup$ randn simply generates pseudo-random numbers from the standard normal distribution (en.wikipedia.org/wiki/…). The values are unbound, but very large and very small values are highly improbable. $\endgroup$
    – Igor F.
    Commented Jan 25, 2021 at 15:37

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Correct, it is possible but wildly improbable. Theoretically, any value is possible but 99.7% of standard normal random variables are between -3 and +3. Only 1 in a billion will be larger than 6 (or less than -6).

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  • $\begingroup$ I'm still a bit confused though. Why is the range not -30 to +30 or -300 to +300? How does the standard normal distribution "know" what range comprises 1,2 or 3 standard deviations, when no actual values are specified? $\endgroup$ Commented Jan 27, 2021 at 9:53
  • $\begingroup$ That's what a distribution does. It specifies precisely the probability that X can be less than -3, greater than 4, etc. This particular distribution just has the property that almost all values are between -3 and 3. There are other distributions that can have large values. For example, normal with mean 0 and variance 100. Now, most of the values will be between -30 and 30 and you will often see numbers like 10.5583, -21.444421, etc. $\endgroup$
    – John L
    Commented Jan 28, 2021 at 0:39
  • $\begingroup$ So there is an implicit variance of 10 in the values produced by the function? $\endgroup$ Commented Jan 28, 2021 at 20:17
  • $\begingroup$ No: by definition, the variance is $1.$ $\endgroup$
    – whuber
    Commented May 18, 2022 at 13:53

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