Calculating the expected range of normal samples

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The previous post looked at the expected IQ range in a jury of 12. This post will look more generally at computing the expected range of n samples from a N(0, 1) random variable. This will give the expected range in units of σ, i.e. multiply the results by σ if your σ isn’t 1.

As mentioned in the previous post, the expected range is given by

d_n = 2n int_{-infty}^infty Phi(x)^{n-1} , x,phi(x) , dx

where φ and Φ are the PDF and CDF of a standard normal. The integral can be calculated in closed form for n ≤ 5, but in general it requires numerical integration [1].

The following Python code can compute dn.

from scipy.stats import norm
from scipy.integrate import quad
import numpy as np

def d(n):
    integrand = lambda x: x*norm.pdf(x)*norm.cdf(x)**(n-1)
    res, info = quad(integrand, -np.inf, np.inf)
    return 2*n*res

For large n we have the asymptotic approximation

d_n = 2 Phi^{-1}left( frac{n ,–, 0.375}{ n + 0.25} right)

which we could implement in Python by

def approx(n):
    return 2*norm.ppf((n - 0.375)/(n + 0.25))

For very large n the asymptotic expression may be more accurate than the integral due to numerical integration error.

Here are a few example values.

|-----+-------|
|   n |   d_n |
|-----+-------|
|   2 | 1.128 |
|   3 | 1.693 |
|   5 | 2.326 |
|  10 | 3.078 |
|  12 | 3.258 |
|  23 | 3.858 |
|  50 | 4.498 |
| 100 | 5.015 |
|-----+-------|

[1] Order Statistics by H. A. David. John Wiley & Sons. 1970.

The post Calculating the expected range of normal samples first appeared on John D. Cook.

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