View source for Range (of variation of a sample)

The difference
$$
w_n=x_\mathrm{max}-x_\mathrm{min}
$$
between the largest $x_\mathrm{max}=x_n$ and smallest $x_\mathrm{min}=x_1$ values in the ordered sample
$$
(x_1,\ldots,x_n),\quad x_1\leq\ldots\leq x_n\,,
$$
obtained by taking $n$ independent measurements of the same random variable $X$. Let $F(x) = \mathbf{P}\{X \le x\}$ be the distribution function of the random variable $X$. Then the probability distribution for the range is
$$
\mathbf{P}\{w_n \le t\} = n \int_{-\infty}^\infty (F(x+t)-F(x))^{n-1} dF(x)\,,\ \ \ 0 \le t \le \infty \ .
$$


====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  B.L. van der Waerden,   "Mathematische Statistik" , Springer  (1957)</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top">  L.N. Bol'shev,   N.V. Smirnov,   "Tables of mathematical statistics" , ''Libr. math. tables'' , '''46''' , Nauka  (1983)  (In Russian)  (Processed by L.S. Bark and E.S. Kedrova)</TD></TR>
</table>



====Comments====
The range of variation of a sample is also called the sample range.

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  D.B. Owen,   "Handbook of statistical tables" , Addison-Wesley  (1962)</TD></TR>
</table>

{{TEX|done}}