Difference between revisions of "Range (of variation of a sample)"
From Encyclopedia of Mathematics
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The difference | The difference | ||
| − | + | $$ | |
| − | $$w_n=x_\mathrm{max}-x_\mathrm{min}$$ | + | w_n=x_\mathrm{max}-x_\mathrm{min} |
| − | + | $$ | |
between the largest $x_\mathrm{max}=x_n$ and smallest $x_\mathrm{min}=x_1$ values in the ordered sample | between the largest $x_\mathrm{max}=x_n$ and smallest $x_\mathrm{min}=x_1$ values in the ordered sample | ||
| + | $$ | ||
| + | (x_1,\dotsc,x_n),\quad x_1\leq\dotsb\leq x_n\,, | ||
| + | $$ | ||
| + | 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 | ||
| + | $$ | ||
| + | \mathbf{P}\{w_n \le t\} = n \int_{-\infty}^\infty (F(x+t)-F(x))^{n-1} dF(x)\,,\ \ \ 0 \le t \le \infty \ . | ||
| + | $$ | ||
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====References==== | ====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> | + | <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> | ||
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====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> D.B. Owen, "Handbook of statistical tables" , Addison-Wesley (1962)</TD></TR></table> | + | <table> |
| + | <TR><TD valign="top">[a1]</TD> <TD valign="top"> D.B. Owen, "Handbook of statistical tables" , Addison-Wesley (1962)</TD></TR> | ||
| + | </table> | ||
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| + | {{TEX|done}} | ||
Latest revision as of 12:46, 14 February 2020
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 $$ (x_1,\dotsc,x_n),\quad x_1\leq\dotsb\leq x_n\,, $$ 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
| [1] | B.L. van der Waerden, "Mathematische Statistik" , Springer (1957) |
| [2] | 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) |
Comments
The range of variation of a sample is also called the sample range.
References
| [a1] | D.B. Owen, "Handbook of statistical tables" , Addison-Wesley (1962) |
How to Cite This Entry:
Range (of variation of a sample). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Range_(of_variation_of_a_sample)&oldid=32664
Range (of variation of a sample). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Range_(of_variation_of_a_sample)&oldid=32664
This article was adapted from an original article by T.Yu. Popova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article