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<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN"
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en" lang="en"><head>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
<meta name="Author" lang="en" content="Miguel Casquilho" />
<title>M·C, IST: sigma from sums, detail</title>
<link rel="stylesheet" type="text/css"
href="http://web.tecnico.ulisboa.pt/~mcasquilho/depot/x_commonstyle.css" />
<style type="text/css"> .h {background-color: #a0ffa0}
.hl {background-color: #a0c0ff} .hil {background-color: #a0c0ff}</style>
</head>
<body class="sm">
<form method="post" action=
"http://web.tecnico.ulisboa.pt/~mcasquilho/cgi-bin/interface_png.php?weightedvariance">
<input name="executable" type="hidden" value="weightedvariance.exe" />
<table width="90%" bgcolor="#eeeeee" align="center" border="1" class="sm">
<tr><td colspan="3">
<table width="100%" class="sm">
src="http://web.tecnico.ulisboa.pt/~mcasquilho/depot/MC.gif" border='0'
width='40' height='41' /></a>
</td>
<td><div style="font-size: 13.5pt; text-align: center"><b
onmouseout="this.style.color='blue', this.style.fontStyle='normal'"
onmouseover="this.style.color='red',
this.style.fontStyle='bold'">
σ
from sums: detail</div><center>Finds σ
from sums of (un)equal samples (a conjecture).<!--b>UN DER
CONSTRUCTION</b--></center></td>
<td width="112px"><table width='100%'>
<tr><td align='center' bgcolor='white'>
<input type="submit"
value="  Execute  " style="background-color: #00bfff"
onmouseout="this.style.fontStyle='normal', this.style.fontWeight='normal'"
onmouseover="this.style.fontStyle='normal', this.style.fontWeight='bold'" /></td>
</tr><tr><td align='center'><span style='font-size:
xx-small'>2019.Dec.20 18:46:49</td></tr>
</table>
</td></tr></table>
</td></tr>
<tr><td width="15%" align="center">x</td>
<td width="45%"><textarea name="xvec" rows="1" cols="45"
> 25,01 24,97 25,02 24,9867</textarea><input
name="eodx" type="hidden" value="EOD" /></td>
<td width="40%">xi,
individual or average,
i = 1..n. •</td></tr>
<tr><td align="center">w</td>
<td><textarea name="lots" rows="1" cols="45"
> 10 20 100 150</textarea><input
name="eodn" type="hidden" value="EOD" /></td>
<td>Weight or sample size, wi,
i = 1..n (respectively; see above). •</td></tr>
<tr><td align="center">Show values</td>
<td><select name="ishow"><option value="0" selected="selected">No</option>
<option value="1">Yes</option></select></td>
<td>Show the graph coordinates.</td></tr>
<tr><td colspan="3"><input name="gnuplot" type="hidden"
value="wid = 600
set terminal png enhanced font 'Times, 16' size wid, wid/1.618
set timestamp offset 0, +0.75 font 'Helvetica-Bold, 9'
set key top left font 'Times-Italic, 13'
set title 'Weighted variance, detail' offset 0, -1.
set xlabel '{/Times-Italic i}' offset 16, +0.85
set ylabel '{/Times-Italic x}' rotate by 0 offset +3.25, +4.
set xrange [0.9:*]; set xtics 1
set ytics # nomirror
# set y2label '{/Times-Italic F}' rotate by 0 offset -2.9, +4.25
# set y2range [0:1]; set y2tics 0.2 nomirror
set grid xtics # y2tics
set tics font 'Times, 13'
set style line 11 lc rgb 'red' lw 2 pt 7 ps 1.5
set style line 12 lc rgb '#09ad00' lw 2 pt 2 ps 1 # green
set style line 13 lc rgb '#0000ff' lw 1 # blue
set style line 14 lt 0 lw 2 # dotted
plot anyname.dat \
usi 1:2 title '{/Times-Italic x}' with linespoints ls 11, \
'' usi 1:3 title '{/Times average}' with linespoints ls 12, \
'' usi 1:4 notitle ls 14, \
'' usi 1:5 notitle ls 14" />Computes the
weighted variance,
σ2 W(x),
of a vector of values,
x = (xi, i = 1..n),
and corresponding weights,
w = (wi, i = 1..n),
assuming successively that
(a) x are individual values</i
> with weights w, and
(b) x are averages</i
> (below, a) from samples of sizes w,
in order to estimate the underlying (individual)
μ and σ.
Case '(b)' relates to estimating
from sums of unequal (or equal) size samples,
assuming that the individual values are not known.
The estimation is done through a linear weighting criterion,
a conjecture. Thus,
σ2 ^ = nW
s²W(a). (Below.)
(→ Excel <a href=
"./,bagsSigma/detailSigma.xlsx">file</a>.xlsx)
Example with T = 4 trucks, and procedure
(as mentioned, only nt and Lt
are assumed known):
<table align="center" border="1">
<tr style="text-align: center">
<td width="45px">t</td>
<td width="85px">nt</td>
<td width="85px">Lt</td>
<td width="85px">at</td>
<td width="85px">wt</td>
<td width="85px">wt(at −
xbarW)²</td></tr>
<tr style="text-align: center">
<td>1</td><td>10</td><td>250.1</td><td>25.01</td><td>0.03571</td>
<td>4.81E-6</td></tr>
<tr style="text-align: center">
<td>2</td><td>20</td><td>499.4</td><td>24.97</td>
<td>0.07143</td><td>57.58E-6</td></tr>
<tr style="text-align: center">
<td>3</td><td>100</td><td>2502</td><td>25.02</td>
<td>0.35714</td><td>166.74E-6</td></tr>
<tr style="text-align: center">
<td>4</td><td>150</td><td>3748</td><td>24.987</td>
<td>0.53571</td><td>69.53E-6</td></tr>
<tr style="text-align: center"><td colspan="4" align="right">Sum:</td>
<td>1</td> <td>S = 298.76E-6</td></tr>
</table>
<center>at = Lt &frasl;
nt;
wt = nt &frasl;
Σnt;
xbarW,
weighted average of at =
Σwt at = 24.998 kg;
weighted size, nW =
Σwt nt = 117.86;
weighted variance of a, s²W(a) =
[T &frasl; (T − 1)] S = 398.0E-6 kg²
σ²^ = nW .
s²W(a) = 117.86 × 398.0E-6 =
0.04693 kg² → σ^ = 0.2166 kg.</center>
From the base data, simulated from a population
with μ = 25 and σ = 0.2, the method gives
('wIstdev' =) 0.2166 &cong; σ .
Draws a simple plot of x and
the estimated average (lower and upper lines are
this average −/+ 'wIstdev', est. σ.).</td></tr>
<tr><td align="center"><input type="reset" style="font-size: smaller"
value=" Reset " /></td><td>References:</td>
<td align="center">Plate: SigmaFSdetail</td></tr>
<tr><td colspan="3"><!--p class="ref">• #Wikipedia: <a href=
"https://en.wikipedia.org/wiki/Variance">variance</a></p-->
• Google search '<a href=
"https://www.google.com/search?q=%22different+sample+size%22+-unequal&oq=
%22different+sample+size%22+-unequal"
>"different sample size" -unequal</a>',   '<a href=
"https://www.google.com/search?q=%22unequal+sample+size%22+-different&oq=
%22unequal+sample+size%22+-different"
>"unequal sample size" -different</a>'
• NIST: <a href="http://www.itl.nist.gov/
div898/software/dataplot/refman2/ch2/weighvar.pdf"
>weighted variance</a>.pdf
(← <a href="http://www.itl.nist.gov/div898/software/dataplot/refman2/"
>Dataplot, vol. 2</a> ←
<a href="https://www.nist.gov/itl/sed/">Statistical Eng.ing Div.</a>
← <a href="https://www.nist.gov/itl/">IT Lab.</a>)
• Google search "<a href=
"https://www.google.com/search?q=%22weighted+variance%22&oq=
%22weighted+variance%22"
>weighted variance</a>"
• 1864-01-13: <a href=
"http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Wien.html"
>Wien</a>, Wilhelm Carl Werner Otto Fritz Franz
(&dagger; 1928-08-30).</td></tr>
</table>
</form>
<div style="line-height: 12pt"> </div>
<div style="line-height: 1.5pt; color: black; background-color: red"> </div>
<table width="100%" class="sm"><tr>
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http://web.tecnico.ulisboa.pt/~mcasquilho/compute/qc/Fx-sigmaFSdetail.php;verbose=1;ss=1"><img
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<td style="width: 70%" align="right">http://web.tecnico.ulisboa.pt/~mcasquilho/compute/qc/Fx-sigmaFSdetail.php
Created: 2018-01-13 — Last modified:
2018-02-11</td></tr></table>
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