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Added paragraph on confidence interval versus observations

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[SVN r39482]
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Paul A. Bristow
2007-09-22 20:19:45 +00:00
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doc/distributions/chi_squared_examples.qbk
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@@ -105,6 +105,67 @@ _____________________________________________
So at the 95% confidence level we conclude that the standard deviation
is between 0.00551 and 0.00729.
[h4 Confidence intervals as a function of the number of observations]
Similarly, we can also list the confidence intervals for the standard deviation
for the common confidence levels 95%, for increasing numbers of observations.
(The standard deviation is here assumed unity,
so we can simply multiply a particular standard deviation,
0.0062789 in the example above, by these values to get the confidence limits).
[pre'''
____________________________________________________
Confidence level (two-sided) = 0.0500000
Standard Deviation = 1.0000000
________________________________________
Observations Lower Upper
Limit Limit
________________________________________
2 0.4461 31.9102
3 0.5207 6.2847
4 0.5665 3.7285
5 0.5991 2.8736
6 0.6242 2.4526
7 0.6444 2.2021
8 0.6612 2.0353
9 0.6755 1.9158
10 0.6878 1.8256
15 0.7321 1.5771
20 0.7605 1.4606
30 0.7964 1.3443
40 0.8192 1.2840
50 0.8353 1.2461
60 0.8476 1.2197
100 0.8780 1.1617
120 0.8875 1.1454
1000 0.9580 1.0459
10000 0.9863 1.0141
50000 0.9938 1.0062
100000 0.9956 1.0044
1000000 0.9986 1.0014
''']
With just 2 observations the limits are from *0.445* up to to *31.9*,
so the standard deviation might be about *half*
the observed value up to *30 times* the observed value!
Estimating a standard deviation with just a handful of values leaves a very great uncertainty,
especially the upper limit.
Note especially how far the upper limit is skewed from the most likely standard deviation.
Even for 10 observations, normally considered a reasonable number,
the range is still from 0.69 to 1.8, about a range of 0.7 to 2,
and is still highly skewed with an upper limit *twice* the median.
When we have 1000 observations, the estimate of the standard deviation is starting to look convincing,
with a range from 0.95 to 1.05 - now near symmetrical, but still about + or - 5%.
Only when we have 10000 or more repeated observations can we start to be reasonably confident
(provided we are sure that other factors like drift are not creeping in).
For 10000 observations, the interval is 0.99 to 1.1 - finally a really convincing + or -1% confidence.
[endsect][/section:chi_sq_intervals Confidence Intervals on the Standard Deviation]
[section:chi_sq_test Chi-Square Test for the Standard Deviation]