From: pbristow_at_[hidden]
Date: 2007-09-22 16:19:48
Author: pbristow
Date: 2007-09-22 16:19:45 EDT (Sat, 22 Sep 2007)
New Revision: 39482
URL: http://svn.boost.org/trac/boost/changeset/39482
Log:
Added paragraph on confidence interval versus observations
Text files modified:
sandbox/math_toolkit/libs/math/doc/distributions/chi_squared_examples.qbk | 61 ++++++++++++++++++++++++++++++++++++++++
1 files changed, 61 insertions(+), 0 deletions(-)
Modified: sandbox/math_toolkit/libs/math/doc/distributions/chi_squared_examples.qbk
==============================================================================
--- sandbox/math_toolkit/libs/math/doc/distributions/chi_squared_examples.qbk (original)
+++ sandbox/math_toolkit/libs/math/doc/distributions/chi_squared_examples.qbk 2007-09-22 16:19:45 EDT (Sat, 22 Sep 2007)
@@ -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]