An example of using the `jackknife' and `bootstrap' estimation:
  Finding bias-corrected estimates and confidence intervals for
  the true standard deviation from a sample.
The starting random-number seed is 23456

A sample of 30 standard normal R.V.s is:
  -0.8720   0.1020   0.0895  -0.3867  -0.2771   1.4472   0.1625   0.5456
   1.7043   0.3594   0.0773  -0.8783   0.3176  -0.3142  -1.6018   0.7049
   1.0378   1.2692  -1.0916   1.6173  -0.0572  -1.4082   0.7606  -0.0179
   0.1951   0.6653  -0.6595   0.2604  -0.4153   1.6192

A sample of 30 normal variables with mean=3 and st.dev.=5 is
    4.775    5.059    8.711   -1.764    8.123    3.321    3.286    9.796
    6.324   -8.932   -0.061    0.058   -2.141    0.997   -1.786   -1.282
    7.753   14.489    1.595    5.068    8.989    8.860   13.986    8.685
    4.464    1.951   -0.106   10.779    3.735    0.561
OBSERVED sample mean: 4.1763  Sample standard deviation: 5.2550

CLASSICAL NORMAL-THEORY CONFIDENCE INTERVAL:
The 0.025, 0.500, and 0.975 quantiles of ChiSquare(29) are
  16.0471  28.3361  45.7223
The classical symmetric 95% CI for sdev is ( 4.1852,  7.0644)

(This is an exact CI, but assumes normal variables. It is based
  on quantiles of the chi-square distribution with
  df=29 degrees of freedom.)

JACKKNIFE CONFIDENCE INTERVALS:
For the jackknife of sdev itself:
Est Sdev=5.2550: The sdev delete-1 jackknife values are:
    5.347    5.345    5.277    5.225    5.294    5.346    5.345    5.238
    5.332    4.717    5.286    5.289    5.208    5.313    5.224    5.244
    5.304    4.967    5.325    5.345    5.267    5.272    5.005    5.277
    5.348    5.331    5.284    5.195    5.347    5.303
The sdev jackknife pseudo-values are:
    2.593    2.636    4.631    6.134    4.126    2.631    2.637    5.754
    3.020   20.849    4.367    4.266    6.608    3.573    6.160    5.572
    3.844   13.603    3.226    2.637    4.895    4.771   12.516    4.607
    2.566    3.054    4.405    6.987    2.577    3.872
The mean and sample sdev of the 30 pseudovalues are:  5.30392  3.9253
The pseudovalue 95% symmetric CI for sdev is ( 3.8993,  6.7086)

Using Student-t (df=29) 95% CI instead: ( 3.8382,  6.7697)

For the jackknife of log(sdev):
Log(sdev)=0.0000: The 30 logsdev delete-1 jackknife values are:
    1.677    1.676    1.663    1.653    1.667    1.676    1.676    1.656
    1.674    1.551    1.665    1.666    1.650    1.670    1.653    1.657
    1.668    1.603    1.672    1.676    1.662    1.662    1.610    1.663
    1.677    1.674    1.665    1.648    1.677    1.668
The log(sdev) jackknife pseudo-values are:
    1.157    1.165    1.541    1.827    1.445    1.164    1.165    1.754
    1.237    4.790    1.491    1.472    1.918    1.341    1.832    1.720
    1.392    3.293    1.276    1.165    1.591    1.567    3.075    1.536
    1.152    1.243    1.498    1.991    1.154    1.397
The mean and sample sdev of the 30 log pseudovalues are:  1.67824  0.773893
The pseudovalue 95% symmetric CI for logsdev is ( 1.4013,  1.9552)
The exponentiated mean and exponentiated 95% symmetric CI are
   5.3561  ( 4.0605,  7.0652)

Using Student-t (df=29) exponentiated 95% CI instead: ( 4.0119,  7.1507)

BOOTSTRAP CONFIDENCE INTERVAL:
Doing 1000 bootstrap replications for the sample standard deviation:
Sorting the 1000 bootstrap replications:
The first 30 sorted bootstrap sdev values are (#1-#30):
    3.160    3.212    3.223    3.288    3.363    3.510    3.529    3.537
    3.544    3.546    3.565    3.566    3.634    3.646    3.662    3.681
    3.744    3.762    3.800    3.816    3.826    3.839    3.853    3.862
    3.865    3.870    3.872    3.886    3.894    3.905
The last  30 sorted bootstrap sdev values are (#971-#1000):
    6.350    6.352    6.361    6.377    6.402    6.412    6.472    6.478
    6.508    6.516    6.517    6.532    6.557    6.579    6.592    6.594
    6.619    6.621    6.636    6.674    6.680    6.707    6.788    6.873
    6.888    6.921    6.982    7.043    7.138    7.167
Bootstrap median bias (sdev=5.25505):  566/1000 = 0.566
Bootstrap median: 5.1333   95% Percentile CI: ( 3.8647,  6.4117)

SUMMARY  (Estimator of sdev, 95% symmetric CI for sdev):
  NormalTheory        5.3162  ( 4.1852,  7.0644)
  Sdev Jackknife      5.3039  ( 3.8993,  6.7086)
  LogSdev Jackknife   5.3561  ( 4.0605,  7.0652)
  Bootstrap           5.1333  ( 3.8647,  6.4117)
Student-t CIs:
  Sdev Jackknife      5.3039  ( 3.8382,  6.7697)
  LogSdev Jackknife   5.3561  ( 4.0119,  7.1507)

Sample size: 30   Number of bootstrap replications: 1000
Starting random-number seed: 23456

The `Sdev Jackknife' CI seems to track the exact CI best,
  even for different starting seeds and different numbers of
  bootstrap replication.

Random numbers used:  30,070