Populations, Parameters, Statistics, and Sampling

• E[S 2 ] ≠ σ 2
Bias in the Sample Variance
– proof on p. 216 E[S2 ] = σ2 - σ 2
M
• this is because the mean used to calculate S will in general be
different than the population mean
– the sample mean is always exactly centered on the particular sample
and so deviations from it underestimate true population deviations
2
2 2 2 2 σ ⎛ N −1
⎞ 2
ES [ ] = σ − σM= σ − = ⎜ ⎟σ
N ⎝ N ⎠
– the sample variance is too small by a factor of (N-1)/N
• S 2 is the uncorrected variance and s 2 is corrected
– the corrected standard deviation is still slightly biased (only variance
is unbiased)
» the amount of remaining bias is small though, so it’s typical to
use s to estimate σ
• unbiased estimate of standard error of the mean
ˆ σ M =
ˆ σ
=
N
s
=
N
S
N −1