A statistic is biased if the long-term average
value of the statistic is not the parameter it is
estimating. More formally, a statistic is biased if the mean
of the sampling distribution of the statistic is not equal to
the parameter. The mean of the sampling distribution of a
statistic is sometimes referred to as the expected value
of the statistic.

As we saw in the section on the sampling distribution of the
mean, the mean of the sampling distribution of the (sample)
mean is the population mean (
m
m
). Therefore the sampling
distribution of the mean is an unbiased estimate of
m
m
. Any given sample mean may underestimate or overestimate
m
m, but,
there is no systematic tendency for sample means to either
under or overestimate
m
m

In the section on variability,
we saw that the formula for
the variance in a population is

σ2=∑X−μ2N
σ
2
X
μ
2
N

(1)
whereas the formula to estimate the variance from a sample is

s2=∑X−M2N−1
s
2
X
M
2
N
1

(2)
Notice that the denominators of the formulas are different:

NN
for the population and

N−1
N
1
for the sample. We saw in the
"Estimating Variance Simulation" that if

NN is used in the
formula for

s2
s
2
, then the estimates tend to be too low and
therefore biased. The formula with

N−1
N
1
in the denominator
gives an unbiased estimate of the population variance.