# Confidence Interval for μ, Standard Deviation Estimated (1 of 3)

It is very rare for a researcher wishing to estimate the

mean of a

population to already know its

standard
deviation. Therefore, the construction of a

confidence interval almost always involves
the estimation of both μ and σ.

When σ is known, the

formula:

M - zσ

_{M} ≤ μ ≤ M + zσ

_{M
}
is used for a confidence interval. When σ is not
known,

(N
is the

sample
size)

is used as an estimate of σ

_{M}. Whenever
the standard deviation is estimated, the

t rather
than the normal (z) distribution should be used. The values of t are
larger than the values of z so confidence intervals when σ
is estimated are wider than confidence intervals when σ
is known

The formula for a confidence interval for μ
when σ is estimated is:

M - t s

_{M} ≤ μ ≤ M + t s

_{M}
where M is the sample

mean,
s

_{M} is an estimate of σ

_{M}, and t depends on the

degrees of freedom and the level of confidence.