Definition 1: The standard normal distribution is N(0, 1).
To convert a random variable x with normal distribution N(μ, σ) to standard normal form you use the following linear transformation:
The resulting random variable is called a z-score. Thus z = STANDARDIZE(x, μ, σ), as described in Definition 3 and Excel Functions in Expectation.
The z-score provides a standard way to compare statistics based on different normal distributions.
Property 1: If x is a random variable with normal distribution N(μ, σ) then the corresponding z-score has normal distribution N(0, 1)
Proof: Since is a linear transformation of the form
by Property 1 of Characteristics of Normal Distribution, it follows that z has normal distribution of type
Excel Functions: Excel provides the following functions for the standard normal distribution:
NORMSDIST(x) = NORMDIST(x, 0, 1, TRUE); the standard normal version of NORMDIST
NORMSINV(p) = NORMINV(p, 0, 1); the inverse of NORMSDIST
Note that NORMSINV(p) = the value x such that NORMSDIST(x, TRUE) = p
Excel 2010/2013 provide the following additional functions: NORM.S.DIST(x, cum), where cum takes the value TRUE or FALSE and NORM.S.DIST(x, cum) = NORM.DIST(x, 0, 1, cum), as well as NORM.S.INV which is equivalent to NORMSINV.