Standard Normal Distribution

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:

Standard normal variable

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.

Standard normal curve

Figure 1 – Standard normal curve

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.

4 Responses to Standard Normal Distribution

  1. ja says:

    Dear Dr. Charles,
    Thank you for your helpful website.

  2. Sanitha Sivadas says:

    Hi Charles!

    Your site is very helpful. Thankx for the reply on ANOVA. I have confusion regarding normal distribution of data.
    If i collect data from 10 sites (10 replicates each), if i have to check the normal distribution of the data , i check for each site separately or overall data i.e all 10 sites together, or
    for an experiment (treatment v control) and with 6 replicates, the normality of the data, is checked for the control and treatment separately, or both together.


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