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# Standard Normal Distribution — Hypothesis Testing

The normal distribution X∼N(0,1), μ = 0 and σ = 1, is called the standard normal distribution, and its distribution function is:

$$f(x)=\frac{e^{-x^2/2}}{\sqrt{2π}}$$

A value from any normal distribution can be transformed into its corresponding value on a standard normal distribution using the following formula:

$$Z =\frac{X - μ}{σ}$$

where Z is the value on the standard normal distribution, X is the value on the original distribution, μ is the mean of the original distribution, and σ is the standard deviation of the original distribution.

$$E(Z)= E\biggl(\frac{X - μ}{σ}\biggr)=\frac{E(X) - μ}{σ}=0$$ $$Var(Z)=Var\biggl(\frac{X - μ}{σ}\biggr)=\frac{Var(X)}{σ^2}=1$$

To computing probabilities for normal distributions, convert to standard value and lookup the distribution function tables for the standard normal distribution:

$$P(X≤b)→P\biggl(Z≤\frac{b - μ}{σ}\biggr)$$

Note: The probabilities may also be obtained using the NORM.DIST function in Excel. The Excel NORM.INV function returns the variable value, given the parameters and the probability.

MarketingMind’s content is sourced from the Marketing Analytics Practitioner’s Guide (MAPG), a comprehensive 4-volume compendium on Marketing, Marketing Analytic and Market Research available in both physical and eGuide formats. (Click to learn more).

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