When we convert normal distributions to the standard?

Question

normal distribution, we are essentially making them all alike. How is this possible and what are the implications?

Answer 1

For any normal random variable X with mean μ and standard deviation σ, X ~ Normal(μ, σ), { note that in most textbooks and literature the notation is with the variance, i.e., X ~ Normal(μ, σ²).  Most software denotes the normal with just the standard deviation.}

You can translate into standard normal units by:

Z = (X - μ) / σ

form the Standard normal to X by:

X = μ + Z σ

Where Z ~ Normal(μ = 0, σ = 1).  You can then use the standard normal cdf tables to get probabilities.

If you are looking at the mean of a sample, then remember that for any sample with a large enough sample size the mean will be normally distributed.  This is called the Central limit theorem.

If a sample of size is is drawn from a population with mean μ and standard deviation σ then the sample average xBar is normally distributed

with mean μ and standard deviation σ / √(n)

An applet for finding the values

http://www-stat.stanford.edu/~naras/jsm/...

calculator

http://stattrek.com/Tables/normal.aspx

how to read the tables

http://rlbroderson.tripod.com/statistics...