# What is the difference between normal distribution and standard normal distribution?

Two terms normal distribution and standard normal distribution are used in statistics. Does standard term contribute to the normal distribution anything? Please give a simple-however a substantive reasoning in the back of these terms.

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Standard normal distribution stands for the case $\mu=0$, $\sigma=1$. Please, have a look at the wikipedia entry. –  user10525 Jul 7 '12 at 15:35
Is it possible to think of normal distribution without zero,1. For example, N(mu,variance) –  subhash c. davar Jul 8 '12 at 7:03

A normal distribution is determined by two parameters the mean and the variance. Often in statistics we refer to an arbitrary normal distribution as we would in the case where we are collecting data from a normal distribution in order to estimate these parameters. Now the standard normal distribution is a specific distribution with mean $0$ and variance $1$. This is the distribution that is used to construct tables of the normal distribution.

Conveniently if $X$ has the a normal distribution with mean $m$ and variance $s^2$ then if we define

$$Z=\frac{X-m}{s}$$ then $Z$ has the standard normal distribution. So for any specific normal distribution we can calculate probabilities of the form $P[a < X < b]$ from the tables for $Z$. This is because we can write $X=sZ+m$. So the standard normal plays a special role with respect to the general family of normal distributions.

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@ Michael Chernik This is a statistical way to explain. Assume that I have a small understanding of the discipline. –  subhash c. davar Jul 8 '12 at 6:51
@subhashdavar, maybe you could be more specific in your question; what are your thoughts exactly? What part confuses you? I think Michael's answer and Procastinator's comment are pretty clear. –  Néstor Jul 8 '12 at 20:20
@subhashdavar My answer is not very heavy on statistics. Maybe the math formulas cause you some confusion. Let me try again. When we say that a variable has a nromal distribution we are talking about a family of distributions. The standard normal distribution is a specific one with mean 0 and variance 1. So you can compute a table of values for the standard normal. Another normal distribution has mean 1 and variance 4. If you take a variable with a normal distribution with mean 1 and variance 4 subtract 1 and divide by 2 and you have a variable that is a standard normal. –  Michael Chernick Jul 8 '12 at 20:56
By subtracting the mean and dividing by the standard deviation you transform one normally distributed variable to a standard normal and in that way you can determine probabilities for any normal distribution based on the standard normal tables. –  Michael Chernick Jul 8 '12 at 20:59
@ Michael Thanks for the statistical interpretation of the distributions. If I have a scale-free variable, do various observations represent Z quantities and we may say it has a standard normal distribution. –  subhash c. davar Jul 13 '12 at 15:21
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