Q: What is the difference of a normal distribution and a stardard normal distribution?

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The domain of the normal distribution is infinite.

It means distribution is flater then [than] a normal distribution and if kurtosis is positive[,] then it means that distribution is sharper then [than] a normal distribution. Normal (bell shape) distribution has zero kurtosis.

No. Normal distribution is a special case of distribution.

The normal distribution can have any real number as mean and any positive number as variance. The mean of the standard normal distribution is 0 and its variance is 1.

No. The binomial distribution (discrete) or uniform distribution (discrete or continuous) are symmetrical but they are not normal. There are others.

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The standard normal distribution is a normal distribution with mean 0 and variance 1.

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If the paired differences are normal in a test of mean differences, then the distribution used for testing is the

Actually the normal distribution is the sub form of Gaussian distribution.Gaussian distribution have 2 parameters, mean and variance.When there is zero mean and unit variance the Gaussian distribution becomes normal other wise it is pronounced as Gaussian.Wrong! The standard normal distribution has mean 0 and variance 1, but a normal distribution is the same as the Gaussiand, and can have any mean and variance. Google stackexcange "what-is-the-difference-between-a-normal-and-a-gaussian-distribution"

A normal distribution is defined by two parameters: the mean, m, and the variance s2, (or standard deviation, s).The standard normal distribution is the special case of the normal distribution in which m = 0 and s = 1.

standard normal is for a lot of data, a t distribution is more appropriate for smaller samples, extrapolating to a larger set.

The uniform distribution is limited to a finite domain, the normal is not.

Suppose you could call it the Gaussian Distribution or the Laplace-Gauss (not to be confused with the Laplace distribution which takes an absolute difference from the mean rather than a squared error)... however the Brits had no one to name this distribution after (not the German and French names) and because it is the ubiquitous distribution they just called it... well the NORMAL!!

le standard normal distribution is a normal distribution who has mean 0 and variance 1

A standard normal distribution has a mean of zero and a standard deviation of 1. A normal distribution can have any real number as a mean and the standard deviation must be greater than zero.

When its probability distribution the standard normal distribution.

No, the normal distribution is strictly unimodal.