Because each phenomenon is made up of many smaller phenomena which are often independent, each of which has a random element associated with it. The sum of such random variations tends towards the Gaussian (normal) distribution. Also, the distribution has well known properties.
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You can use a normal distribution to approximate a binomial distribution if conditions are met such as n*p and n*q is > or = to 5 & n >30.
The F distribution is used to test whether two population variances are the same. The sampled populations must follow the normal distribution. Therefore, as the sample size increases, the F distribution approaches the normal distribution.
A researcher wants to go from a normal distribution to a standard normal distribution because the latter allows him/her to make the correspondence between the area and the probability. Though events in the real world rarely follow a standard normal distribution, z-scores are convenient calculations of area that can be used with any/all normal distributions. Meaning: once a researcher has translated raw data into a standard normal distribution (z-score), he/she can then find its associated probability.
If a random variable X has a Normal distribution with mean m and standard deviation s, then z = (X - m)/s has a Standard Normal distribution. That is, Z has a Normal distribution with mean 0 and standard deviation 1. Probabilities for a general Normal distribution are extremely difficult to obtain but values for the Standard Normal have been calculated numerically and are widely tabulated. The z-transformation is, therefore, used to evaluate probabilities for Normally distributed random variables.
The Normal distribution is a probability distribution of the exponential family. It is a symmetric distribution which is defined by just two parameters: its mean and variance (or standard deviation. It is one of the most commonly occurring distributions for continuous variables. Also, under suitable conditions, other distributions can be approximated by the Normal. Unfortunately, these approximations are often used even if the required conditions are not met!