Carefully define a standard normal distribution Why does a r

Carefully define a standard normal distribution. Why does a researcher want to go from a normal distribution to a standard normal distribution? Explain. (250 words)

Solution

Standard Normal Distribution

The standard normal distribution is a special case of the normal distribution. It is the distribution that occurs when a normal random variable has a mean of zero and a standard deviation of one.

Standard Score (aka, z Score)

The normal random variable of a standard normal distribution is called a standard score or a z-score. Every normal random variable X can be transformed into a z score via the following equation:

z = (X - ) /

where X is a normal random variable, is the mean of X, and is the standard deviation of X.

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.