Recall that a normal distribution is symmetrical. What's the probability that the return on an asset falls between -3.52% and 50.96%? The rates of return on the assets in a portfolio are normally distributed, with a mean of 20% and a standard deviation of 12%. Note that the z distribution will only be a normal distribution if the original distribution (X) is normal. So, a z score of 2 means the original score was 2 standard deviations above the mean. Converting the test scores to z scores, an X of 70 would be: z = (70 - 50) / 10 = 2. For instance, if a person scored 70 on a test with a mean of 50 and a standard deviation of 10, that person scored 2 standard deviations above the mean. It is in fact a standard normal random variable. A z score (also called z-value or z-statistic) is the distance between a selected value (X) and the population mean, divided by the population standard deviation. The standard normal distribution is sometimes called the z distribution. Where X is a score from the original normal distribution, μ is the mean of the original normal distribution, and σ is the standard deviation of original normal distribution. Normal distributions can be transformed to standard normal distributions by the formula: 
However, the actual distribution for a normal random variable to a standard normal distribution can be standardized. Therefore, it's impractical to provide a table of probabilities for each combination of mean and standard deviations. There is an unlimited number of normal distributions, each with a different mean or standard deviation. Below are some confidence intervals for the standard normal distribution: The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. The way to get around this problem is to standardize a normal random variable, which involves converting it to a general scale for which probability tables exist. A computer is needed to calculate areas under the graph this is required in order to calculate probabilities.
The problem with working with a normal distribution is that its formula is very complicated.
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