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Probability and the standard normal distribution
Comparing data from two different distributions can be very challenging. It is therefore important to convert the scores from the different distributions into a common metric so as to conform it to the standard distribution. In standard normal distribution the standard deviation and the means always have fixed values. The standard deviation is 1.0 and the mean is always 0 thus making it possible to directly compare the two distributions. In a standard distribution nearly all the proportions of its population occurring in any area of the distribution are already determined. This means that when any normal distribution is made to conform to the standard distribution then we can establish what’s likely to occur in almost any area of the distribution.
The z-scores are normally used to convert ordinary normal variables with a standard deviation s and a mean m to a normal variable with a standard deviation 1 and a mean of 0. The Z-score therefore clarifies how far a specific value lies from the mean of standard normal distribution in terms of the standard deviation. Transforming values to a Z-score thus allows us to use one table to evaluate probabilities. When transforming data from multiple distributions, if x represents the normal variable in the problem then the Z-score denotes the corresponding variable in the standard distribution.
A proportion of 1.0 can be translated to mean 100% of the population. A Z-score can simple be transformed into a percentage. For instance a value 0.1985 of a normal distribution can occur between the point Z=-0.52 and the point where Z=0. If we multiply 0.1985 by 100 we get 19.85%. This implies that 19.85% of the normal distribution occurs between the two points. The Z-score represents the proportion of the area under the standard curve better than the percentages. The Z-score is very easy to use and can readily be programmed to be used in excel in the case of large data sets. The percentage on the other hand can be quite confusing. For instance when working with a sample where the highest score is 25 and the lowest is 5, it is very easy to assume that the percentage of the population between the two points is 100%. This is, however, never the case since this is only a sample size and there is a possibility someone from the population scored above 25 or below 5. In a standard distribution there will never be a value of z that represents 100% of the distribution.