The probabilities for normal distribution are found from the area under normal curve. This means that a table relating the variable values to the corresponding areas is of great use in determining the probabilities real quick. But this would mean that we require separate tables for distributions defined by different parameter values.

In order to make the method of using tables efficient, a standard normal distribution of variable Z is defined with mean = 0 and standard deviation 1. Any normal variable X can be transformed to z scores and the table of areas of Z distribution can be used for all normal variables.

The z-score for any normal variable X with mean μ and standard deviation σ can be found using the formula,

z = $\frac{x-\mu }{\sigma }$There are two types of normal distribution tables are used for determining the probabilities.

Positive z-score table gives the area between a positive z value and mean. Probability of any interval value can be found using this table and applying the property of symmetry of the normal curve. The picture of the table below shows the area corresponding to z = 2.35 as 0.4906.

When we say that the area corresponding to z = 2.35 is equal to 0.4906, if means P(0 < Z < 2.35 ) = 0.4906 and not

P( Z = 2.35) = 0.4906, as the normal variables assumes only intervals as values and not any specific numerical value.

Indeed the probability of the variable assuming any numerical value is zero and hence P(Z = 2.35) = 0

Left area table, gives the area to the left of any z-score both positive and negative. This table can also be used for finding any probability applying the property of symmetry of the normal curve.