Why is only one normal distribution table needed to find any probability under the normal curve?

Answer 1

The normal distribution is symmetric about its mean; it increases from an asymptote with the x-axis below the mean until the mean whereupon it decreases until another asymptote with the x-axis the same distance above the mean. (This is not a linear increase/decrease, but a "bell" shape.)

As the distribution is symmetric about its mean, only tables up to the mean need be calculated/given in a table. The area under the curve between any two points can then be calculated.

For a normal distribution with mean µ and standard deviation σ, a z value is calculated for a given point x:

z = (x - µ) / σ

This z value is then used to look up the area in the given "half" tables, giving the area (probability) of the value lying between the mean and the given z value. If negative, z is below the mean, but for the table, the sign is ignored.

This can be expressed as:

area = normal(|z|)

where normal(z) is the value in the normal table at the given (positive) z value.

To calculate the area between two points (ie the probability that a value lies between two given values), their corresponding z values (z₁ and z₂) are first calculated and then combined viz:

• If they are both on the same side of the mean (ie z₁ and z₂ have the same sign) then the area is given by:

area = | normal(|z₁|) - normal(|z₂|) |

• If they are on opposite sides of the mean (ie z₁ and z₂ have different signs) then the area is given by:

area = normal(|z₁|) + normal(|z₂|)

Almost all of the normal distribution lies between ±4 standard deviations of the mean.

Answer 2

Because given any variable X with Normal distribution with mean = mu and variance = sigma-square, the Z transform given byZ = (X - mu)/sigma

has the Standard Normal distribution ie it is distributed with mean 0 and variance 1.