Statistics The Standard Normal Distribution
The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1.
Recall that all normally distributed variables can be transformed into the standard normally distributed variable by using the formula for the standard score: x z
If each data value of a normally distributed random variable x is transformed into a z-score, the result will be the standard normal distribution.
In theory, we can convert any normal distribution into a standard normal distribution giving us a universal scale to find the probability of any interval within our normal distribution.
Using left-tail style standard normal distribution table 1. For areas to the left of a specified z value, use the table entry directly.
Example 1. Blah, blah, blah, word problem, blah, blah, blah. z score is 1.58 determined by using our z formula. Yay. Find the probability of z < 1.58. For areas to the left of a specified z value, use the table entry directly. Or we can use that wonderful graphing utility.
2. For areas to the right of a specified z value, look up the table entry for z and subtract the area from 1. Another way to find the same area is to use the symmetry of the normal curve and look up the table entry for â&#x20AC;&#x201C;z.
Example 2. Blah, blah, blah, another word problem. z score is â&#x20AC;&#x201C;1.23. Find the probability of z > â&#x20AC;&#x201C;1.23. For areas to the right of a specified z value, look up the table entry for z and subtract the area from 1.
3. For areas between two z values, z1 and z2 (where z2 > z1), subtract the table area for z1 from the table area for z2.
Example 3. That’s right, another word problem. z scores –2.00 and 1.50. Find the probability of –2.00 < z < 1.50. For areas between two z values, z1 and z2 (where z2 > z1), subtract the table area for z1 from the table area for z2.