$z$ -Value
The functional form for a normal distribution is a bit complicated. It can also be difficult to compare two variables if their mean and or standard deviations are different. For example, heights in centimeters and weights in kilograms, even if both variables can be described by a normal distribution. To get around both of these conflicts, we can define a new variable:
This variable gives a measure of how far the variable is from the mean (
Areas Under the Curve
To calculate the probability that a variable is within a range, we have to find the area under the curve. Normally, this would mean we'd need to use calculus. However, statisticians have figured out an easier method, using tables, that can typically be found in your textbook or even on your calculator.
Standard Normal Table
This table can be used to find the cumulative probability up to the standardized normal value
These tables can be a bit intimidating, but you simply need to know how to read them. The leftmost column tells you how many sigmas above the the mean to one decimal place (the tenths place).The top row gives the second decimal place (the hundredths).The intersection of a row and column gives the probability.
For example, if we want to know the probability that a variable is no more than 0.51 sigmas above the mean,
A common mistake is to look up a
There is another note of caution to take into consideration when using the table: The table provided only gives values for positive
Symmetrical Normal Curve
This images shows the symmetry of the normal curve. In this case, P(z2.01).
We may even wish to find the probability that a variable is between two z-values, such as between 0.50 and 1.50, or
68-95-99.7 Rule
Although we can always use the
68-95-99.7 Rule
Dark blue is less than one standard deviation away from the mean. For the normal distribution, this accounts for about 68% of the set, while two standard deviations from the mean (medium and dark blue) account for about 95%, and three standard deviations (light, medium, and dark blue) account for about 99.7%.