All normal curves share certain characteristics. It is possible to find areas under any normal curve by working with only the standard normal curve, which has a mean of 0 and a standard deviation of 1, which is tabulated in the back of the textbook. All normal curves are described by two parameters: the mean, mu, and the standard deviation, sigma.
z = (x - mu) / sigmaThe z-score tells how many standard deviations an observation x is from the mean. Positive z-scores are greater than the mean, and negative z-scores are below the mean.
It is also useful to use this formula in reverse.
x = mu + z*sigmaThis says explicitly (in algebra) that x is z standard deviations above the mean.
You will need to be able to use the table to find areas when the numbers on the axis are known, and to be able to use the table to find numbers on the axis when areas are known.
Always draw a sketch of a normal curve in working out problems.
Examples
To find the area to the left of 0.57.
Use the table:
z | 0.00 0.01 0.07 ------------------------------------------------------- 0.00 | 0.10 | 0.50 | .7157Since the total area is 1, the area to the right of 0.57 must be 1 - .7157 = .2843.
The table may also be used backwards. Suppose that you want to find a value z so that the area to the right of z is .6000.
Then the area to the left of z is .4000. Areas are in the middle of the table. Try to find the area closest to .4000.
z | -0.09 -0.05 ------------------------------------------ -3.80 | -0.20 | .4013 .4013 is the closest, so z is about -0.25.
Bret Larget, larget@mathcs.duq.edu