Download Normal Density Curves and Z-Scores Statistics S. Lohse Now that

Normal Density Curves and Z-Scores
Statistics
S. Lohse
Now that you have taken a look at how to calculate probabilities for normally distributed variables on your calculator, I'd
like to relate that back to z-scores, and something called the Standard Normal Distribution.
Turning a Normal Probability Density Curve into a Standard Normal Density Curve
In Graph A we see a normal curve with mean of 849 and a standard deviation of 32.
Graph A:
753
785
817
849
881
913
945
Recall that each individual piece of data has a z-score. The z-score represents the distance between that piece of data
and the mean, as measured in standard deviations. Data values that are above the mean have positive z-scores, those
below the mean have negative z-scores.
The formula for calculating a z-score is given by z =
x-µ
. But in the graph above we can see that each of thelabeled
numbers on the x-axis are a whole number of standard deviations from the mean, so the corresponding z-scores are as
follows:
Z735 = -3
Z785 = -2
Z817 = -1
Z849 = 0
Z881 = 1
Labeling the x-axis with z-scores, the graph now looks like this.
-3
-2
-1
0
1
2
3
1
Z913 = 2
Z945 = 3
Calculating probabilities using Standard Normal Density Curves and a Table
Thie distribution pictured on the previous page using z-scores, with a mean of 0 and a standard deviation of 1
has a special name, called the Standard Normal Dsitribution. In the days before the easy availability of graphing
calculators, and other technology to perform complicated calculations, calculus was needed to find the area
under a curve, and it turns out that those calculus calculations were much easier to do if the distribution in
question was a Standard Normal Curve.
So, another method of calculating probabilities (the olden way), is to change every Normal Distribution into a
Standard Normal Distribution using z-scores, then look the probability values up in a table.
For instance, on our calculator we can find P(X < 881) = normalcdf(0, 881, 849, 32) = .8413
But, using Z-scores, this is the same as P(Z < 1), since 881 was 1 standard deviation above the mean. Then
we would look in a table like the one on appendix page A-12 of your paper or ebook. In the left hand column,
we look down to a z-score of 1.0, then across to the first clumn under .00, and we see the very same answer
.8413
2