Standard Normal Distribution
The Classic Bell-Shaped curve is
symmetric, with
mean = median = mode = midpoint
Standard Normal Distribution
Standard Normal Distribution
.34
0.40
0.30
.50
.135
Probability Density
0.20
0.10
.025
0.00
-4 -3 -2 -1 0 1 2 3 4
Standard Score (z)
Probabilities in the
Normal Distribution
The distribution is symmetric, with a mean
of zero and standard deviation of 1.
The probability of a score between 0 and 1
is the same as the probability of a score
between 0 and –1: both are .34.
Thus, in the Normal Distribution, the
probability of a score falling within one
standard deviation of the mean is .68.
More Probabilities
The area under the Normal Curve from 1 to
2 is the same as the area from –1 to –
2: .135.
The area from 2 to infinity is .025, as is the
area from –2 to negative infinity.
Therefore, the probability that a score falls
within 2 standard deviations of the mean
is .95.
Normal Distribution Problems
Suppose the SAT Verbal exam has a mean
of 500 and a standard deviation of 100.
Joe wants to be accepted to a journalism
program that requires that applicants score
at or above the 84th percentile. In other
words, Joe must be among the top 16% to
be admitted.
What score does Joe need on the test?
To solve these problems, start by drawing
the standard normal distribution.
Next, formula for z:
Standard Normal Distribution
.34 X i − X X i − 500
zX i = =
.50 sX 100
.135 +
.025 =
.16
-4 -3 -2 -1 0 1 2 3 4
€
Standard Score (z)
Next: Label the Landmarks
zX –2 –1 0 1 2
X 300 400 500 600 700
Now Check the Normal Areas
We now know that:
2.5% score below 300; i.e., z = –2
16% score below 400
50% score below 500; i.e., z = 0
84% score below 600
97.5% score below 700; i.e., z = 2
Solution Summary
Joe had to be among the top 16% to be
accepted.
That means his z-score must be +1.
Thus, his raw score must be at least 600,
which is one standard deviation (100) above
the mean (500).
Therefore, Joe needs to score at least 600.
Next Topic: Correlation
We have seen that the z-score
transformation allows us to convert any
normal distribution to a standard normal
distribution.
The z-score formula is also useful for
calculating the correlation coefficient,
which measures how well one can predict
from one variable to another, as you learn in
the next lesson.
