1. The Abnormal Distribution
I’m just
misunderstood.

2. The Normal Distribution
The normal distribution is a very common and useful distribution
Describes many real variables
Height
Weight IQ
SAT
Can be used to calculate probabilities

3. Characteristics of the Normal Distribution
Bell-shaped
Symmetrical
Mean, median, and mode are equal
Tails extend indefinitely
Area under the curve equals 1.00

4. Standard Scores
It is often desirable to compare scores from different distributions
For example,
X = X = 60
S = S = 30
A direct comparison of those scores is not possible because each distribution has a different mean and standard deviation

5. Standard Scores
The problem can be solved by converting the original scores to standard scores
Standard scores are scores with a set mean and standard deviation
Examples of standard scores IQ
t scores
GRE

6. z Scores A very useful standard score is the z score
A z score states how many standard deviations the original score is above or below the mean of a distribution
X = X = 60
S = S = 30 1s 2s
z =
X - X S

7. Properties of z Scores The mean of any set of z scores equals to 0.0
The standard deviation of any set of z scores equals 1.00
The shape of a distribution of z scores maintains the shape of the original score distribution from which the z scores were derived

8. Finding Areas Under the Curve
The proportion of area between two scores of a frequency distribution is equal to the proportion of cases between those two scores
The relation between area and proportion permits us to determine probabilities
It is necessary, then, that we be able to determine the area under the curve

9. The z Score Table
A great deal of the work has been done for us in the z table
Column 1, z
Column 2, Area between mean and z
Column 3, Area beyond z

10. Finding the Area
Given a mean of 100 and a standard deviation of 20, what is the area between the mean and a score of 120?
z =
X - X S 20 =
= +1.00
100
.3413
120
+1.00

11. Finding the Area
Given a mean of 100 and a standard deviation of 20, what is the area above a score of 130?
z =
X - X S 20 =
= +1.50
100
.0668
130
+1.50

12. Finding the Area
Given a mean of 100 and a standard deviation of 20, what is the area above a score of 85?
z =
X - X S 20 =
= -.75
100
.2734
.5000 85
= .7734
-.75

13. Finding the Area
Given a mean of 100 and a standard deviation of 20, what is the area between a score of 75 and 90 ?
z =
X - X S 20 = 20 = =
= -.50
.1915
100 90 =
.2029
.3944 75
-1.25
-.50

14. Finding Scores When the Area is Known
Given a mean of 100 and a standard deviation of 20, what is the cutoff score for the lower 10% ?
X = (z x S) + X
= (-1.28 x 20) = 74.4
100
.1000 ?
-1.28

15. Finding Scores When the Area is Known
Given a mean of 100 and a standard deviation of 20, below what score would 85% of the cases fall ?
X = (z x S) + X
= (1.04 x 20) = 120.8
100
.5000
.3500 ?
85%
1.04