Chapter 4: Standardized Scores
and the Normal Distribution
The simplest standardized scores are
called z scores. They tell you the distance
of a raw score from the mean in terms of
standard deviations (the sign of the z score
tells you whether a score is above or below
the mean). Here is the formula:
z
Properties of z scores:
–
–
–
Xµ

The mean of a complete set of z scores is
always 0, and its standard deviation will
always be 1.0.
Converting a set of raw scores into z
scores will not change the shape of the
distribution at all.
However, z scores are especially helpful
when used in conjunction with a normal
distribution.
Chapter 4
For Explaining Psychological
Statistics, 4th ed. by B. Cohen
1
Try This Example of
Computing z Scores:
Jan just received three midterm grades
(see table below). Which grade is associated with the highest z score?
Psychology
Mathematics
Geology
X
µ
σ
68
77
83
65
77
89
6
9
8
z
To get you started, here is the
calculation of the first z score:
z
Chapter 4
68  65 
6
0.50
For Explaining Psychological
Statistics, 4th ed. by B. Cohen
2
T Scores
– Transform z scores into T scores, which
follow a distribution with mean = 50 and
σ = 10, using this formula:
T  10 z  50
– In the table below, first find the z score
for each subject, and then the T score.
We will get you started by calculating z
and T for Alice:
Subj
Alice
Bob
Carol
Ted
X
10
9
3
5
z
T Score
1.26
62.6
Given: µ = 6.0, σ = 3.18
z 
10  6   1.26
3.18
T  10z  50  101.26  50  62.6
Chapter 4
For Explaining Psychological
Statistics, 4th ed. by B. Cohen
3
SAT Scores
– Transform z scores into SAT scores,
which follow a distribution with mean =
500 and σ = 100, using this formula:
SAT  100 z  500
– In the table below, find the SAT score for
each subject. You can use the z scores
you calculated for the previous slide.
Subj
Alice
Bob
Carol
Ted
X
10
9
3
5
z
1.26
SAT
626
Given: µ = 6.0, σ = 3.18
– Note that, traditionally, T scores have been used
as a way to standardize the scores from
psychological tests, whereas SAT and related
scores have been used by ETS, and other
companies that administer standardized aptitude
and achievement/ability tests.
Chapter 4
For Explaining Psychological
Statistics, 4th ed. by B. Cohen
4
IQ Scores
– You can transform z scores into StanfordBinet IQ scores, which follow a distribution with mean = 100 and σ = 16, using
this formula:
IQ  16 z  100
– In the table below, find the IQ score for
each subject. You can use the z scores
you calculated for the T or SAT scores.
Subj
Alice
Bob
Carol
Ted
X
10
9
3
5
z
1.26
IQ
120.2
Given: µ = 6.0, σ = 3.18
– To find WAIS IQ scores instead, multiply
the z score by 15, instead of 16. For
Alice, the WAIS IQ score would be 118.9.
Chapter 4
For Explaining Psychological
Statistics, 4th ed. by B. Cohen
5
The Normal Distribution
• It is the most commonly used theoretical distribution in psychological
research.
• It is unimodal and symmetric, and
its tails extend infinitely in both the
positive and negative directions.
• It is described as bell-shaped, but it
is defined by a specific mathematical equation.
• It arises frequently, in an approximate form, in nature.
• When the area under the normal
curve is used to determine proportions of the population, the whole
area is defined as 1.0 (or 100%).
Chapter 4
For Explaining Psychological
Statistics, 4th ed. by B. Cohen
6
Parameters of the Normal
Distribution
The normal distribution is actually a
family of distributions each
defined by two parameters: the
mean and the standard deviation.
•
•
•
Two normal distributions can have
different means and the same
standard deviation; the same mean
and different standard deviations; or
they can differ on both parameters.
However, all normal distributions
have the same shape.
The standard normal distribution has
the following parameters: μ = 0, and
σ = 1. It arises whenever you transform all the scores in any normal
distribution into z scores.
Chapter 4
For Explaining Psychological
Statistics, 4th ed. by B. Cohen
7
Areas Under the Normal
Distribution
• There is a table of the standard normal
distribution that tells us the exact area
under the normal curve, as a proportion
of the total, between the mean and any z
score, to two decimal places.
• In the figure below, you can see the
areas corresponding to z scores of 1, 2,
and 3, on both sides of the distribution.
– There is a table to determine the exact
area under the curve
– This area is the probability of a
particular event
z scores
Chapter 4
For Explaining Psychological
Statistics, 4th ed. by B. Cohen
8
Finding the Area Above a
Positive z Score or Below a
Negative z Score:
–
Find the percent area between the
mean and the given z score in Table A of
your text.
–
Subtract that percentage from 100.
–
Divide by 100 to convert to a proportion
or probability.
Finding the Area Below a
Positive z Score or Above a
Negative z Score:
–
–
–
Find the percent area between the
mean and the given z score in Table A of
your text.
Add 50 to that percentage.
Divide by 100 to convert to a proportion
or probability.
Chapter 4
For Explaining Psychological
Statistics, 4th ed. by B. Cohen
9
Finding the z Score Corresponding
to a Given Area in the Positive
Tail of the Normal Distribution
–
–
–
–
Ex., What z score does a student
need in order to be among the highest
12% of scores?
The area you want is not between the
mean and z, it is above z, so subtract
the given percent from 50.00. For this
example, the area you are looking for is
50.00 – 12.00 = 38.00.
Look in the body of Table A for the
percent area that comes closest to the
area calculated in the previous step,
and read off the corresponding z score.
For this example, the z score that cuts
off the top 12% falls midway between
1.17 and 1.18, so the answer is that the
student needs a z score that is at least
1.175 to be in the top 12% of the distribution.
Chapter 4
For Explaining Psychological
Statistics, 4th ed. by B. Cohen
10
Finding the z Score Corresponding
to a Given Area in the Negative
Tail of the Normal Distribution
–
–
–
–
Ex., Below what z score does the
lowest 8% of the normal distribution
fall?
The area you want is not between the
mean and z, it is below z, so subtract
the given percent from 50.00. For this
example, the area you are looking for is
50.00 – 8.00 = 42.00.
For this example, the z score that cuts
off the top 8% falls midway between
1.40 and 1.41, so the answer for the
top 8% would be 1.405.
Because you are looking for the
bottom 8%, place a minus sign in front
of the z score you found in the previous
step – i.e., 8% of the normal distribution
is below a z score of –1.405.
Chapter 4
For Explaining Psychological
Statistics, 4th ed. by B. Cohen
11
Areas in the Middle of the
Normal Distribution
–
It is often useful to know the z
scores that enclose a given
amount of area in the middle of
the normal distribution— e.g., the
middle 80%.
– The area you need to find in the
body of Table A is half of that area
— in this case, 40.00.
–
The z score that comes closest
to containing 40.00% between the
mean and z is 1.28.
– Because of the symmetry of the
normal distribution, the two z
scores that enclose the middle
80% are (approximately) –1.28
and +1.28.
Chapter 4
For Explaining Psychological
Statistics, 4th ed. by B. Cohen
12
•
Sampling Distribution of the
Mean
–
The mean of the sampling
distribution of the mean is the same
as the population mean.
The standard deviation of the
sampling distribution of the mean:
–
standard error of the mean
•
Is smaller than the standard deviation
of the population distribution
X 
•
•
Chapter 4

N
Extreme scores are more likely than
extreme means, so the distribution of
means will be less variable than the
population.
As N increases, sample means are
clustered more closely and the
standard error gets smaller.
For Explaining Psychological
Statistics, 4th ed. by B. Cohen
13
•
Sampling Distribution
Versus Population
Distribution
–
–
–
Chapter 4
The sampling distribution is
normal if the population
distribution is normal.
The sampling distribution will
approach normal even if the
population distribution is not
normal (if N is large enough—
Central Limit Theorem).
Mean will be the same as
population distribution, but the
variability is less (smaller
standard error).
For Explaining Psychological
Statistics, 4th ed. by B. Cohen
14
•
Describing Groups
–
You can use z scores to find
the location of one group with
respect to all other groups of
the same size.
•
•
Must use the sampling
distribution of the mean for
groups.
Need to determine the z score
and then find the area
(proportion).
z
Chapter 4
X 
X
For Explaining Psychological
Statistics, 4th ed. by B. Cohen
15
• Ex., Suppose we know that the
average weight of 6th grade boys is
normally distributed with  = 99 lbs
and  = 4 lbs. What is the
probability of obtaining a sample
mean (with N = 25) of less than 98
lbs?
• Ex., A beverage company sets its
bottling machinery to put a  = 16.5
oz of ice tea in each bottle, with a
 = 0.5 oz. The ice tea is sold by in
6-packs. We assume that each of
the bottles in a 6-pack is a random
sample. What percentage of 6packs have a mean bottle amount
of 17 oz or more?
• These are questions about means,
not individuals.
Chapter 4
For Explaining Psychological
Statistics, 4th ed. by B. Cohen
16