Download CHAPTER SIX The Normal Curve, Standardization, and z Scores

CHAPTER SIX
The Normal Curve,
Standardization, and z Scores
NOTE TO INSTRUCTORS
A number of essential statistical concepts are
covered in this chapter, including the two most
important concepts: the standardization process
using z scores and the central limit theorem. In
order to get students more comfortable with
standardization and z scores, it is essential that
they have sufficient practice transforming raw
scores. Make sure you go through several examples
from the textbook during your lecture, and use
Classroom Activity 6-3, “Comparing Apples and
Oranges,” to help teach the material. Students can
find that the central limit theorem is a little
tricky to grasp conceptually. Again, use the
examples from the textbook and the Classroom
Activity 6-5, “Distribution of Scores vs. Means,”
to illustrate this theorem.
OUTLINE OF RESOURCES
I.
The Normal Curve
 Classroom Activity 6-1: Make It Your Own: Normal
Distributions
(p. 50)
 Classroom Activity 6-2: Observing the Normal
Curve (p. 50)
II.
Standardization, z Scores, and the Normal Curve
 Discussion Question 6-1 (p. 51)
 Classroom Activity 6-3: Comparing Apples and
Oranges (p. 51)
 Discussion Question 6-2 (p. 51)
 Classroom Activity 6-4: Make It Your Own: z
Scores (p. 51)
III. The Central Limit Theorem
 Discussion Question 6-3
 Classroom Activity 6-5:
vs. Means (p. 52)
 Discussion Question 6-4
 Discussion Question 6-5
 Discussion Question 6-6
 Classroom Activity 6-6:
Distributions (p. 53)
(p. 52)
Distribution of Scores
(p. 52)
(p. 53)
(p. 53)
Make It Your Own: Normal
IV.
Next Steps: The Normal Curve and Catching Cheaters
 Additional Readings (p. 54)
 Online Resources (p. 54)
V.
Handouts
 Handout 6-1: Observing the Normal Curve (p. 55)
 Handout 6-2: Comparing Apples and Oranges (p. 56)
 Handout 6-3: Distribution of Scores vs. Means (p.
57)
CHAPTER GUIDE
I.
The Normal Curve
1. The normal curve is a specific bell-shaped
curve that is unimodal, symmetric, and defined
mathematically.
Classroom Activity 6-1
Make It Your Own: Normal Distributions
 Use
the height and weight data that were
anonymously collected previously (see Chapter 3,
Classroom Activity 3-1).
 Using
the graph function in SPSS, generate
histograms for each of these variables.
 Ask your students to describe the distribution.
Classroom Activity 6-2
Observing the Normal Curve
 Have
students observe the normal curve in the
world around them.
 For
this activity, it would be best if students
continue to try to find normal curves in their
everyday life outside of the classroom and bring
in their observations to discuss in class.
II.
Standardization, z Scores, and the Normal Curve
1. Standardization converts individual scores to
standard scores for that which we know the
mean, standard deviation, and percentiles. To
standardize, we convert individual scores from
different normal distributions to a shared
normal distribution with a known mean, standard
deviation, and percentiles.
2. To do this, we will convert our raw scores to
z scores. A z score is the number of standard
deviations a particular score is from the mean.
3. The z score is part of the z distribution, a
normal distribution of z scores or standardized
scores.
4. The standardized z distribution allows us to
transform raw scores into standardized scores
called z scores. We can also transform z scores
back into raw scores. The standardization also
allows us to compare z scores to each other
even when the z scores represent raw scores on
different scales. In addition, we can use the z
distribution
to
transform
z
scores
into
percentiles that are more easily understood.
> Discussion Question 6-1
What is the purpose of standardization? How is
standardization useful?
Your students’ answers should include:
 The
purpose of standardization is to compare
scores from different normal distributions to a
shared normal distribution with a known mean,
standard deviation, and percentiles.
 Standardization is useful because it allows us to
compare the raw scores on two different scales by
converting them to z scores.
Classroom Activity 6-3
Comparing Apples and Oranges
 Have
students use Handout 6-2, found at the end
of this chapter, to examine the usefulness of
standardization.
 Have
them also use standardization and the
handout to compare amounts that use different
units.
5. The normal curve is symmetric such that 50% of
scores lie below the mean and 50% lie above the
mean.
6. We also know that 34% of scores fall between
the mean and a z score of 1 as well as between
the mean and a z score of –1.
7. Looking between z scores of 1 and 2 (or
between –1 and –2), we know that 14% of scores
will lie between these z scores.
8. Lastly, 2% of scores fall between the z scores
of 2 and 3 (or –2 and –3).
9. Putting this together, we know that 68% of
scores lie within 1 standard deviation of the
mean, 96% lie within 2 standard deviations of
the mean, and nearly all scores lie within 3
standard deviations of the mean.
> Discussion Question 6-2
What percentage of scores lies within 1 standard deviation
of the mean? within 2? within 3?
Your students’ answers should include:
 The
percentage of scores that lie within 1
standard deviation of the mean is 34%.
 The
percentage of scores that lie within 2
standard deviations of the mean is 14%.
 The
percentage of scores that lie within 3
standard deviations of the mean is 2%.
Classroom Activity 6-4
Make It Your Own: z Scores
 Use the data from Classroom Activities 6-1 and 66 (height, weight, SAT, and GPA data).
 Assume
that your class is the population and
provide the students with the population mean.
 Then
have the students calculate their own z
scores.
III. The Central Limit Theorem
1. The central limit theorem refers to how a
distribution of sample means is a more normal
distribution than a distribution of scores,
even when the population distribution is not
normal.
As
sample
size
increases,
a
distribtuion of sample means more closely
approximates a normal curve.
2. The central limit theorem demonstrates two
important principles. First, repeated sampling
will approximate a normal curve, even when the
original
population
is
not
normally
distributed. Second, a distribution of means is
less variable than a distribution of individual
scores. It is a distribution composed of many
means that are calculated from all possible
samples of a given size, all taken from the
same population.
> Discussion Question 6-3
What is the central limit theorem? What are two principles
of the central limit theorem?
Your students’ answers should include:
 The central limit theorem states that regardless
of the shape of the population, the shape of the
sampling distribution of the mean approximates a
normal curve if the sample size is adequately
large.
 Two
principles of the central limit theorem are
that repeated sampling will more closely resemble
a normal distribution, and that a distribution of
means is less variable than a distribution of raw
scores.
Classroom Activity 6-5
Distribution of Scores vs. Means
 Have
students work in small groups, and give
each group two six-sided dice.
 Have
students observe how the distribution
changes from rolling the two dice a different
number of times.
 Next, have students take a distribution of means
to further observe how distribution changes.
3. The distribution of means (which is symbolized
as µM) is a distribution composed of many means
that are calculated from all possible samples
of a given size, all taken from the same
population.
4. When creating a distribution of means rather
than a distribution of scores, the spread of
the distribution decreases.
> Discussion Question 6-4
What is a distribution of means, and why does the spread of
the distribution decrease with the distribution of means?
Your students’ answers should include:
 A distribution of
means is composed of the means
of all possible random samples, and the mean of
the sampling distribution of the mean always
equals the mean of the population.
 The spread of the distribution decreases with the
distribution
of
means because extreme scores
averaged out by other scores.
are
balanced and
5. In addition, larger sample sizes will shift a
skewed distribution to a normal distribution of
means from that population.
6. The standard error is the name for the
standard deviation of a distribution of means
and is symbolized by M.
7. The standard error is calculated by the
formula: M = (/N).
> Discussion Question 6-5
What is the standard error? How is it calculated?
Your students’ answers should include:
 The
standard error is the standard deviation of
the distribution of means.
 The
standard error is calculated by dividing the
standard deviation of the population by the
square root of the sample size, N.
8. As sample size increases, the mean of a
distribution of means approaches the mean of
the population of individual scores.
9. The standard error is smaller than the
standard deviation of a distribution of scores.
As sample size increases, the standard error
becomes even smaller.
10. The shape of the distribution of means will
have an approximately normal shape if the
distribution of the population of individual
scores has a normal shape or if the size of
each sample that comprises the distribution is
sufficiently large (at least 30).
> Discussion Question 6-6
Why is it that the shape of the distribution of means will
have an approximately normal shape if the size of each
sample that comprises the distribution is sufficiently
large?
Your students’ answers should include:
 The shape of the distribution of means will have
an approximately normal shape if the size of each
sample
that
comprises
the
distribution
is
sufficiently large because of the central limit
theorem, which indicates that the shape of the
sampling distribution of the mean approximates a
normal curve if the sample size is adequately
large.
11. The formula for
distribution
z = (M – M)/M.
the
of
z
statistic
means
for
a
is:
Classroom Activity 6-6
Make It Your Own: Normal Distributions
 Anonymously collect SAT and GPA data.
 Graph the data and have the students discuss the
distributions.
This activity can also be used as a homework
assignment.
IV.
Next Steps: The Normal Curve and Catching Cheaters
1. Statisticians can use principles based on the
normal curve to determine when certain patterns
are extreme as in the case of standardized
testing in the Chicago Public School system or
researchers who “play” with their data to beat
the 5% cutoff.
Additional Readings
Jegerski, J. (1999). Probability distributions
with real social judgment data. In Ludy, B.,
Nodine, B., Ernst, R. M., & Broeker, C. B. (Eds.),
Activities handbook for the teaching of psychology
(Vol. 4, pp. 77-79). Washington, DC: American
Psychological Association.
This chapter provides some useful background
information on teaching the normal curve along
with a classroom activity.
Lehmann, E. L., & Romano, J. P. (2005). Testing
Statistical
Hypotheses
(Springer
Tests
in
Statistics). New York: Springer Science + Business
Media Inc.
First published in 1959, this is a classic text
on hypothesis testing, providing the fundamentals
of the theory of hypothesis testing. The book is
geared toward graduate students and researchers.
Online Resources
For demonstrations of the normal distribution
using different means and standard deviations,
see:
http://www.stat.stanford.edu/~naras/jsm/NormalDens
ity/NormalDensity.html or
http://psych.colorado.edu/~mcclella/java/zcalc.htm
l
PLEASE NOTE: Due to formatting, the Handouts are only available in Adobe
PDF®.