Download Describing Data with z-scores and the Normal

Chapter 5
DESCRIBING DATA WITH Z-SCORES
AND THE NORMAL CURVE
Going Forward
Your goals in this chapter are to learn:
• What a z-score is
• What a z-distribution is and how it indicates a score’s
relative standing
• How the standard normal curve is used with z-scores
to determine relative frequency, simple frequency,
and percentile
• What the sampling distribution of means is and what
the standard error of the mean is
• How to compute z-scores for sample means and then
determine their relative frequency
Understanding z-Scores
A Frequency Distribution
z-Scores
• A z-score is a location on the distribution
• A z-score automatically communicates its
distance from the mean
• A z-score indicates how far a raw score is
above or below the mean when measured in
standard deviations
z-Score Formula
The formula for transforming a raw score into a
z-score is
XX
z
SX
Computing a Raw Score
When a z-score and the associated S X and X
are known, this information can be used to
calculate the original raw score. The formula for
this is
X  ( z )( S X )  X
Using the z-Distribution to
Interpret Scores
A z-Distribution
A z-distribution is the distribution produced by
transforming all raw scores in the data into zscores.
z-Distribution of
Attractiveness Scores
Characteristics of the
z-Distribution
1. A z-distribution always has the same shape
as the raw score distribution
2. The mean of any z-distribution is 0
3. The standard deviation of any z-distribution
is 1
Using the z-Distribution to
Compare Different Variables
Comparison of Distributions
Using the z-Distribution to
Compute Relative Frequency
Relative Frequency
• The relative frequency of a particular z-score
will be the same on all normal z-distributions
• Relative frequency can be computed using the
proportion of the total area under the curve
The Standard Normal Curve
The standard normal curve is a perfect normal
z-distribution that serves as our model of any
approximately normal z-distribution.
Area Under the Standard
Normal Curve
Using the z-Table
Using z-Scores to Describe
Sample Means
Sampling Distribution of Means
A frequency distribution of all possible sample
means occurring when an infinite number of
samples of the same size N are selected from
one raw score distribution is called the
sampling distribution of means.
Central Limit Theorem
The central limit theorem tells us
1. A sampling distribution is always an approximately
normal distribution
2. The mean of the sampling distribution equals the
mean of the underlying raw score population used
to create the sampling distribution
3. The standard deviation of the sampling distribution
is mathematically related to the standard deviation
of the raw score population
Standard Error of the Mean
The standard deviation of the sampling
distribution of means is called the standard
error of the mean. The formula for the true
standard error of the mean is
X 
X
N
z-Score Formula for
a Sample Mean
The formula for computing a z-score for a
sample mean is
z
X 
X
Example
Using the following data set, what is the z-score
for a raw score of 13?
What is the raw score for a z-score of -2?
14
14
13
15
11
15
13
10
12
13
14
13
14
15
17
14
14
15
Example z-Score
XX
z
SX
We know (from prior chapters)
and S X  1.56 for these data.
X  13.67
13  13.67
z
  0.43
1.56
Example
Raw Score from a z-Score
X  ( z )( S X )  X
We know (from prior chapters)
and S X  1.56 for these data.
X  13.67
X  (2)(1.56)  13.67  10.55
Example
z-Score for a Sample Mean
We know X  13.67
. . If m = 12 and
 X = 2.5, what is the z-score for this
sample mean?
X 
X
N
2.5
X 
 0.589
18
z
X 
X
13.67  12
z
 2.84
0.589