Comparisons across normal distributions
Z-Scores
Overview
 Plan for the night
 Z-scores



Definition
Calculation
Use
 Graphing Data/Distributions


Frequencies/Percentages
Charts/Graphs
Last time…
 Last week we covered
 Measures of Central Tendency

Mean, Mode, Median
 Measures of Variability

Range, IQR, SIQR, Standard Deviation
 The most commonly used of the above are Mean (SD)
 These two measures can be combined to further
describe the “position” of a score/datapoint
Is that a good score?
 Mean and SD are useful, but sometimes we need to
make comparisons between different measures
 Example (w/ same units of measure):
 SAT vs. ACT vs. GRE
 10-yd dash time vs. 40-yd dash time
 Free-throw% vs. FG% vs. 3-Point%
 Example (w/different unit of measure):
 ERA vs. WHIP
 VO2max vs. Vertical Jump
 BMI vs. %BodyFat vs. Waist Circumference
Minimal Statistics
 Mean
 SD
m
Describe the “typical” score,
the “spread” of scores, and
the number of cases
Z-scores
 Combine the mean w/ SD to create a new unit of
measurement (Standardizes Scores)
 Clearly identifies a score as above or below the mean
AND expresses a score in units of SD
 Examples:
 z-score = 1.00 (1 SD above mean)
 z-score = -2.00 (2 SD below mean)
Z-score = 1.0: GRAPHICALLY
84% of scores smaller than this
Z=1
Recall – 50% of scores are below the mean + 34% of
scores between the mean and 1 SD above
Calculating z-scores
X  X OR Deviation Score
Ζ
SD
SD
Calculate Z for each of the following
situations:
X  20, SD  3, X  32
X  9, SD  2, X  6
Other features of z-scores
 1) The Mean of a distribution of z-scores = 0
 Recall the mean is the balance point of a distribution,
where deviation scores sum to 0
 A z-score of 0 is equivalent to scoring the mean
Here is our normal distribution example from last week
X = 70
SD = 10
If a subject scored
70, their z-score
would be 0
34.1% 34.1%
13.6%
13.6%
2.3%
2.3%
40
50
60
70
80
90
100
Z = -3
-2
-1
0
1
2
3
Other features of z-scores
 1) The Mean of a distribution of z-scores = 0
 Recall the mean is the balance point of a distribution,
where deviation scores sum to 0
 A z-score of 0 is equivalent to scoring the mean
 2) The SD of a distribution of z-scores = 1
 Since SD is unit of measurement, when the mean is z=0
then the mean + 1 SD = a z-score of 1
Here is our normal distribution example from last week
X = 70
SD = 10
34.1%
What is the z-score of
a subject that got:
80?
50?
100?
34.1%
13.6%
13.6%
2.3%
2.3%
40
50
60
70
80
90
100
Z = -3
-2
-1
0
1
2
3
Other features of z-scores
 1) The Mean of a distribution of z-scores = 0
 Recall the mean is the balance point of a distribution, where
deviation scores sum to 0
 A z-score of 0 is equivalent to scoring the mean
 2) The SD of a distribution of z-scores = 1
 Since SD is unit of measurement, when the mean is z=0 then
the mean + 1 SD = a z-score of 1
 3) A z-score distribution is same shape as raw score
distribution
 Even though you are changing the unit of measurement, this
does not change the “look” of the distribution when plotted
Here is our normal distribution example from last week
34% of scores still
fall between 0 and 1
z-score
X = 70
SD = 10
34.1% 34.1%
13.6%
13.6%
2.3%
2.3%
40
50
60
70
80
90
100
Z = -3
-2
-1
0
1
2
3
Z-score Comparison
 As stated, z-scores standardize different distributions
allowing you to make comparisons regardless of the
unit of measure
 Bart’s score
 SAT Exam 450 (mean 500, SD 100)
 Lisa’s score
 ACT Exam 24 (mean 18, SD 6)
 Who scored higher?
Bart: (450 – 500)/100 = - 0.5
Lisa: (24 – 18)/6 = 1
Z-scores & the normal curve
 For any z-score, we can calculate the percentage of
scores between it and the mean; all scores below it
& all above it
 Tons of online calculators:

http://www.measuringusability.com/normal_curve.php
Example: Mean BMI and WC in
elementary school boys
What upper and lower limits include 95% of BMI scores?
If one boy’s BMI is 22 kg/m2 and another’s WC is 70 cm,
which of the two has the highest adiposity?
Nomenclature/Terminology
 Frequency: number of cases or subjects or occurrences
in a distribution
 Represented with f
 i.e. f = 12 for a score of 25
 12 occurrences of 25 in the sample
Nomenclature/Terminology
 Percentage: Number of cases or subjects or
occurrences expressed per 100
 Represented with P or %
 Ex. f=12 for a score of 25 when n=25
 P = 12/25*100 = 48% (of scores were 25)
Warning
 Should report the f when presenting percentages
 i.e. 80% of the elementary students came from a family
with an income < $25,000

different interpretation if n=5 compared to n=100
 Reported in literature as
 f = 4 (80%) OR
 80% (f = 4) OR
 80% (n = 4)
Numerator Monster
Pantagraph reported that State Farm paid out over 1
Billion in dividends to customers in the United States
Pantagraph, 6/13/00
Numerator Monster
How much do you pay in car insurance every 6 months?
So…how much is State Farm keeping?
Frequency Distributions
 Graphically displaying the data should ALWAYS come
before any type of statistical analysis
 Measures of central tendency and variability will give
you a feeling for the distribution of the data – but it’s
always easier to visually examine it
 Check for normality (are data normally distributed?)
 Check for outliers (are any subjects sticking out as odd?)
 Check of potential associations (might two variables
relate to each other?)
Frequency Distribution of Math Test
Scores: SPSS Output
t
,
m
u
l
P
r
u
c
c
e
e
e
e
V
2
4
1
0
0
0
2
5
1
0
0
1
2
8
2
1
1
1
2
9
2
1
1
2
3
0
1
0
0
2
3
1
1
0
0
2
3
2
3
1
1
3
3
3
1
0
0
4
3
4
6
2
2
5
3
5
3
1
1
6
3
6
4
1
1
8
3
7
8
2
2
0
T
3
0
0
 40 items on
exam
 Most students
>34
 skewed (more
scores at one
end of the
scale)
Cumulative frequencies &,
Cumulative percentages
t
,
m
 Cumulative
u
l
P
r
u
c
c
e
e
e
e
V
2
4
1
0
0
0
2
5
1
0
0
1
2
8
2
1
1
1
2
9
2
1
1
2
3
0
1
0
0
2
3
1
1
0
0
2
3
2
3
1
1
3
3
3
1
0
0
4
3
4
6
2
2
5
3
5
3
1
1
6
3
6
4
1
1
8
3
7
8
2
2
0
T
3
0
0
Percentage: how
many subjects at
and below a given
score?
 i.e., 33.3% of
students scored a 32
or lower
Eyeball check of data: Intro to
(brute force) graphing with SPSS
 Stem and Leaf Plot: quick viewing of data
distribution
 Boxplot: visual representation of many of the
descriptive statistics discussed last week
 Bar Chart: frequency of all cases
 Histogram: malleable bar chart
 Scatterplot: displays all cases based on two
values of interest (X & Y)
 Note: compare to our previous discussion of
distributions (normal, positively skewed, etc…)
Stem and Leaf Plots
Frequency Stem & Leaf
2.00
Extremes (=<25.0)
2.00
28 . 00
2.00
29 . 00
1.00
30 . 0
1.00
31 . 0
3.00
32 . 000
1.00
33 . 0
6.00
34 . 000000
3.00
35 . 000
4.00
36 . 0000
8.00
37 . 00000000
Stem width: 1
Each leaf:
1 case
 Fast look at shape of
distribution
 shows f numerically
& graphically
 stem is value, leaf is f
Stem and Leaf Plots
 Another way of doing
a stemplot
 Babe Ruth’s home
runs in each of 14
seasons with the NY
Yankees
 54, 59, 35, 41, 46, 25,
47, 60, 54, 46, 49, 46,
41, 34, 22
2
3
4
5
6
25
45
1166679
449
0
Stem and Leaf Plots
 Back-to-back stem
plots allow you to
visualize two data
sets at the same time
 Babe Ruth vs. Roger
Maris
Maris
8
643
863
93
1
Ruth
0
1
2
3
4
5
6
25
45
1166679
449
0
Boxplots
180
Maximum
160
Q3
140
Median
Q1
120
Minimum
100
80
N=
16
Weight (in pounds)
Note: we can also do sideby-side boxplots for a visual
comparison of data sets
Format of Bar Chart
Y axis
f
X axis
scores/categories
Test score data as Bar Chart
10
8
6
4
Count
2
0
24
25
28
29
math test, max = 40
30
31
32
33
34
35
36
37
Format of Histogram (similar to Bar)
Y axis
f
Can be
manipulated
X axis
scores/categories
Test score data as Histogram
10
8
6
4
2
Std. Dev = 3.62
Mean = 33.4
N = 33.00
0
24.0
26.0
28.0
math test, max = 40
30.0
32.0
34.0
36.0
38.0
Test score data as Revised Histogram
14
12
10
8
6
4
Std. Dev = 3.62
2
Mean = 33.4
N = 33.00
0
24.6
27.8
math test, max = 40
31.0
34.2
37.4
Scatterplot


Quick way to visualize
the data & see trends,
patterns, etc…
This plot visually shows
the relationship
between BMI and WC
in a group of
elementary school boys
Scatterplot
Somebody shook their
pedometer for 2 hours a day…

Here’s the
relationship
between females
Steps/day and
waist
circumference
Scatterplot


Outlier removed
This will impact
any statistical
tests you run
(correlations,
regression,
etc…)
Take home message
 Z-scores:
 A simple combination of Mean and SD
 Allow comparisons regardless of unit of measurement
 Always plot your data first!
 Descriptive statistics (like Mean/SD) are generally
presented along with graphical representations of the
distribution
 A histogram (for single variable) and scatterplot (for
paired variables) are most commonly used

Check for outliers! Is the value plausible?
Upcoming…
 Homework = Cronk 3.5 & all of Chapter 4
 Blackboard description upcoming
 We will examine relationships between variables
next week
 Think about those scatterplots…do statistical
relationships exist between those variables? How
strong? In what direction?
 In-class activity 3…