HPHE 3150
Dr. Ayers

Introduction Review
• Terminology
• Reliability
•
Validity
• Objectivity
•
Formative vs Summative evaluation
• Norm- vs Criterion-referenced standards

Scales of Measurement
• Nominal
• name or classify
•
Major, gender, yr in college
•
Ordinal
• order or rank
• Sports rankings
• Continuous
• Interval equal units, arbitrary zero
• Temperature, SAT/ACT score
• Ratio equal units, absolute zero (total absence of characteristic)
• Height, weight

Summation Notation
• S is read as "the sum of"
•
X is an observed score
• N = the number of observations
•
Complete ( ) operations first
• Exponents then * and / then + and -

Operations Orders
65
26
-5
4 2 -3

Summation Notation Practice:
Mastery Item 3.2
Scores:
3, 1, 2, 2, 4, 5, 1, 4, 3, 5
Determine:
∑
X
(
∑
X) 2
∑
X 2
30
900
110

Percentile
•
The percent of observations that fall at or below a given point
•
Range from 0% to 100%
•
Allows normative performance comparisons
If I am @ the 90 th percentile, how many folks did better than me?

Test Score Frequency Distribution
Figure 3.1
(p.42 explanation)
53
54
55
Total
48
49
50
51
52
Valid
41
43
44
45
46
47
1
65
3
2
7
6
11
8
3
Frequency
1
3
3
5
5
7
16.9
12.3
10.8
9.2
4.6
4.6
3.1
1.5
100.0
Percent
1.5
4.6
4.6
7.7
7.7
10.8
Valid Percent
1.5
4.6
4.6
7.7
7.7
10.8
16.9
12.3
10.8
9.2
4.6
4.6
3.1
1.5
100.0
Cumulative Percent
1.5
6.2
10.8
18.5
26.2
36.9
53.8
66.2
76.9
86.2
90.8
95.4
98.5
100.0

Central Tendency
Where do the scores tend to center?
• Mean sum scores / # scores
• Median (P
50
) exact middle of ordered scores
• Mode most frequent score

• Mean
• Median
(P
50
)
• Mode
Raw scores
2
7
5
5
1
5
5
7
Rank order
1
2
•
Mean: 4 (20/5)
•
Median: 5
•
Mode: 5

Distribution Shapes
Figure 3.2
So what? OUTLIERS
Direction of tail = +/-

Distribution of Initial CRF
Mean = 11.7
SD = 2.0
Normal Density
Superimposed
5 7 9 11 13 15
CRF at Initial Examination (METs)
17
Based on 15,242 maximal GXT
19
Kampert, MSSE, Suppl. 2004, p. S135

Histogram of Skinfold Data
60
50
40
30
20
10
0
10 15 20 25 30 35 40 45 50 55 60 65 70 75 80

Three Symmetrical Curves
Figure 3.3
The difference here is the variability;
Fully normal
More heterogeneous
More homogeneous

Descriptive Statistics I
•
What is the most important thing you learned today?
•
What do you feel most confident explaining to a classmate?

Descriptive Statistics I
REVIEW
•
Measurement scales
•
Nominal, Ordinal, Continuous (interval, ratio)
•
Summation Notation:
3, 4, 5, 5, 8 Determine:
∑
X, (
∑
X) 2 , ∑
X 2
9+16+25+25+64 25 625 139
•
Percentiles: so what?

•
Measures of central tendency
•
3, 4, 5, 5, 8
•
Mean (?), median (?), mode (?)
•
Distribution shapes

Variability
•
Range
Hi – Low scores only (least reliable measure; 2 scores only )
•
Variance (s 2 ) inferential stats
Spread of scores based on the squared deviation of each score from mean
Most stable measure of variability
Error
True
Variance
Total variance
•
Standard Deviation (S) descriptive stats
Square root of the variance
S

Most commonly used measure of variability
S
2

Variance
(Table 3.2)
The didactic formula
S
2 
 
X n


1
M

2
4+1+0+1+4=10 10 = 2.5
5-1=4 4
The calculating formula
S
2 

X
2
 
2
 n

1 n
55 - 225 = 55-45=10 = 2.5
5 4 4
4

Standard Deviation
The square root of the variance
S

S
2
Nearly 100% scores in a normal distribution are captured by the mean + 3 standard deviations
M + S
100 + 10

The Normal Distribution
M + 1s = 68.26% of observations
M + 2s = 95.44% of observations
M + 3s = 99.74% of observations

Calculating Standard Deviation
S
 

X

M

2
N Raw scores
3
7
4
5
1
∑
20
(XM )
-1
3
0
1
-3
0
Mean: 4
(XM ) 2
1
9
0
1
9
20
S= √20
5
S= √4
S=2

Coefficient of Variation (V)
Relative variability
Relative variability around the mean OR determine homogeneity of two data sets with different units S / M
Relative variability accounted for by the mean when units of measure are different (ht, hr, running speed, etc.)
Helps more fully describe different data sets that have a common std deviation (S) but unique means (M)
Lower V=mean accounts for most variability in scores
.1 - .2=homogeneous >.5=heterogeneous

Descriptive Statistics II
• What is the “muddiest” thing you learned today?

Descriptive Statistics II
REVIEW
Variability
•
Range
•
Variance : Spread of scores based on the squared deviation of each score from mean Most stable measure
•
Standard deviation Most commonly used measure
Coefficient of variation
•
Relative variability around the mean (homogeneity of scores)
•
Helps more fully describe relative variability of different data sets
50+10
What does this tell you?

Standard Scores
Z or t
•
Set of observations standardized around a given M and standard deviation
Z

X

•
S scores in the group
M
• Converting scores to Z scores expresses a score’s distance from its own mean in sd units
• Use of standard scores: determine composite scores from different measures (bball: shoot, dribble); weight?

• Z-score
M=0, s=1
Standard Scores
•
T-score
T = 50 + 10 * (Z)
M=50, s=10
•
Percentile p = 50 + Z (%ile)
Z

X

M
S
T

50

10

X

M

S p
Z


50
X
 z  (
S
)

Conversion to Standard Scores
Raw scores
3
7
4
5
1
•
Mean: 4
•
St. Dev: 2
0
1
-3
X-M
-1
3
Z
-.5
1.5
0
.5
-1.5
Z

X

M
S
SO WHAT?
You have a Z score but what do you do with it? What does it tell you?
Allows the comparison of scores using different scales to compare “apples to apples”

Normal distribution of scores
Figure 3.6
99.9

Descriptive Statistics II
REVIEW
Standard Scores
• Converting scores to Z scores expresses a score’s distance from its own mean in sd units
• Value?
Coefficient of variation
•
Relative variability around the mean ( homogeneity of scores )
•
Helps more fully describe relative variability of different data sets
100+20
What does this tell you?
Between what values do 95% of the scores in this data set fall?

Normal-curve Areas
Table 3.4
• Z scores are on the left and across the top
•
Z=1.64: 1.6 on left , .04 on top=44.95
•
Since 1.64 is +, add 44.95 to 50 (mean) for 95 th percentile
•
Values in the body of the table are percentage between the mean and a given standard deviation distance
• ½ scores below mean, so + 50 if Z is +/-
• The "reference point" is the mean
•
+Z=better than the mean
• -Z=worse than the mean

p. 51

Area of normal curve between 1 and
1.5 std dev above the mean
Figure 3.7

Normal curve practice
•
Z score Z = (X-M)/S
• T score T = 50 + 10 * (Z)
• Percentile P = 50 + Z percentile ( +: add to 50, -: subtract from 50 )
• Raw scores
•
Hints
•
Draw a picture
• What is the z score?
• Can the z table help?

•
Assume M=700, S=100
Percentile
64
T score
53.7
43
z score
.37
–1.23
17
68
68
.57
Raw score
737
618
835

Descriptive Statistics III
•
Explain one thing that you learned today to a classmate
• What is the “muddiest” thing you learned today?