Transcript Descriptive Statistics

Descriptive Statistics
and the
Normal Distribution
HPHE 3150
Dr. Ayers
Introduction Review
• Terminology
•
•
•
•
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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
30
(∑ X)2
900
∑ X2
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 90th percentile,
how many folks did better than me?
Test Score Frequency Distribution
Figure 3.1 (p.42 explanation)
Valid
Frequency
Percent
Valid Percent
Cumulative Percent
41
1
1.5
1.5
1.5
43
3
4.6
4.6
6.2
44
3
4.6
4.6
10.8
45
5
7.7
7.7
18.5
46
5
7.7
7.7
26.2
47
7
10.8
10.8
36.9
48
11
16.9
16.9
53.8
49
8
12.3
12.3
66.2
50
7
10.8
10.8
76.9
51
6
9.2
9.2
86.2
52
3
4.6
4.6
90.8
53
3
4.6
4.6
95.4
54
2
3.1
3.1
98.5
55
1
1.5
1.5
100.0
Total
65
100.0
100.0
Central Tendency
Where do the scores tend to center?
• Mean
sum scores / # scores
• Median (P50)
exact middle of ordered scores
• Mode
most frequent score
• Mean
• Median
(P50)
Raw scores
2
7
5
5
1
Rank order
1
2
5
5
7
• Mode
• Mean: 4
• Median: 5
• Mode: 5
(20/5)
Distribution Shapes
Figure 3.2
So what? OUTLIERS
Direction of tail = +/-
.2
Distribution of Initial CRF
Normal Density
Superimposed
0
.05
.1
.15
Mean = 11.7
SD = 2.0
5
7
9
11
13
15
17
CRF at Initial Examination (METs)
19
Based on 15,242 maximal GXT
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, ∑X2
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 (s2) 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
Most commonly used measure of variability
S 
S2
Variance (Table 3.2)
The didactic formula
S
2

 X
 M
2
n  1
4+1+0+1+4=10
5-1=4
10 = 2.5
4
The calculating formula
 X 
2
S2 
X
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
Raw scores (X-M)
3
-1
7
3
4
0
5
1
1
-3
∑ 20
0
Mean: 4
(X-M)2
1
9
0
1
9
20
S 

S= √20
5
S= √4
S=2
X  M 
2
N
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
• Standard deviation
Most stable measure
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
X M
Z

•Score transformed based on its magnitude relative
S to other
scores in the group
•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?
Standard Scores
• Z-score
M=0, s=1
• T-score
T = 50 + 10 * (Z)
M=50, s=10
• Percentile
p = 50 + Z (%ile)
X M
Z 
S
10  X  M 
T  50 
S
p  50X 
z ( percentile
)
M
Z 
S
Conversion to Standard Scores
Raw scores
3
7
4
5
1
• Mean: 4
• St. Dev: 2
X-M
-1
3
0
1
-3
Z
-.5
1.5
0
.5
-1.5
X M
Z 
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 95th 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
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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
T score
z score
Raw score
64
53.7
.37
737
43
–1.23
618
17
68
68
835
.57
Descriptive Statistics III
• Explain one thing that you learned today
to a classmate
• What is the “muddiest” thing you learned
today?