Descriptive Statistics, Histograms, and Normal Approximations
Descriptive Statistics and the Normal Distribution
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Descriptive Statistics and the
Normal Distribution
HPHE 3150Dr. Ayers
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Introduction Review
• Terminology• Reliability• Validity• Objectivity• Formative vs Summative evaluation• Norm- vs Criterion-referenced standards
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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
• Ratioequal units, absolute zero (total absence of
characteristic)• Height, weight
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Summation Notation• is read as "the sum of"
• X is an observed score
• N = the number of observations
• Complete ( ) operations first
• Exponents then * and / then + and -
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Operations Orders
65
26
-5
2 -34
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Summation Notation Practice:Mastery Item 3.2
Scores:3, 1, 2, 2, 4, 5, 1, 4, 3, 5
Determine:∑ X
(∑ X)2
∑ X2
30
900
110
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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?
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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
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Central Tendency
• Meansum scores / # scores
• Median (P50)exact middle of ordered scores
• Modemost frequent score
Where do the scores tend to center?
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• Mean
• Median (P50)
• Mode
Raw scores27551
Rank order12557
• Mean: 4 (20/5)• Median: 5• Mode: 5
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Distribution Shapes Figure 3.2
So what? OUTLIERS
Direction of tail = +/-
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Mean = 11.7 SD = 2.0
Normal DensitySuperimposed
0.0
5.1
.15
.2D
ensi
ty
5 7 9 11 13 15 17 19CRF at Initial Examination (METs)
Based on 15,242 maximal GXT
Distribution of Initial CRF
Kampert, MSSE, Suppl. 2004, p. S135
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Histogram of Skinfold Data
0
10
20
30
40
50
60
10 15 20 25 30 35 40 45 50 55 60 65 70 75 80
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Three Symmetrical Curves Figure 3.3
The difference here is the variability;
Fully normal
More heterogeneous
More homogeneous
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Descriptive Statistics I
• What is the most important thing you learned today?
• What do you feel most confident explaining to a classmate?
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Descriptive Statistics IREVIEW
• 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?
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• Measures of central tendency• 3, 4, 5, 5, 8• Mean (?), median (?), mode (?)
• Distribution shapes
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Variability
• RangeHi – Low scores only (least reliable measure; 2 scores only)
• Variance (s2) inferential statsSpread of scores based on the squared
deviation of each score from meanMost stable measure of variability
• Standard Deviation (S) descriptive statsSquare root of the variance
Most commonly used measure of variability
True Var-iance
Totalvariance
Error
2SS
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Variance (Table 3.2)
The didactic formula
The calculating formula
1
22
nMX
S
1
2
2
2
nnX
XS
4+1+0+1+4=10 10 = 2.5 5-1=4 4
55 - 225 = 55-45=10 = 2.5 5 4 4
4
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Standard Deviation
The square root of the variance
Nearly 100% scores in a normal distribution are captured by the mean + 3 standard deviations
M + S100 + 10
2SS
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The Normal Distribution
M + 1s = 68.26% of observationsM + 2s = 95.44% of observationsM + 3s = 99.74% of observations
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Calculating Standard Deviation
Raw scores37451
∑ 20
Mean: 4
(X-M)-1301-30
S= √20 5
S= √4
S=2
NMXS
2
(X-M)2
1901920
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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
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Descriptive Statistics II
• What is the “muddiest” thing you learned today?
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Descriptive Statistics IIREVIEW
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+10What does this tell you?
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Standard ScoresZ or t
SMXZ
•Set of observations standardized around a given M and standard deviation
•Score transformed based on its magnitude relative 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?
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Standard Scores• Z-scoreM=0, s=1
• T-scoreT = 50 + 10 * (Z)
M=50, s=10
• Percentilep = 50 + Z (%ile)
SMXZ
SMXT
1050
SMXZ
)(50 percentilezp
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Conversion to Standard Scores
Raw scores37451
• Mean: 4• St. Dev: 2
SMXZ
X-M-1 3 0 1-3
Z-.5 1.5 0 .5-1.5 Allows the comparison of
scores using different scales to compare “apples to apples”
SO WHAT? You have a Z score but what
do you do with it? What does it tell you?
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Normal distribution of scores Figure 3.6
99.9
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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+20What does this tell you?Between what values do 95% of the scores in this data set fall?
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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
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p. 51
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Area of normal curve between 1 and 1.5 std dev above the mean
Figure 3.7
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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?
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• 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
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Descriptive Statistics III
• Explain one thing that you learned today to a classmate
• What is the “muddiest” thing you learned today?