Kin 304 Descriptive Statistics & the Normal Distribution Normal Distribution Measures of Central...
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![Page 1: Kin 304 Descriptive Statistics & the Normal Distribution Normal Distribution Measures of Central Tendency Measures of Variability Percentiles Z-Scores.](https://reader033.fdocuments.net/reader033/viewer/2022061616/56649e8f5503460f94b92b74/html5/thumbnails/1.jpg)
Kin 304Descriptive Statistics & the Normal Distribution
Normal DistributionMeasures of Central TendencyMeasures of VariabilityPercentiles
Z-ScoresArbitrary Scores & ScalesSkewnes & Kurtosis
![Page 2: Kin 304 Descriptive Statistics & the Normal Distribution Normal Distribution Measures of Central Tendency Measures of Variability Percentiles Z-Scores.](https://reader033.fdocuments.net/reader033/viewer/2022061616/56649e8f5503460f94b92b74/html5/thumbnails/2.jpg)
Descriptive Statistics
Descriptive
Describe the characteristics of your sample Descriptive statistics
– Measures of Central Tendency Mean, Mode, Median
– Skewness, Kurtosis
– Measures of Variability Standard Deviation & Variance
– Percentiles, Scores in comparison to the Normal distribution
![Page 3: Kin 304 Descriptive Statistics & the Normal Distribution Normal Distribution Measures of Central Tendency Measures of Variability Percentiles Z-Scores.](https://reader033.fdocuments.net/reader033/viewer/2022061616/56649e8f5503460f94b92b74/html5/thumbnails/3.jpg)
Descriptive Statistics
Normal Frequency Distribution
-4 -3 -2 -1 0 1 2 3 4
34.13% 34.13%
13.59%13.59%2.15% 2.15%
Mean Mode Median
68.26%
95.44%
![Page 4: Kin 304 Descriptive Statistics & the Normal Distribution Normal Distribution Measures of Central Tendency Measures of Variability Percentiles Z-Scores.](https://reader033.fdocuments.net/reader033/viewer/2022061616/56649e8f5503460f94b92b74/html5/thumbnails/4.jpg)
Descriptive Statistics
Measures of Central Tendency
When do you use mean, mode or median?– Height– Skinfolds (positively skewed)– House prices in Vancouver– Measurements (criterion value)
Objective testAnthropometryVertical jump100m run time
![Page 5: Kin 304 Descriptive Statistics & the Normal Distribution Normal Distribution Measures of Central Tendency Measures of Variability Percentiles Z-Scores.](https://reader033.fdocuments.net/reader033/viewer/2022061616/56649e8f5503460f94b92b74/html5/thumbnails/5.jpg)
Descriptive Statistics
Measures of Variability
Standard DeviationVariance = Standard Deviation2
Range (approx. = ±3 SDs)Nonparametric
– Quartiles (25%ile, 75%ile)– Interquartile distance
![Page 6: Kin 304 Descriptive Statistics & the Normal Distribution Normal Distribution Measures of Central Tendency Measures of Variability Percentiles Z-Scores.](https://reader033.fdocuments.net/reader033/viewer/2022061616/56649e8f5503460f94b92b74/html5/thumbnails/6.jpg)
Descriptive Statistics
Central Limit Theorem
If a sufficiently large number of random samples of the same size were drawn from an infinitely large population, and the mean (average) was computed for each sample, the distribution formed by these averages would be normal.
![Page 7: Kin 304 Descriptive Statistics & the Normal Distribution Normal Distribution Measures of Central Tendency Measures of Variability Percentiles Z-Scores.](https://reader033.fdocuments.net/reader033/viewer/2022061616/56649e8f5503460f94b92b74/html5/thumbnails/7.jpg)
Using the Normal Distribution
Standard Error of the Mean
SEMSD
n
Describes how confident you are that the mean of the sample is the mean of the population
![Page 8: Kin 304 Descriptive Statistics & the Normal Distribution Normal Distribution Measures of Central Tendency Measures of Variability Percentiles Z-Scores.](https://reader033.fdocuments.net/reader033/viewer/2022061616/56649e8f5503460f94b92b74/html5/thumbnails/8.jpg)
Using the Normal Distribution
Z- Scores
s
XXZ
)( Score = 24
Norm Mean = 30
Norm SD = 4
Z-score = (24 – 30) / 4 = -1.5
![Page 9: Kin 304 Descriptive Statistics & the Normal Distribution Normal Distribution Measures of Central Tendency Measures of Variability Percentiles Z-Scores.](https://reader033.fdocuments.net/reader033/viewer/2022061616/56649e8f5503460f94b92b74/html5/thumbnails/9.jpg)
Using the Normal Distribution
Internal or External Norm
Internal Norm
A sample of subjects are measured. Z-scores are calculated based upon mean and sd of the sample.
Mean = 0, sd = 1
External Norm
A sample of subjects are measured. Z-scores are calculated based upon mean and sd of an external normative sample (national, sport specific etc.)
Mean = ?, sd = ? (depends upon how the sample differs from the external norm.
![Page 10: Kin 304 Descriptive Statistics & the Normal Distribution Normal Distribution Measures of Central Tendency Measures of Variability Percentiles Z-Scores.](https://reader033.fdocuments.net/reader033/viewer/2022061616/56649e8f5503460f94b92b74/html5/thumbnails/10.jpg)
Testing Normality
Standardizing Data
Transforming data into standard scores Useful in eliminating units of measurements
HT
186.0182.0
178.0174.0
170.0166.0
162.0158.0
154.0150.0
146.0
800
600
400
200
0
Std. Dev = 6.22
Mean = 161.0
N = 5782.00
Zscore(HT)
4.003.50
3.002.50
2.001.50
1.00.50
0.00-.50
-1.00-1.50
-2.00-2.50
700
600
500
400
300
200
100
0
Std. Dev = 1.00
Mean = 0.00
N = 5782.00
![Page 11: Kin 304 Descriptive Statistics & the Normal Distribution Normal Distribution Measures of Central Tendency Measures of Variability Percentiles Z-Scores.](https://reader033.fdocuments.net/reader033/viewer/2022061616/56649e8f5503460f94b92b74/html5/thumbnails/11.jpg)
Testing Normality
Standardizing Data
Standardizing does not change the distribution of the data.
WT
125.0120.0
115.0110.0
105.0100.0
95.090.0
85.080.0
75.070.0
65.060.0
55.050.0
45.0
700
600
500
400
300
200
100
0
Std. Dev = 11.14
Mean = 61.9
N = 5704.00
Zscore(WT)
5.505.00
4.504.00
3.503.00
2.502.00
1.501.00
.500.00
-.50-1.00
-1.50
800
600
400
200
0
Std. Dev = 1.00
Mean = 0.00
N = 5704.00
![Page 12: Kin 304 Descriptive Statistics & the Normal Distribution Normal Distribution Measures of Central Tendency Measures of Variability Percentiles Z-Scores.](https://reader033.fdocuments.net/reader033/viewer/2022061616/56649e8f5503460f94b92b74/html5/thumbnails/12.jpg)
Z-scores allow measurements from tests with different units to be combined. But beware: Higher z-scores are not necessarily better performances
Variablez-scores for
profile Az-scores for
profile B
Sum of 5 Skinfolds (mm) 1.5 -1.5*
Grip Strength (kg) 0.9 0.9
Vertical Jump (cm) -0.8 -0.8
Shuttle Run (sec) 1.2 -1.2*
Overall Rating 0.7 -0.65
*Z-scores are reversed because lower skinfold and shuttle run scores are regarded as better performances
![Page 13: Kin 304 Descriptive Statistics & the Normal Distribution Normal Distribution Measures of Central Tendency Measures of Variability Percentiles Z-Scores.](https://reader033.fdocuments.net/reader033/viewer/2022061616/56649e8f5503460f94b92b74/html5/thumbnails/13.jpg)
-1 0 1 2
Sum of 5Skinfolds
(mm)
GripStrength
(kg)
VerticalJump(cm)
ShuttleRun (sec)
OverallRating
z-scores
-2 -1 0 1 2
Sum of 5Skinfolds
(mm)
GripStrength
(kg)
VerticalJump(cm)
ShuttleRun (sec)
OverallRating
z-scores
Test Profile A Test Profile B
![Page 14: Kin 304 Descriptive Statistics & the Normal Distribution Normal Distribution Measures of Central Tendency Measures of Variability Percentiles Z-Scores.](https://reader033.fdocuments.net/reader033/viewer/2022061616/56649e8f5503460f94b92b74/html5/thumbnails/14.jpg)
Descriptive Statistics
Percentile: The percentage of the population that lies at or below that score
-4 -3 -2 -1 0 1 2 3 4
34.13% 34.13%
13.59%13.59%2.15% 2.15%
Mean Mode Median
68.26%
95.44%
![Page 15: Kin 304 Descriptive Statistics & the Normal Distribution Normal Distribution Measures of Central Tendency Measures of Variability Percentiles Z-Scores.](https://reader033.fdocuments.net/reader033/viewer/2022061616/56649e8f5503460f94b92b74/html5/thumbnails/15.jpg)
Using the Normal Distribution
Cumulative Frequency Distribution
![Page 16: Kin 304 Descriptive Statistics & the Normal Distribution Normal Distribution Measures of Central Tendency Measures of Variability Percentiles Z-Scores.](https://reader033.fdocuments.net/reader033/viewer/2022061616/56649e8f5503460f94b92b74/html5/thumbnails/16.jpg)
Area under the Standard Normal Curve
What percentage of the population is above or below a given z-score or between two given z-scores?
Percentage between 0 and -1.5
43.32%
Percentage above -1.5
50 + 43.32% = 93.32%
-4 -3 -2 -1 0 1 2 3 4
![Page 17: Kin 304 Descriptive Statistics & the Normal Distribution Normal Distribution Measures of Central Tendency Measures of Variability Percentiles Z-Scores.](https://reader033.fdocuments.net/reader033/viewer/2022061616/56649e8f5503460f94b92b74/html5/thumbnails/17.jpg)
Using the Normal Distribution
Predicting Percentiles from Mean and sdassuming a normal distribution
PercentileZ-score for
Percentile
Predicted Percentile value
based upon Mean = 170Sd = 10
5 -1,645 153.55
25 -0.675 163.25
50 0 170
75 +0.675 176.75
95 +1.645 186.45
![Page 18: Kin 304 Descriptive Statistics & the Normal Distribution Normal Distribution Measures of Central Tendency Measures of Variability Percentiles Z-Scores.](https://reader033.fdocuments.net/reader033/viewer/2022061616/56649e8f5503460f94b92b74/html5/thumbnails/18.jpg)
Using the Normal Distribution
T-scores– Mean = 50, sd
= 10
Hull scores– Mean = 50, sd
= 14
Arbitrary Scores & Scales
![Page 19: Kin 304 Descriptive Statistics & the Normal Distribution Normal Distribution Measures of Central Tendency Measures of Variability Percentiles Z-Scores.](https://reader033.fdocuments.net/reader033/viewer/2022061616/56649e8f5503460f94b92b74/html5/thumbnails/19.jpg)
Using the Normal Distribution
All Scores based upon z-scores
Z-score = +1.25
T-Score = 50 + (+1.25 x 10) = 62.5
Hull Score = 50 + (+1.25 x 14) = 67.5
Z-score = -1.25
T-Score = 50 + (-1.25 x 10) = 37.5
Hull Score = 50 + (-1.25 x 14) = 32.5
![Page 20: Kin 304 Descriptive Statistics & the Normal Distribution Normal Distribution Measures of Central Tendency Measures of Variability Percentiles Z-Scores.](https://reader033.fdocuments.net/reader033/viewer/2022061616/56649e8f5503460f94b92b74/html5/thumbnails/20.jpg)
![Page 21: Kin 304 Descriptive Statistics & the Normal Distribution Normal Distribution Measures of Central Tendency Measures of Variability Percentiles Z-Scores.](https://reader033.fdocuments.net/reader033/viewer/2022061616/56649e8f5503460f94b92b74/html5/thumbnails/21.jpg)
Skewness & Kurtosis
Skewness is a measure of symmetry, or more accurately, the lack of symmetry. A distribution, or data set, is symmetric if it looks the same to the left and right of the center point.
Kurtosis is a measure of whether the data are peaked or flat relative to a normal distribution. That is, data sets with a high kurtosis tend to have a distinct peak near the mean, decline rather rapidly, and have heavy tails. Data sets with low kurtosis tend to have a flat top near the mean rather than a sharp peak. A uniform distribution would be the extreme case
![Page 22: Kin 304 Descriptive Statistics & the Normal Distribution Normal Distribution Measures of Central Tendency Measures of Variability Percentiles Z-Scores.](https://reader033.fdocuments.net/reader033/viewer/2022061616/56649e8f5503460f94b92b74/html5/thumbnails/22.jpg)
Many variables in measured in BPK are positively skewed
Skewness
Kurtosis
![Page 23: Kin 304 Descriptive Statistics & the Normal Distribution Normal Distribution Measures of Central Tendency Measures of Variability Percentiles Z-Scores.](https://reader033.fdocuments.net/reader033/viewer/2022061616/56649e8f5503460f94b92b74/html5/thumbnails/23.jpg)
Coefficient of Skewness
Where: X = mean, N = sample size, s = standard deviation
Normal Distribution: Skewness = 0
Significant skewness: >1 or <-1
31
3
)1(
)(
sN
XXskewness
N
i i
![Page 24: Kin 304 Descriptive Statistics & the Normal Distribution Normal Distribution Measures of Central Tendency Measures of Variability Percentiles Z-Scores.](https://reader033.fdocuments.net/reader033/viewer/2022061616/56649e8f5503460f94b92b74/html5/thumbnails/24.jpg)
Coefficient of Kurtosis
41
4
)1(
)(
sN
XXkurtosis
N
i i
Where: X = mean, N = sample size, s = standard deviation
![Page 25: Kin 304 Descriptive Statistics & the Normal Distribution Normal Distribution Measures of Central Tendency Measures of Variability Percentiles Z-Scores.](https://reader033.fdocuments.net/reader033/viewer/2022061616/56649e8f5503460f94b92b74/html5/thumbnails/25.jpg)
HT
186.0182.0
178.0174.0
170.0166.0
162.0158.0
154.0150.0
146.0
Height (Women)800
600
400
200
0
Std. Dev = 6.22
Mean = 161.0
N = 5782.00
![Page 26: Kin 304 Descriptive Statistics & the Normal Distribution Normal Distribution Measures of Central Tendency Measures of Variability Percentiles Z-Scores.](https://reader033.fdocuments.net/reader033/viewer/2022061616/56649e8f5503460f94b92b74/html5/thumbnails/26.jpg)
WT
125.0120.0
115.0110.0
105.0100.0
95.090.0
85.080.0
75.070.0
65.060.0
55.050.0
45.0
Weight (Women)700
600
500
400
300
200
100
0
Std. Dev = 11.14
Mean = 61.9
N = 5704.00
![Page 27: Kin 304 Descriptive Statistics & the Normal Distribution Normal Distribution Measures of Central Tendency Measures of Variability Percentiles Z-Scores.](https://reader033.fdocuments.net/reader033/viewer/2022061616/56649e8f5503460f94b92b74/html5/thumbnails/27.jpg)
Descriptive Statistics
5704 61.9210 .1474 11.1361 1.297 .032 2.643 .065
5782 161.0457 8.183E-02 6.2225 .092 .032 .090 .064
5362 75.7820 .3961 29.0066 1.043 .033 1.299 .067
5347
WT
HT
S5SF
Valid N (listwise)
Statistic Statistic Std. Error Statistic Statistic Std. Error Statistic Std. Error
N Mean Std.Deviation
Skewness Kurtosis
S5SF
220.0200.0
180.0160.0
140.0120.0
100.080.0
60.040.0
20.0
Sum of 5 Skinfolds (Women)1000
800
600
400
200
0
Std. Dev = 29.01
Mean = 75.8
N = 5362.00
![Page 28: Kin 304 Descriptive Statistics & the Normal Distribution Normal Distribution Measures of Central Tendency Measures of Variability Percentiles Z-Scores.](https://reader033.fdocuments.net/reader033/viewer/2022061616/56649e8f5503460f94b92b74/html5/thumbnails/28.jpg)
Testing Normality
Normal Probability Plots
Correlation of observed with expected cumulative probability is a measure of the deviation from normal
Normal P-P Plot of WT
Observed Cum Prob
1.00.75.50.250.00
Exp
ect
ed
Cu
m P
rob
1.00
.75
.50
.25
0.00
Normal P-P Plot of HT
Observed Cum Prob
1.00.75.50.250.00
Exp
ect
ed
Cu
m P
rob
1.00
.75
.50
.25
0.00
![Page 29: Kin 304 Descriptive Statistics & the Normal Distribution Normal Distribution Measures of Central Tendency Measures of Variability Percentiles Z-Scores.](https://reader033.fdocuments.net/reader033/viewer/2022061616/56649e8f5503460f94b92b74/html5/thumbnails/29.jpg)
T-scores and Osteoporosis
To diagnose osteoporosis, clinicians measure a patient’s bone mineral density (BMD) and then express the patient’s BMD in terms of standard deviations above or below the mean BMD for a “young normal” person of the same sex and ethnicity.
![Page 30: Kin 304 Descriptive Statistics & the Normal Distribution Normal Distribution Measures of Central Tendency Measures of Variability Percentiles Z-Scores.](https://reader033.fdocuments.net/reader033/viewer/2022061616/56649e8f5503460f94b92b74/html5/thumbnails/30.jpg)
Although they call this standardized score a T-score, it is really just a Z-score where the reference mean and standard deviation come from an external population (i.e., young normal adults of a given sex and ethnicity).
Descriptive Statistics
lyoungnorma
lyoungnormapatient
SD
BDMBDMscoreT
)(
![Page 31: Kin 304 Descriptive Statistics & the Normal Distribution Normal Distribution Measures of Central Tendency Measures of Variability Percentiles Z-Scores.](https://reader033.fdocuments.net/reader033/viewer/2022061616/56649e8f5503460f94b92b74/html5/thumbnails/31.jpg)
Classification using t-scores
T-scores are used to classify a patient’s BMD into one of three categories: – T-scores of -1.0 indicate normal bone density– T-scores between -1.0 and -2.5 indicate low bone
mass (“osteopenia”)– T-scores -2.5 indicate osteoporosis
Decisions to treat patients with osteoporosis medication are based, in part, on T-scores.