application of statistical techniques to interpretation of water monitoring data eric smith, golde...
TRANSCRIPT
![Page 1: Application of Statistical Techniques to Interpretation of Water Monitoring Data Eric Smith, Golde Holtzman, and Carl Zipper](https://reader036.vdocuments.us/reader036/viewer/2022062404/5517aefb5503460e6e8b618d/html5/thumbnails/1.jpg)
Application of Statistical Techniques to Interpretation
of Water Monitoring Data
Eric Smith, Golde Holtzman,
and Carl Zipper
![Page 2: Application of Statistical Techniques to Interpretation of Water Monitoring Data Eric Smith, Golde Holtzman, and Carl Zipper](https://reader036.vdocuments.us/reader036/viewer/2022062404/5517aefb5503460e6e8b618d/html5/thumbnails/2.jpg)
OutlineI. Water quality data: program design (CEZ, 15 min)
II. Characteristics of water-quality data (CEZ, 15 min)
III. Describing water quality(GIH, 30 min)
IV. Data analysis for making decisions
A, Compliance with numerical standards (EPS, 45 min)
Dinner Break
B, Locational / temporal comparisons (“cause and effect”) (EPS, 45)
C, Detection of water-quality trends (GIH, 60 min)
![Page 3: Application of Statistical Techniques to Interpretation of Water Monitoring Data Eric Smith, Golde Holtzman, and Carl Zipper](https://reader036.vdocuments.us/reader036/viewer/2022062404/5517aefb5503460e6e8b618d/html5/thumbnails/3.jpg)
III. Describing water quality(GIH, 30 min)
• Rivers and streams are an essential component of the biosphere
• Rivers are alive• Life is characterized by variation• Statistics is the science of variation• Statistical Thinking/Statistical Perspective • Thinking in terms of variation• Thinking in terms of distribution
![Page 4: Application of Statistical Techniques to Interpretation of Water Monitoring Data Eric Smith, Golde Holtzman, and Carl Zipper](https://reader036.vdocuments.us/reader036/viewer/2022062404/5517aefb5503460e6e8b618d/html5/thumbnails/4.jpg)
The present problem is multivariate
• WATER QUALITY as a function of • TIME, under the influence of co-variates like• FLOW, at multiple • LOCATIONS
![Page 5: Application of Statistical Techniques to Interpretation of Water Monitoring Data Eric Smith, Golde Holtzman, and Carl Zipper](https://reader036.vdocuments.us/reader036/viewer/2022062404/5517aefb5503460e6e8b618d/html5/thumbnails/5.jpg)
WQ variable versus time
Time in Years
Wat
er V
aria
ble
![Page 6: Application of Statistical Techniques to Interpretation of Water Monitoring Data Eric Smith, Golde Holtzman, and Carl Zipper](https://reader036.vdocuments.us/reader036/viewer/2022062404/5517aefb5503460e6e8b618d/html5/thumbnails/6.jpg)
Bear Creek below Town of Wise STP
6.5
7
7.5
8
8.5
9
PH
1973/12/14 1978/12/14 1983/12/14 1988/12/14 1993/12/14
DATE
![Page 7: Application of Statistical Techniques to Interpretation of Water Monitoring Data Eric Smith, Golde Holtzman, and Carl Zipper](https://reader036.vdocuments.us/reader036/viewer/2022062404/5517aefb5503460e6e8b618d/html5/thumbnails/7.jpg)
Univariate WQ Variable
Time
Wat
er Q
ual
ity
![Page 8: Application of Statistical Techniques to Interpretation of Water Monitoring Data Eric Smith, Golde Holtzman, and Carl Zipper](https://reader036.vdocuments.us/reader036/viewer/2022062404/5517aefb5503460e6e8b618d/html5/thumbnails/8.jpg)
Univariate WQ Variable
Time
Wat
er Q
ual
ity
Wat
er Q
ual
ity
Water Quality
Wat
er Q
ual
ity
Water Quality
Wat
er Q
ual
ity
Wat
er Q
ual
ity
Wat
er Q
ual
ity
Wat
er Q
ual
ity
Wat
er Q
ual
ity
Wat
er Q
ual
ity
Wat
er Q
ual
ity
![Page 9: Application of Statistical Techniques to Interpretation of Water Monitoring Data Eric Smith, Golde Holtzman, and Carl Zipper](https://reader036.vdocuments.us/reader036/viewer/2022062404/5517aefb5503460e6e8b618d/html5/thumbnails/9.jpg)
Univariate Perspective, Real Data (pH below STP)
6.5 7 7.5 8 8.5 9
6.5
7
7.5
8
8.5
9
![Page 10: Application of Statistical Techniques to Interpretation of Water Monitoring Data Eric Smith, Golde Holtzman, and Carl Zipper](https://reader036.vdocuments.us/reader036/viewer/2022062404/5517aefb5503460e6e8b618d/html5/thumbnails/10.jpg)
The three most important pieces of information in a sample:
• Central Location– Mean, Median, Mode
• Dispersion– Range, Standard Deviation,
Inter Quartile Range
• Shape– Symmetry, skewness, kurtosis
![Page 11: Application of Statistical Techniques to Interpretation of Water Monitoring Data Eric Smith, Golde Holtzman, and Carl Zipper](https://reader036.vdocuments.us/reader036/viewer/2022062404/5517aefb5503460e6e8b618d/html5/thumbnails/11.jpg)
Central Location: Sample Mean
• (Sum of all observations) / (sample size)• Center of gravity of the distribution• depends on each observation• therefore sensitive to outliers
![Page 12: Application of Statistical Techniques to Interpretation of Water Monitoring Data Eric Smith, Golde Holtzman, and Carl Zipper](https://reader036.vdocuments.us/reader036/viewer/2022062404/5517aefb5503460e6e8b618d/html5/thumbnails/12.jpg)
Central Location: Sample Mean
• (Sum of all observations) / (sample size)• Center of gravity of the distribution• depends on each observation• therefore sensitive to outliers
![Page 13: Application of Statistical Techniques to Interpretation of Water Monitoring Data Eric Smith, Golde Holtzman, and Carl Zipper](https://reader036.vdocuments.us/reader036/viewer/2022062404/5517aefb5503460e6e8b618d/html5/thumbnails/13.jpg)
Central Location: Sample Mean
• (Sum of all observations) / (sample size)• Center of gravity of the distribution• depends on each observation• therefore sensitive to outliers
![Page 14: Application of Statistical Techniques to Interpretation of Water Monitoring Data Eric Smith, Golde Holtzman, and Carl Zipper](https://reader036.vdocuments.us/reader036/viewer/2022062404/5517aefb5503460e6e8b618d/html5/thumbnails/14.jpg)
Central Location: Sample Mean
• (Sum of all observations) / (sample size)• Center of gravity of the distribution• depends on each observation• therefore sensitive to outliers
![Page 15: Application of Statistical Techniques to Interpretation of Water Monitoring Data Eric Smith, Golde Holtzman, and Carl Zipper](https://reader036.vdocuments.us/reader036/viewer/2022062404/5517aefb5503460e6e8b618d/html5/thumbnails/15.jpg)
Central Location: Sample Mean
• (Sum of all observations) / (sample size)• Center of gravity of the distribution• depends on each observation• therefore sensitive to outliers
![Page 16: Application of Statistical Techniques to Interpretation of Water Monitoring Data Eric Smith, Golde Holtzman, and Carl Zipper](https://reader036.vdocuments.us/reader036/viewer/2022062404/5517aefb5503460e6e8b618d/html5/thumbnails/16.jpg)
Central Location: Sample Mean
• (Sum of all observations) / (sample size)• Center of gravity of the distribution• depends on each observation• therefore sensitive to outliers
![Page 17: Application of Statistical Techniques to Interpretation of Water Monitoring Data Eric Smith, Golde Holtzman, and Carl Zipper](https://reader036.vdocuments.us/reader036/viewer/2022062404/5517aefb5503460e6e8b618d/html5/thumbnails/17.jpg)
Central Location: Sample Median• Center of the ordered array
• I.e., the (½)(n + 1) observation in the ordered array.
If sample size n is odd, then the
median is the middle value in the
ordered array.
Example A:
1, 1, 0, 2 , 3
Order:
0, 1, 1, 2, 3
n = 5, odd
(½)(n + 1) = 3
Median = 1
If sample size n is even, then the
median is the average of the two
middle values in the ordered array.
Example B:
1, 1, 0, 2, 3, 6
Order:
0, 1, 1, 2, 3, 6
n = 6, even,
(½)(n + 1) = 3.5
Median = (1 + 2)/2 = 1.5
![Page 18: Application of Statistical Techniques to Interpretation of Water Monitoring Data Eric Smith, Golde Holtzman, and Carl Zipper](https://reader036.vdocuments.us/reader036/viewer/2022062404/5517aefb5503460e6e8b618d/html5/thumbnails/18.jpg)
Central Location: Sample Median
• Center of the ordered array• depends on the magnitude of the central
observations only• therefore NOT sensitive to outliers
![Page 19: Application of Statistical Techniques to Interpretation of Water Monitoring Data Eric Smith, Golde Holtzman, and Carl Zipper](https://reader036.vdocuments.us/reader036/viewer/2022062404/5517aefb5503460e6e8b618d/html5/thumbnails/19.jpg)
Central Location: Sample Median
• Center of the ordered array• depends on the magnitude of the central
observations only• therefore NOT sensitive to outliers
![Page 20: Application of Statistical Techniques to Interpretation of Water Monitoring Data Eric Smith, Golde Holtzman, and Carl Zipper](https://reader036.vdocuments.us/reader036/viewer/2022062404/5517aefb5503460e6e8b618d/html5/thumbnails/20.jpg)
Central Location: Sample Median
• Center of the ordered array• depends on the magnitude of the central
observations only• therefore NOT sensitive to outliers
![Page 21: Application of Statistical Techniques to Interpretation of Water Monitoring Data Eric Smith, Golde Holtzman, and Carl Zipper](https://reader036.vdocuments.us/reader036/viewer/2022062404/5517aefb5503460e6e8b618d/html5/thumbnails/21.jpg)
Central Location: Sample Median
• Center of the ordered array• depends on the magnitude of the central
observations only• therefore NOT sensitive to outliers
![Page 22: Application of Statistical Techniques to Interpretation of Water Monitoring Data Eric Smith, Golde Holtzman, and Carl Zipper](https://reader036.vdocuments.us/reader036/viewer/2022062404/5517aefb5503460e6e8b618d/html5/thumbnails/22.jpg)
Central Location: Sample Median
• Center of the ordered array• depends on the magnitude of the central
observations only• therefore NOT sensitive to outliers
![Page 23: Application of Statistical Techniques to Interpretation of Water Monitoring Data Eric Smith, Golde Holtzman, and Carl Zipper](https://reader036.vdocuments.us/reader036/viewer/2022062404/5517aefb5503460e6e8b618d/html5/thumbnails/23.jpg)
Central Location: Sample Median
• Center of the ordered array• depends on the magnitude of the central
observations only• therefore NOT sensitive to outliers
![Page 24: Application of Statistical Techniques to Interpretation of Water Monitoring Data Eric Smith, Golde Holtzman, and Carl Zipper](https://reader036.vdocuments.us/reader036/viewer/2022062404/5517aefb5503460e6e8b618d/html5/thumbnails/24.jpg)
Central Location: Mean vs. Median
• Mean is influenced by outliers• Median is robust against (resistant to) outliers• Mean “moves” toward outliers• Median represents bulk of observations almost always
Comparison of mean and median tells us about outliers
![Page 25: Application of Statistical Techniques to Interpretation of Water Monitoring Data Eric Smith, Golde Holtzman, and Carl Zipper](https://reader036.vdocuments.us/reader036/viewer/2022062404/5517aefb5503460e6e8b618d/html5/thumbnails/25.jpg)
Dispersion
• Range• Standard Deviation• Inter-quartile Range
![Page 26: Application of Statistical Techniques to Interpretation of Water Monitoring Data Eric Smith, Golde Holtzman, and Carl Zipper](https://reader036.vdocuments.us/reader036/viewer/2022062404/5517aefb5503460e6e8b618d/html5/thumbnails/26.jpg)
Dispersion: Range
• Maximum - Minimum
• Easy to calculate
• Easy to interpret
• Depends on sample size (biased)
• Therefore not good for statistical inference
![Page 27: Application of Statistical Techniques to Interpretation of Water Monitoring Data Eric Smith, Golde Holtzman, and Carl Zipper](https://reader036.vdocuments.us/reader036/viewer/2022062404/5517aefb5503460e6e8b618d/html5/thumbnails/27.jpg)
Dispersion: Standard Deviation
1
2
n
YY-0 5
-1+1
SD = 10
0 5
-2+2
SD = 2
1 2
-1 1 3
![Page 28: Application of Statistical Techniques to Interpretation of Water Monitoring Data Eric Smith, Golde Holtzman, and Carl Zipper](https://reader036.vdocuments.us/reader036/viewer/2022062404/5517aefb5503460e6e8b618d/html5/thumbnails/28.jpg)
Dispersion: Properties of SD• SD > 0 for all data
• SD = 0 if and only if all observations the same (no variation)
• For a normal distribution, – 68% expected within 1 SD,– 95% expected within 2 SD,– 99.6% expected within 3 SD,
• For any distribution, nearly all observations lie within 3 SD
![Page 29: Application of Statistical Techniques to Interpretation of Water Monitoring Data Eric Smith, Golde Holtzman, and Carl Zipper](https://reader036.vdocuments.us/reader036/viewer/2022062404/5517aefb5503460e6e8b618d/html5/thumbnails/29.jpg)
Interpretation of SD
6.5 7 7.5 8 8.5 9
n = 200
SD = 0.41
Median = 7.6
Mean = 7.6
![Page 30: Application of Statistical Techniques to Interpretation of Water Monitoring Data Eric Smith, Golde Holtzman, and Carl Zipper](https://reader036.vdocuments.us/reader036/viewer/2022062404/5517aefb5503460e6e8b618d/html5/thumbnails/30.jpg)
Quantiles, Five Number Summary, Boxplot
Maximum 4th quartile 100th percentile 1.00 quantile
3rd quartile 75th percentile 0.75 quantile
Median 2nd quartile 50th percentile 0.50 quantile
1st quartile 25th percentile 0.25 quantile
Minimum 0th quartile 0th percentile 0.00 quantile
![Page 31: Application of Statistical Techniques to Interpretation of Water Monitoring Data Eric Smith, Golde Holtzman, and Carl Zipper](https://reader036.vdocuments.us/reader036/viewer/2022062404/5517aefb5503460e6e8b618d/html5/thumbnails/31.jpg)
Quantile Location and Quantiles
Quantile Rank Quantile Location Quartile
0.75 = 3/4
0.50 = 2/4
0.25 = 1/4
Example: 0, − 3.1, − 0.4, 0, 2.2, 5.1, 3.8, 3.8, 3.9, 2.3, n = 10
Value Rank
5.1 10
3.9 9
3.8 8
3.8 7
2.3 6
2.2 5
0 4
0 3
−0.4 2
−3.1 1
0.75 1 8.3n 3
3.8 3.93.85
2Q
0.50 1 5.5n 2
2.2 2.32.25
2Q
0.25 1 2.8n 1
0.4 00.2
2Q
Minimum = −3.1
Maximum = 5.1
![Page 32: Application of Statistical Techniques to Interpretation of Water Monitoring Data Eric Smith, Golde Holtzman, and Carl Zipper](https://reader036.vdocuments.us/reader036/viewer/2022062404/5517aefb5503460e6e8b618d/html5/thumbnails/32.jpg)
5-Number Summary and Boxplot
Min Q1 Q2 Q3 Max
−3.10 −0.20 2.25 3.85 5.10
2 2.25Median Q
5.10 3.10 8.20Range Max Min
3 1 3.85 0.20 4.05IQR Q Q
![Page 33: Application of Statistical Techniques to Interpretation of Water Monitoring Data Eric Smith, Golde Holtzman, and Carl Zipper](https://reader036.vdocuments.us/reader036/viewer/2022062404/5517aefb5503460e6e8b618d/html5/thumbnails/33.jpg)
Dispersion: IQRInter-Quartile Range
• (3rd Quartile - (1st Quartile)
• Robust against outliers
![Page 34: Application of Statistical Techniques to Interpretation of Water Monitoring Data Eric Smith, Golde Holtzman, and Carl Zipper](https://reader036.vdocuments.us/reader036/viewer/2022062404/5517aefb5503460e6e8b618d/html5/thumbnails/34.jpg)
Interpretation of IQR
6.5 7 7.5 8 8.5 9
n = 200
SD = 0.41
Median = 7.6
Mean = 7.6
IQR = 0.54
For a Normal distribution, Median 2IQR includes 99.3%
![Page 35: Application of Statistical Techniques to Interpretation of Water Monitoring Data Eric Smith, Golde Holtzman, and Carl Zipper](https://reader036.vdocuments.us/reader036/viewer/2022062404/5517aefb5503460e6e8b618d/html5/thumbnails/35.jpg)
Shape: Symmetry and Skewness
• Symmetry mean bilateral symmetry
![Page 36: Application of Statistical Techniques to Interpretation of Water Monitoring Data Eric Smith, Golde Holtzman, and Carl Zipper](https://reader036.vdocuments.us/reader036/viewer/2022062404/5517aefb5503460e6e8b618d/html5/thumbnails/36.jpg)
Shape: Symmetry and Skewness
• Symmetry mean bilateral symmetry
• Positive Skewness (asymmetric “tail” in positive direction)
![Page 37: Application of Statistical Techniques to Interpretation of Water Monitoring Data Eric Smith, Golde Holtzman, and Carl Zipper](https://reader036.vdocuments.us/reader036/viewer/2022062404/5517aefb5503460e6e8b618d/html5/thumbnails/37.jpg)
Shape: Symmetry and Skewness• “Symmetry” mean bilateral
symmetry, skewness = 0• Mean = Median (approximately)
• Positive Skewness (asymmetric “tail” in positive direction)
• Mean > Median
• Negative Skewness (asymmetric “tail” in negative direction)
• Mean < Median
Comparison of mean and median tells us about shape
![Page 38: Application of Statistical Techniques to Interpretation of Water Monitoring Data Eric Smith, Golde Holtzman, and Carl Zipper](https://reader036.vdocuments.us/reader036/viewer/2022062404/5517aefb5503460e6e8b618d/html5/thumbnails/38.jpg)
6.5 7 7.5 8 8.5 9
6.5
7
7.5
8
8.5
9
Bear Creek below Town of Wise STP
![Page 39: Application of Statistical Techniques to Interpretation of Water Monitoring Data Eric Smith, Golde Holtzman, and Carl Zipper](https://reader036.vdocuments.us/reader036/viewer/2022062404/5517aefb5503460e6e8b618d/html5/thumbnails/39.jpg)
6.5
7
7.5
8
8.5
9
Outlier Box Plot
Outliers
Whisker
Whisker
Median
75th %-tile = 3rd Quartile
25th %-tile = 1st Quartile
IQR
![Page 40: Application of Statistical Techniques to Interpretation of Water Monitoring Data Eric Smith, Golde Holtzman, and Carl Zipper](https://reader036.vdocuments.us/reader036/viewer/2022062404/5517aefb5503460e6e8b618d/html5/thumbnails/40.jpg)
Wise, VA, below STP
6.5
7
7.5
8
8.5
9
0
2
4
6
8
1011
13
pH
TK
N m
g/l
![Page 41: Application of Statistical Techniques to Interpretation of Water Monitoring Data Eric Smith, Golde Holtzman, and Carl Zipper](https://reader036.vdocuments.us/reader036/viewer/2022062404/5517aefb5503460e6e8b618d/html5/thumbnails/41.jpg)
Wise, VA below STP
10
20
30
40
50
60
70
80
90
100
110
120
130
0
5
10
15
20
25
DO
(%
sat
ur)
BO
D
(mg/
l)
![Page 42: Application of Statistical Techniques to Interpretation of Water Monitoring Data Eric Smith, Golde Holtzman, and Carl Zipper](https://reader036.vdocuments.us/reader036/viewer/2022062404/5517aefb5503460e6e8b618d/html5/thumbnails/42.jpg)
0
1
2
3
4
5
Wise, VA below STPT
ot P
hosp
horo
us (
mg/
l
0
10000
20000
30000
40000
50000
60000
Fecal Coliforms