Application of Statistical Techniques to Interpretation

of Water Monitoring Data

Eric Smith, Golde Holtzman,

and Carl Zipper

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)

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

The present problem is multivariate

• WATER QUALITY as a function of • TIME, under the influence of co-variates like• FLOW, at multiple • LOCATIONS

WQ variable versus time

Time in Years

Wat

er V

aria

ble

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

Univariate WQ Variable

Time

Wat

er Q

ual

ity

Univariate WQ Variable

Time

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er Q

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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

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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

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

Central Location: Sample Mean

• (Sum of all observations) / (sample size)• Center of gravity of the distribution• depends on each observation• therefore sensitive to outliers

Central Location: Sample Mean

• (Sum of all observations) / (sample size)• Center of gravity of the distribution• depends on each observation• therefore sensitive to outliers

Central Location: Sample Mean

• (Sum of all observations) / (sample size)• Center of gravity of the distribution• depends on each observation• therefore sensitive to outliers

Central Location: Sample Mean

• (Sum of all observations) / (sample size)• Center of gravity of the distribution• depends on each observation• therefore sensitive to outliers

Central Location: Sample Mean

• (Sum of all observations) / (sample size)• Center of gravity of the distribution• depends on each observation• therefore sensitive to outliers

Central Location: Sample Mean

• (Sum of all observations) / (sample size)• Center of gravity of the distribution• depends on each observation• therefore sensitive to outliers

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

Central Location: Sample Median

• Center of the ordered array• depends on the magnitude of the central

observations only• therefore NOT sensitive to outliers

Central Location: Sample Median

• Center of the ordered array• depends on the magnitude of the central

observations only• therefore NOT sensitive to outliers

Central Location: Sample Median

• Center of the ordered array• depends on the magnitude of the central

observations only• therefore NOT sensitive to outliers

Central Location: Sample Median

• Center of the ordered array• depends on the magnitude of the central

observations only• therefore NOT sensitive to outliers

Central Location: Sample Median

• Center of the ordered array• depends on the magnitude of the central

observations only• therefore NOT sensitive to outliers

Central Location: Sample Median

• Center of the ordered array• depends on the magnitude of the central

observations only• therefore NOT sensitive to outliers

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

Dispersion

• Range• Standard Deviation• Inter-quartile Range

Dispersion: Range

• Maximum - Minimum

• Easy to calculate

• Easy to interpret

• Depends on sample size (biased)

• Therefore not good for statistical inference

Dispersion: Standard Deviation

1

2

n

YY-0 5

-1+1

SD = 10

0 5

-2+2

SD = 2

1 2

-1 1 3

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

Interpretation of SD

6.5 7 7.5 8 8.5 9

n = 200

SD = 0.41

Median = 7.6

Mean = 7.6

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

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

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

Dispersion: IQRInter-Quartile Range

• (3rd Quartile - (1st Quartile)

• Robust against outliers

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%

Shape: Symmetry and Skewness

• Symmetry mean bilateral symmetry

Shape: Symmetry and Skewness

• Symmetry mean bilateral symmetry

• Positive Skewness (asymmetric “tail” in positive direction)

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

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

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

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

Wise, VA below STP

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20

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130

0

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25

DO

(%

sat

ur)

BO

D

(mg/

l)

0

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Wise, VA below STPT

ot P

hosp

horo

us (

mg/

l

0

10000

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30000

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50000

60000

Fecal Coliforms