Statistics in Hydrology
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Transcript of Statistics in Hydrology
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Statistics in Hydrology
• Mean, median and mode (central tendency)
• Dispersion: the spread of the items in a data set around its central value
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Statistics in Hydrology
Measure of central tendency Measure of dispersion
Mode RangeMedian Quartile deviationMean Standard dev.
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Statistics in Hydrology
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Statistics in Hydrology
Why do we need to include variance/SD?
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Probability
• We need to know its probability of occurrence (the level of peak Q likely to occur/100 years (moving to inferential stats)
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Probability
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Probability
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Probability
Example 1
What is the probability that an individual value will be more than 1.5 S.D. below the mean in normally distributed data?
a. Draw a diagram
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Probability
b. If the area under the curve = 100%, then the area below 1.5 S.D. below the mean represents the probability we require
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Probability
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Probability
c. With z = 1.5 then p = 0.9932%. Look at the table again! It lists p values > 50% (not always required value); be careful!
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Probability
d. the probability we require is 1 - 0.9332 = 0.0668 e. The probability that an individual value will be more than 1.5 S.D. below the mean in the data set is 6.68% At home: What is the probabilityof getting less than 500 mm of rainfall in any one year inEdinburgh, Scotland given a mean annual rainfall of 664 mm and a S.D. of 120 mm?
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Risk
Risk = probability * consequence
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Probability
Fi = m/(n + 1) * 100
where Fi = cum. % frequency
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Hypothesis Testing
• Sampling from a larger population
Null hypothesis: no significant difference between the figures (H0)
Alternative hypothesis: is a significant difference between the figures (H1)
• Level of significance (0.05 and 0.01)
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Hypothesis Testing
Daily Q: Mean = 200 l/day S.D. = 30 l/day
Sample = 128 litres
SD = 30 * 1.96 = 58.8 L
95% of obs. should lie between 141.2and 258.8 L
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Correlation and Regression
Observation Oil consumption (gallons)
Temperature (oC)
1 11.5 11.5
2 13.5 11.0
3 13.8 10.5
4 15.0 7.5
5 16.2 8.0
6 17.0 7.0
7 18.5 7.5
8 22.0 3.5
9 22.3 3
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Correlation and Regression
25
0 2 4 6 8 10 12
Temperature (C)
Oil
consu
mp
tion
(g
)
20
15
10
5
0
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Correlation and Regression
Correlation coefficient(r)
Coefficient of determination(r2) Lies between 0 and 1 Proportion of variation of Y associated with variations in X r = 0.96; r2 = 0.92
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Correlation and Regression
y = a + bx (y = mx + c)
y = 1 + 0.5x
0
0.5
1
1.5
2
2.5
3
0 0.5 1 1.5 2 2.5 3 3.5 4
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Correlation and Regression
• Least squares
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Non-linear Regression
0
2
4
6
8
10
0 50 100 150Discharge (m^3/s)
SS
C (
t/d
ay)
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Non-linear Regression
logy = a + b logx (y = 0.8393x - 1.2253)r^2 = 0.8534
-2.0-1.5-1.0-0.50.00.51.01.5
0.0 1.0 2.0 3.0log Discharge (m^3/s)
log
SS
C (
t/d
ay)
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Non-linear Regression
y = 0.0595x0.8393
R2 = 0.8534
0.1
1.0
10.0
1 10 100 1000Discharge (m^3/s)
SS
C (
t/d
ay)