Transcript Document

Statistics in Hydrology
• Mean, median and mode (central tendency)
• Dispersion: the spread of the items in a data set
around its central value
Statistics in Hydrology
Measure of central tendency Measure of dispersion
Mode
Median
Mean
Range
Quartile deviation
Standard dev.
Statistics in Hydrology
Statistics in Hydrology
Why do we need to include variance/SD?
Probability
• We need to know its probability of occurrence (the
level of peak Q likely to occur/100 years (moving to
inferential stats)
Probability
Probability
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
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
Probability
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!
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 probability
of getting less than 500 mm of
rainfall in any one year in
Edinburgh, Scotland given a
mean annual rainfall of 664 mm
and a S.D. of 120 mm?
Risk
Risk = probability * consequence
Probability
Fi = m/(n + 1) * 100
where Fi = cum. % frequency
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)
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 141.2
and 258.8 L
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
Correlation and Regression
25
20
15
10
5
0
0
2
4
6
8
Temperature (C)
10
12
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
Correlation and Regression
y = a + bx (y = mx + c)
y = 1 + 0.5x
3
2.5
2
1.5
1
Correlation and Regression
• Least squares
Non-linear Regression
SSC (t/day)
10
8
6
4
2
0
0
50
100
Discharge (m^3/s)
150
Non-linear Regression
1.5
1.0
0.5
0.0
-0.5
-1.0
-1.5
-2.0
log SSC (t/day)
logy = a + b logx (y = 0.8393x - 1.2253)
r^2 = 0.8534
0.0
1.0
2.0
log Discharge (m^3/s)
3.0
Non-linear Regression
10.0
0.8393
SSC (t/day)
y = 0.0595x
R2 = 0.8534
1.0
0.1
1
10
100
Discharge (m^3/s)
1000