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 RangeMedian Quartile deviationMean 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 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?

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 141.2and 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

0 2 4 6 8 10 12

Temperature (C)

Oil

consu

mp

tion

(g

)

20

15

10

5

0

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  

0

0.5

1

1.5

2

2.5

3

0 0.5 1 1.5 2 2.5 3 3.5 4

Correlation and Regression

• Least squares

Non-linear Regression

0

2

4

6

8

10

0 50 100 150Discharge (m^3/s)

SS

C (

t/d

ay)

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)

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)