1. Two Concepts of Probability
Statistical
Relative frequency in repeated experiments
Inductive
Subjective
Based on incomplete information, judgment and logical reasoning
Bayesian

2. Line Diagram
From Kottegoda and Rosso, p3

3. Dot diagram
From Kottegoda and Rosso, p4

4. Histogram of minimum annual flow in the Po river between 1918 and 1978

5. Minimum annual flow in the Po river between 1918 and 1978
Alternative histogram axis scaling
- Relative Frequency
- Density

6. Po River, Minimum annual flow cumulative relative frequency (number of values ≤ n)/n (KR p 8)
qs=sort(q)
n=length(q)
crf=(0:(n-1))/n
plot(qs,crf)

7. Po River, Minimum annual flow Quantile plot (Q-plot)
qs=sort(q)
n=length(q)
crf=(0:(n-1))/n
plot(crf,qs)
75% Quantile or quartile
Interquartile range IQR
Median
25% Quantile or quartile

8. Quantile Definition pi p qi x
A quantile qi is the random variable value associated with a specific cumulative probability pi

9. Numerical Quantities

10. Helsel and Hirsch page 21

11. Box (Red Lines) enclose 50% of the values
Time Series Box Plot
Median
Box (Red Lines) enclose 50% of the values

12. Box Plot Outliers: beyond 1.5*IQR Whiskers: 1.5*IQR or largest value
Box: 25th %tile to 75th %tile
Line: Median (50th %tile) - not the mean
Note: The range shown by the box is called the “Inter-Quartile Range” or IQR.
This is a robust measure of spread. It is insensitive to outliers since it is based purely on the rank of the values.

13. Horizontal Line is the mean
Seasonality of Flow
“Monthly Subseries Plot” - time series for each month
Outliers
Compare change in mean and median between Aug-Sep. Note Skew in September
Flow (cfs)
Horizontal Line is the mean
Box Plots
Flow (cfs)

14. Scatter Plot - Flow v. Water Level
ASK
How good is this relationship? Is it linear? What would you do next?

15. Multiple Scatterplots
Flow = f(Pumping) Causality?
Co-effect? OR
Pumping = f(Flow)
Water Level = f(Pumping) Logical relationship

16. Scatterplot - between raw x and y data
Q-Q plot - between sorted x and y data
Compares individual X and Y values
Compares the distributions of X and Y

17. Quantiles to compare to theoretical distribution
Rank the data
Theoretical distribution, e.g. Standard Normal x1 x2 x3 . xn pi qi
qi is the distribution specific theoretical quantile associated with ranked data value xi

18. Quantile-Quantile Plots
QQ-plot for Raw Flows
QQ-plot for Log-Transformed Flows
ln(xi) qi xi qi
Used as a basis for finding transformation to make the Raw flows Normally distributed.

19. Quantile plots and Probability Plots
Q-Q Plot
Probability Plot
Theoretical quantile axis is relabeled with corresponding probability values