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

Line Diagram From Kottegoda and Rosso, 1997 p3

Dot diagram From Kottegoda and Rosso, 1997 p4

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

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

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)

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

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

Numerical Quantities

Helsel and Hirsch page 21

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

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.

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)

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

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

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

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

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.

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