Maths & Stats

Foundation Mathematics & Statistics for Hydrologists

Hydro-Logic has provided instruction on the Environment Agency's Hydrometry Training contract since 1996. Stewart Child, who heads the training programme, had been asked to provide training in Maths and Stats at "Foundation" level. It had been found that many Environment Agency staff with degrees in Geography or a similar natural science were struggling with numerical methods, having not studies Maths or Stats since GCSE level. Stewart turned to me to devise a programme of Statistics training and asked another colleague to produce the Mathematics elements of the course. These were designed to be taught on successive days, with the Stats on Day 2.

I designed the Statistics course around a series of exercises in Excel. This provided opportunities to learn techniques in Excel, as well as the statistical tests under consideration. Apart from the virtual tossing of coins, all examples were based on the natural environment, being taken mainly from hydrology. The course was designed for people who had worked with statistical concepts for years but needed refreshing and reminding what it is all about. The day starts with the statement "The majority of Belgians have more than the average number of legs" and works through the material below. Along the way, they learn about randomness and how it influences natural distributions; how distributions are built from binary events and the binomial series; how logarithms of different bases provide a non-arithmetic way to order axes. After lunch, the course looks at standard error and levels of significance; bivariate comparisons and ways to approach them in Excel and finishes by looking at some of the pitfalls and dangers inherent in the subject.

As someone whose formal maths training ended after "O" level, I was concerned at the prospect of teaching people with "A" level Maths and even one or two Maths Graduates have attended the course. To my delight, the course has gone down well with almost all the attendees so far, with the trained mathematicians sometimes more pleased that the others. I attribute this to the different slant that my presentations and exercises put on the subject and the fact that the work is directly relevant to their day jobs, managing environmental data. At the end of the course, the Environment Agency personnel are asked to complete a feedback sheet. Analysis of feedback is available on request.

Outline of the course, as presented by myself
Material		Exercise
Populations and samples Types of Data Populations and Samples Sampling strategies & avoiding bias.		Exercise 1 Investigating the nature of randomness Use RandomPlot.exe to show that inherent in randomness is an element of clustering, consider its implications for sampling strategy.
Measures of central tendency & spread Mean, median & mode, Geometric means. Variance and standard deviation.		Exercise 2 Plot & Calculate some statistics from rainfall data in Excel. Use Excel to solve the standard deviation function manually. Help to understand and solve statistical formulae, warm up Excel skills.
Frequency Distributions and Probability The Binomial and Normal distributions, skew & kurtosis. Poisson and Extreme Event Distributions		Exercise 3 Calculate probabilities using Probability Density Functions Use the spreadsheet "Distributions.xls", which calculates Pascal's Triangle and plots the binomial distribution from it. Other sheets derive and plot Normal distribution and calculate probabilities. A "weighting" cell alters the probability from 0.5, showing the effect of a biassed coin or a skewed distribution.
Data transformations Extreme events & flood frequency Compare plotting positions (Weibull, Gringorten)		Exercise 4 Plotting Flow distributions, with and without transformation. Plot the frequency of daily flow data from a small stream, transform the data by calculating Log10 of each flow and plot the frequency distribution of log values. Discover the meaning and value of a Log-Normal distribution.
Samples and populations, Standard Error of the Mean Uncertainties in flow data Significance and confidence limits		Exercise 5 Investigating the effects of sample size Tossing virtual coins, it becomes clear that a particular distribution may be a consequence of a small sample, rather than particular attributes of the physical system.
Assumptions and Limitations of Statistical Analysis Statistical hypotheses Standard Error of the Difference One & two tailed tests Parametric vs. non-parametric techniques.		Exercise 6 Finding whether a change in Flow is significant. Real flow data from the River Nile is used to determine whether the apparent decrease in flow is real, within specified confidence limits.
Comparing variables Least squares methods, Regression & Correlation analysis. Functional form: lines & curves. Regression in Excel Regression and uncertainty, Standard Error of the Estimate, SMR. Assumptions of least squares techniques.		Exercise 7 Use least squares in Excel to calculate a rainfall / runoff relationship Various techniques are compared to calculate bi-variate relationships and the strength of the correlation. Multi-variate methods are introduced.