Statistics in WR: Lecture 7 Key Themes – Statistics for populations and samples – Suspended sediment sampling – Testing for differences in means and variances Reading: Helsel and Hirsch Chapter 8 Correlation

Question about n and n-1 in statistics A question that came up while working on problem 1: Why for the variance (and thus the standard deviation), does the formula use 1/(N-1) but in the Coefficient of Skewness use 1/N? From reading in the Barnett text, it appears that 1/N could also be used in the variance formula, though using this results in slightly different numbers from the Excel-calculated descriptive statistics. Could you comment on this. James Seppi

Correcting bias

Estimators of the Variance Maximum Likelihood Estimate for Population variance Unbiased estimate from a sample

Bias in the Variance Common sense would suggest to apply the population formula to the sample as well. The reason that it is biased is that the sample mean is generally somewhat closer to the observations in the sample than the population mean is to these observations. This is so because the sample mean is by definition in the middle of the sample, while the population mean may even lie outside the sample. So the deviations from the sample mean will often be smaller than the deviations from the population mean, and so, if the same formula is applied to both, then this variance estimate will on average be somewhat smaller in the sample than in the population.

Suspended Sediment Sampling

Suspended sediment sampler The US P-72 is a cast aluminum sampler having an electrically operated valve for collection of a suspended-sediment sample at any point in a stream cross section or to take a depth-integrated sample over a range of depths. The sampler is streamlined and has tail fins to orient the sampler so that the intake nozzle in the head points directly into the approaching flow. The sampler head is hinged to provide access to the round pint or quart bottle sample container, which is located in a cavity in the sampler body. An exhaust port pointing downstream on the side of the sampler head permits escape of air from the bottle as it is displaced by the sample being collected. A valve mechanism enclosed in the head of the sampler is electrically activated to start and stop the sampling process.

Suspended Sediment Concentration

Suspended Sediment Load (USGS1)

T-test with same variances

T-test with different variances