Statistics in WR: Lecture 7

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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

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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  
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Correcting bias
http//en.wikipedia.org/wiki/Skewness
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Estimators of the Variance
Maximum Likelihood Estimate for Population
variance
Unbiased estimate from a sample
http//en.wikipedia.org/wiki/Variance
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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.
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Suspended Sediment Sampling
http//pubs.usgs.gov/sir/2005/5077/
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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.
http//www.wildco.com/vw_prdct_mdl.asp?prdct_mdl_c
d342
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Suspended Sediment Concentration
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Suspended Sediment Load (USGS1)
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T-test with same variances
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T-test with different variances