Transcript StatWRLecture7

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
http://en.wikipedia.org/wiki/Skewness
Estimators of the Variance
Maximum Likelihood
Estimate for
Population variance
Unbiased estimate
from a sample
http://en.wikipedia.org/wiki/Variance
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
http://pubs.usgs.gov/sir/2005/5077/
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_cd=342
Suspended Sediment
Concentration
C_(mg/L)
100000
Concentration (mg/L)
10000
1000
C_(mg/L)
100
10
1
0.01
0.1
1
10
Discharge (cfs)
100
1000
10000
Suspended Sediment Load (USGS1)
L_(ton/day)
1000000
100000
10000
Load (Tn/day)
1000
100
L_(ton/day)
10
1
0.01
0.1
1
10
0.1
0.01
Discharge (cfs)
100
1000
10000
T-test with same variances
T-test with different variances