Quick and Simple Statistics

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Quick and Simple Statistics

• Peter Kasper

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Basic Concepts

• Variables Distributions
• Mean Standard Deviation
• Estimators Errors
• Comparing Two Sample Results
• Significance
• t-Tests

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Variables Distributions

• A Variable is anything that can be ..
• Measured (continuous variable)
• E.g. The height of vegetation in an area
• Counted (discreet variable)
• E.g. The number of birds in an area
• Categorized (categorical variable)
• E.g. Birds that are grassland specialists or not
  grassland specialists

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Variables Distributions

• Each variable has a natural distribution i.e.
  the frequency at which particular values of the
  variable occur
• A common example is the Bell Curve
• In general can be arbitrary!

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Another Bell Curve distribution
A Bell Curve distribution
Different average value and different width
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Mean Standard Deviation

• The mean of a sample of n measurements of some
  variable is defined as ..
• µ ?i vi / n
• The standard deviation or width of the
  distribution is ..
• s 2 ?i (vi - µ )2 / ( n 1 )

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For a Bell Curve distribution 68 of values are
within 1 s of the mean m 95 of values are within
2 s of the mean m
m
s
s
s
s
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Estimators Errors

• The values for µ and s from finite samples vary
  even if the samples are part of the same
  distribution
• They are Estimators of the true values.
• The error on the estimators indicates how much
  variation is expected

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Estimators Errors

• The Standard Error on the mean of a sample of n
  measurements is defined as ..
• SEµ s / vn
• The error gets smaller as the sample size
  increases
• µ has a Bell Curve distribution with mean
  true mean and SD SEµ

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Comparing Results

• Assume two different samples
• Are they from the same distribution?
• Compare the two means
• The difference in the means will usually not be
  zero
• How do we measure the significance of an observed
  difference
• It will clearly depend on the standard errors

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Comparing Results

• The standard error on the difference between to
  quantities is defined as ..
• SEdiff v( SE12 SE22 )
• If the two results are from the same distribution
  (Null Hypothesis)
• The difference will be a Bell Curve
  distribution with mean zero and Standard
  Deviation SEdiff

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Significance

• Can use the difference of the means, ?, and its
  error to measure the likelihood of an observed
  difference.
• What is the probability of randomly getting a
  bigger difference than we obtained?
• Assuming a Bell Curve
• Probability 32 if ? SE?
• Probability 5 if ? 2 x SE?

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Significance

• This is NOT the same thing as the probability
  that the two distributions are the same.
• Probability 1 if ? 0 !
• But it is clearly not impossible to measure the
  same mean from two similar but different
  distributions

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t-Tests

• When the size of a sample is small (lt30
  measurements), we need to take into account the
  uncertainty in the estimates of the Standard
  Errors
• Instead of comparing ?/SE? with a Normal Bell
  curve distribution, Compare a variable t with a
  tabulated t-distribution.

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t-Tests

• The t variable is defined as
• t ? / v( s2/n1 s2/n2 )
• where
• s2 (n1-1)SE12 (n2-1) SE12 / (n1n2-2)
• The number of degrees of freedom (needed by the
  tables) are (n1-1) and (n2-1)

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Conclusion

• This talk has skipped a LOT of details
• It was designed to give you a feel for concepts
• There are lots of resources on the web and in the
  library