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Lecture 9 Moments of distributions. Body size distribution of European Collembola. Body size distribution of European Collembola. Modus. The histogram of raw data. Three Collembolan weight classes. What is the average body weight ? . Sample mean. Population mean.

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Body size distribution of European Collembola

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Body size distribution of european collembola

Lecture 9

Moments of distributions

Body sizedistribution of EuropeanCollembola


Body size distribution of european collembola

Body sizedistribution of EuropeanCollembola

Modus

The histogram of raw data


Body size distribution of european collembola

ThreeCollembolanweightclasses

Whatistheaverage body weight?

Sample mean

Population mean

Weighedmean


Body size distribution of european collembola

Weighed mean

Discrete distributions

TheaverageEuropeanspringtailhas a body weight of e-1.476 = 023 mg.

Most oftenencountedis a weightaround e-1.23 = 029 mg.

Continuous distributions


Body size distribution of european collembola

Whydid we use log transformedvalues?

Linear data

Log transformed data

Thedistributionisskewed


Body size distribution of european collembola

lbscaledweightclasses

TheaverageEuropeanspringtailhas a body weight of e-1.476 = 023 mg.

Geometricmean

In thecase of exponentiallydistributed data we have to usethegeometricmean.

To make thingseasier we first log-transformour data.


Body size distribution of european collembola

How to usegeometricmeans

A tropical forest is logged during three years:

first year 0.1%, second year 1% and third year 10% of area.

Hence the total decrease in forest area is

11% of area has been logged during three year.

What is the mean logging rate per year?

Arithmetic mean

Geometric mean

In multiplicativeprocesses we shouldusethegeometricmean.


Body size distribution of european collembola

Degrees of freedom

Variance

Continuousdistributions

Mean

1 SD

Standard deviation

The standard deviationis a measure of thewidth of thestatisticaldistributionthathasthe sam dimension as themean.


Body size distribution of european collembola

The standard deviation as a measure of errors

± 1 standard deviationisthe most oftenusedestimator of error.

Theprobablitythatthetruemeaniswithin± 1 standard deviationisapproximately 68%.

Theprobablitythatthetruemeaniswithin± 2 standard deviationsisapproximately 95%.

The precision of derivedmetricsshouldalwaysmatchthe precision of theraw data

± 1 standard deviation


Body size distribution of european collembola

Standard deviation and standard error

The standard deviationisconstantirrespective of samplesize.

The precision of theestimate of themeanshouldincreasewithsamplesize n.

The standard erroris a measure of precision.


Body size distribution of european collembola

Central moments

[E(x)]2

E(x2)

Mathematicalexpectation

First central moment

First moment of central tendency

Thevarianceisthedifferencebetweenthemean of thesquaredvalues and thesquaredmean

k-th central moment


Body size distribution of european collembola

Frequency distributions of resource use or wealth in a population can be described by a power law (the famous Pareto-Zipf law) with exponents that often have values around -5/2. Whatare the mean and thevarianceof such a power function distribution?

Discretedistribution

Most peopleareinthelowestincomeclass and theaverageishalfbetweenthe first and thesecond.


Body size distribution of european collembola

Continuousapproximation

Notethatthey-axisisat log scale.

Upper bound of ten wouldonlycoverhalf of thecolumn

Theestimate of a isimprecise


Body size distribution of european collembola

The Arrhenius probability model assumesthe same probability of an eventirrespective of the time thatelapsedfromthestarting. Whatarethe mean and thevarianceof such a distribution?

Cumulativedensityfunction


Body size distribution of european collembola

Third central moment

Skewness

g>0

g<0

g=0

Leftskeweddistribution

Rightskeweddistribution

Symmetricdistribution

d>0

d=0

Kurtosis


Body size distribution of european collembola

How to getthe modus?

We needthemaximum of thepdf

A probabilitydistributionif

Mode

Mean

Arithmeticmean


Body size distribution of european collembola

Body volumes are estimated from measures of height*length*width. Assume you estimated the thorax volume of insects and usedthisvolume to infer the body weight.

How to gettheparameters a and z?


Body size distribution of european collembola

Body weightsareestimated from speciesweightsagainstthoraxvolume.

The body weight of a newspeciesisestimatedfromtheregressionfunction

Height, length and width could be measured with an accuracy of ± 2%.

Independent measurements

Standarddeviation is a measure of accuracy (error)

Theerror of thethoraxestimateis 3.5%.


Home work and literature

Home work and literature

  • Refresh:

  • Arithmetic, geometric, harmonicmean

  • Cauchyinequality

  • Statisticaldistribution

  • Probabilitydistribution

  • Moments of distributions

  • Error law of Gauß

  • Bootstrap

  • Prepare to thenextlecture:

  • Bionomialdistribution

  • Mean and variance of thebinomialdistribution

  • Poisson distribution

  • Mean and variance of the Poisson distribution

  • Moments of distributions

  • DNA mutations

  • Transitionmatrix

Literature:

Łomnicki: Statystyka dla biologów

Binomialdistribution:

http://www.stat.yale.edu/Courses/1997-98/101/binom.htm

Poisson dstribution:

http://en.wikipedia.org/wiki/Poisson_distribution


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