1. Body size distribution of European Collembola Lecture 9 Moments of distributions

2. Body size distribution of European Collembola The histogram of raw data Modus

3. Weighed mean Three Collembolan weight classes What is the average body weight? Population mean Sample mean

4. Weighed mean Discrete distributions Continuous distributions The average European springtail has a body weight of e -1.476 = 023 mg. Most often encounted is a weight around e -1.23 = 029 mg.

5. Why did we use log transformed values? Log transformed data Linear data The distribution is skewed

6. In the case of exponentially distributed data we have to use the geometric mean. To make things easier we first log-transform our data. Geometric mean The average European springtail has a body weight of e -1.476 = 023 mg. lb scaled weight classes

7. How to use geometric means 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 meanGeometric mean In multiplicative processes we should use the geometric mean.

8. Variance Continuous distributions Standard deviation Mean 1 SD The standard deviation is a measure of the width of the statistical distribution that has the sam dimension as the mean. Degrees of freedom

9. The standard deviation as a measure of errors Environmental pollution StationNOx [ppm] 18.49 21.12 39.11 47.75 50.75 68.23 70.97 86.06 98.48 105.88 118.51 129.62 133.35 147.74 152.03 165.06 177.61 180.99 192.55 208.91 Mean5.66 Variance10.45 Standard deviation 3.23 Distance Average NOx concentration Standard deviation 19.531.70 27.371.18 35.240.86 43.150.26 52.170.18 61.050.09 70.840.14 80.630.10 90.320.03 100.210.02 The precision of derived metrics should always match the precision of the raw data ± 1 standard deviation is the most often used estimator of error. The probablity that the true mean is within ± 1 standard deviation is approximately 68%. The probablity that the true mean is within ± 2 standard deviations is approximately 95%. ± 1 standard deviation

10. Mean Standard deviation 5.444.15 4.495.29 5.553.39 5.563.13 Standard deviation and standard error Environmental pollution Station NOx [ppm] 18.49 21.12 39.11 47.75 50.75 68.23 70.97 86.06 98.48 105.88 118.51 129.62 133.35 147.74 152.03 165.06 177.61 180.99 192.55 208.91 The standard deviation is constant irrespective of sample size. The precision of the estimate of the mean should increase with sample size n. The standard error is a measure of precision. Distance Average NOx concentration Standard deviation Standard error n=20 19.533.320.74 27.372.450.55 35.241.240.28 43.150.670.15 52.170.870.19 61.050.340.08 70.840.140.03 80.630.100.02 90.320.030.01 100.210.020.01

11. E(x 2 ) [E(x)] 2 The variance is the difference between the mean of the squared values and the squared mean k-th central moment Mathematical expectation Central moments First central moment First moment of central tendency

12. 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. What are the mean and the variance of such a power function distribution? Discrete distribution Most people are in the lowest income class and the average is half between the first and the second.

13. Continuous approximation Upper bound of ten would only cover half of the column Note that the y- axis is at log scale. The estimate of a is imprecise

14. The Arrhenius probability model assumes the same probability of an event irrespective of the time that elapsed from the starting. What are the mean and the variance of such a distribution? Cumulative density function

15. Skewness Third central moment Kurtosis  =0  >0  <0 Symmetric distribution Right skewed distribution Left skewed distribution  =0  >0

16. How to get the modus? We need the maximum of the pdf Mode A probability distribution if Arithmetic mean Mean

17. Body volumes are estimated from measures of height*length*width. Assume you estimated the thorax volume of insects and used this volume to infer the body weight. How to get the parameters a and z?

18. Standard deviation is a measure of accuracy (error) Independent measurements Body weights are estimated from species weights against thorax volume. The body weight of a new species is estimated from the regression function Height, length and width could be measured with an accuracy of ± 2%. The error of the thorax estimate is 3.5%.

19. Home work and literature Refresh: Arithmetic, geometric, harmonic mean Cauchy inequality Statistical distribution Probability distribution Moments of distributions Error law of Gauß Bootstrap Prepare to the next lecture: Bionomial distribution Mean and variance of the binomial distribution Poisson distribution Mean and variance of the Poisson distribution Moments of distributions DNA mutations Transition matrix Literature: Łomnicki: Statystyka dla biologów Binomial distribution: http://www.stat.yale.edu/Courses/1997- 98/101/binom.htm Poisson dstribution: http://en.wikipedia.org/wiki/Poisson_distribution