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. 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

8. 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

9. 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

10. 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

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

12. What is the probability that of 10 newborn babies at least 7 are boys? p(girl) = p(boy) = 0.5 Lecture 10 Important statistical distributions Bernoulli distribution

13. The Bernoulli or binomial distribution comes from the Taylor expansion of the binomial Bernoulli or binomial distribution

14. Assume the probability to find a certain disease in a tree population is 0.01. A bio- monitoring program surveys 10 stands of trees and takes in each case a random sample of 100 trees. How large is the probability that in these stands 1, 2, 3, and more than 3 cases of this disease will occur? Mean, variance, standard deviation

15. What happens if the number of trials n becomes larger and larger and p the event probability becomes smaller and smaller. Poisson distribution The distribution or rare events

16. Assume the probability to find a certain disease in a tree population is 0.01. A bio- monitoring program surveys 10 stands of trees and takes in each case a random sample of 100 trees. How large is the probability that in these stands 1, 2, 3, and more than 3 cases of this disease will occur? Poisson solution Bernoulli solution The probability that no infected tree will be detected The probability of more than three infected trees Bernoulli solution

17. Variance, mean Skewness

18. What is the probability in Duży Lotek to have three times cumulation if the first time 14 000 000 people bet, the second time 20 000 000, and the third time 30 000 000? The probability to win is The events are independent: The zero term of the Poisson distribution gives the probability of no event The probability of at least one event:

19. A pile model to generate the binomial. If the number of steps is very, very large the binomial becomes smooth. The normal distribution is the continous equivalent to the discrete Bernoulli distribution Abraham de Moivre (1667-1754)

20. If we have a series of random variates Xn, a new random variate Yn that is the sum of all Xn will for n→∞ be a variate that is asymptotically normally distributed. The central limit theorem

21. The normal or Gaussian distribution Mean:  Variance:  2

22. Important features of the normal distribution The function is defined for every real x. The frequency at x = m is given by The distribution is symmetrical around m. The points of inflection are given by the second derivative. Setting this to zero gives

23. ++ -- 0.68 +2  -2  0.95 Many statistical tests compare observed values with those of the standard normal distribution and assign the respective probabilities to H 1.

24. The Z-transform The variate Z has a mean of 0 and and variance of 1. A Z-transform normalizes every statistical distribution. Tables of statistical distributions are always given as Z- transforms. The standard normal The 95% confidence limit

25. P(  -  < X <  +  ) = 68% P(  - 1.65  < X <  + 1.65  ) = 90% P(  - 1.96  < X <  + 1.96  ) = 95% P(  - 2.58  < X <  + 2.58  ) = 99% P(  - 3.29  < X <  + 3.29  ) = 99.9% The Fisherian significance levels ++ -- 0.68 +2  -2  0.95 The Z-transformed (standardized) normal distribution

26. x,s  The estimation of the population mean from a series of samples The n samples from an additive random variate. Z is asymptotically normally distributed. Confidence limit of the estimate of a mean from a series of samples.  is the desired probability level. Standard error

27. How to apply the normal distribution Intelligence is approximately normally distributed with a mean of 100 (by definition) and a standard deviation of 16 (in North America). For an intelligence study we need 100 persons with an IO above 130. How many persons do we have to test to find this number if we take random samples (and do not test university students only)?

29. One and two sided tests We measure blood sugar concentrations and know that our method estimates the concentration with an error of about 3%. What is the probability that our measurement deviates from the real value by more than 5%?

30. Albinos are rare in human populations. Assume their frequency is 1 per 100000 persons. What is the probability to find 15 albinos among 1000000 persons? =KOMBINACJE(1000000,15)*0.00001^15*(1-0.00001)^999985 = 0.0347