1. Last lecture summary The nature of the normal distribution Non-Gaussian distributions

2. New stuff

3. Lognormal distribution Frazier et al. measured the ability of a drug isoprenaline to relax the bladder muscle. The results are expressed as the EC50, which is the concentration required to relax the bladder halfway between its minimum and maximum possible relaxation.

4. Lognormal distribution

5. Geometric mean Geometric mean – transform all values to their logarithms, calculate the mean of the logarithms, transform this mean back to the units of original data (antilog)

6. The nature of the lognormal distribution Lognormal distributions arise when multiple random factors are multiplied together to determine the value. A typical example: cancer (cell division is multiplicative) Lognormal distributions are very common in many scientific fields. Drug potency is lognormal To analyse lognormal data, do not use methods that assume the Gaussian distribution. You will get misleding results (e.g.,non-existing outliers). Better way is to convert data to logarithm and analyse the converted values.

7. How normal is normal? http://www.nate-miller.org/blog/how-normal-is-normal-a-q-q-plot-approach Checking normality 1.Eyball histograms 2.Eyball QQ plots 3.There are tests

8. QQ plot Q stands for ‘quantile’. Quantiles are values taken at regular intervals from the data. The 2-quantile is called the median, the 3-quantiles are called terciles, the 4-quantiles are called quartiles (deciles, percentiles).

9. Typical normal QQ plot http://emp.byui.edu/BrownD/Stats-intro/dscrptv/graphs/qq-plot_egs.htm

10. QQ plot of left-skewed distribution http://emp.byui.edu/BrownD/Stats-intro/dscrptv/graphs/qq-plot_egs.htm

11. QQ plot of right-skewed distribution http://emp.byui.edu/BrownD/Stats-intro/dscrptv/graphs/qq-plot_egs.htm

12. SAMPLING DISTRIBUTIONS výběrová rozdělení

14. Histogram

16. Sampling distribution of sample mean výběrové rozdělení výběrového průměru

17. Sweet demonstration of the sampling distribution of the mean

18. 2 3 3 1 2 6 5 5 4 3 5 5 4 2 6 3 4 3 1 5

19. 2 3 3 1 2 6 5 5 4 3 5 5 4 2 6 3 4 3 1 5 průměr = 3.3 průměr = 1.7

20. Data 2015 Population: 4,3,3,5,0,4,4,4,3,4,2,6,8,2,4,3,5,7,3,3 25 samples (n=3) and their averages 3,5,3,4,2,3,3,3,5,5,3,4,3,4,5,4,4,4,6,3,4,3,4,3,4 http://blue-lover.blog.cz/1106/lentilky

21. Histogram of 2015 data

22. 2015, n = 3, number of samples = 25

23. Going further So far, we have generated 25 samples with n = 3. To improve our histogram, we need more samples. However, we don’t want to spend ages in the classroom. Thus, I have prepared a simulation for you. In this simulation, I use data from 2014 and I generate all possible samples, n = 3.

24. Sampling distribution, n = 3 1 540 samples

25. Sampling distribution, n = 5 42 504 samples

26. Sampling distribution, n = 10 20 030 010 samples

27. Central limit theorem (CLT)

28. Non-Gaussian distribution 1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,5,5,6,7,7,8,8,8,9,9,9,9,10,10,10,10,10,11,11,11,11,11,11

29. Sampling distribution n = 2

30. Sampling distribution n = 4

31. Sampling distribution n = 6

32. Sampling distribution n = 8

33. Back to CLT Once we know that the sampling distribution of the sample mean is normal, we want to characterize this distribution. By which numbers you characterize a distribution? mean standard deviation

34. Back to CLT

35. M and SE Let’s have a look at our demonstration data: 1. Calculate population mean, population standard deviation and standard error for n=3. 2. Take all our sample means and calculate their mean. It should be close to the population mean. 3. Take all our sample means and calculate their standard deviation. It should be close to the standard error.

36. M and SE pop_mean <- mean(data.set2015) pop_sd <- sd(data.set2015)*sqrt(19/20) se <- pop_sd/sqrt(3) sampl_mean <- mean(prumery2015) sampl_sd <- sd(prumery2015)

37. Quiz As the sample size increases, the standard error increases decreases As the sample size increases, the shape of the sampling distribution gets skinnier wider

38. Sampling distribution applet parent distribution sample data sampling distributions of selected statistics http://onlinestatbook.com/stat_sim/sampling_dist/index.html