1. Last lecture summary Bias, Bessel's correction MAD Normal distribution Empirical rule

2. Standard deviation – empirical rule

5. The nature of the normal distribution In laboratory experiments, results vary due to several factors: imprecise weighting of reagents, imprecise pipetting, nonhomogeneous suspensions of cells or membarnes... Similarly, variation in a clinical value might be caused by many genetic and environmental factors. These random factors are independent and they tend to offset each other. Variation among values will approximate a Gaussian distribution when there are many independent sources of variation when individual sources add up to get the final result

6. New stuff

7. STANDARD NORMAL DISTRIBUTION

8. Who is more popular?

9. Who is more popular s.d. = 36 s.d. = 60 Z = -3.53 Z = -2.57

10. Standardizing

11. Formula

12. Quiz What does a negative Z-score mean? 1. The original value is negative. 2. The original value is less than mean. 3. The original value is less than 0. 4. The original value minus the mean is negative.

13. Quiz II If we standardize a distribution by converting every value to a Z-score, what will be the new mean of this standardized distribution? If we standardize a distribution by converting every value to a Z-score, what will be the new standard deviation of this standardized distribution?

14. Standard normal distribution

15. Z Z – number of standard deviations away from the mean If the Z-value is 1, how many percent are less than that value? cca 84 % 0 +1+2+3-2 -3

16. Proportion of human heights

17. +1-2+20

18. Quiz Approximately what proportion of people is smaller than 168 cm? 173178 183 168163 16%

19. Quiz Approximately what proportion of people is higher than 183 cm? 173178 183 168163 2.5%

20. Quiz Approximately what proportion of people is between 163 cm and 178 cm high? 173178 183 168163 81.5%

21. Quiz Approximately what proportion of people is smaller than 180 cm? 173178 183 168163 ca 91.5%

22. Quiz What is the probability of randomly selecting a height in the sample that is >5 standard deviations above the mean? 1. 0.01 2. 0.3 3. 0.8 4. 0.99

23. Quiz What is the probability of randomly selecting a height in the sample that is <5 standard deviations below the mean? 1. 0.01 2. 0.3 3. 0.8 4. 0.99

24. Quiz What proportion of the data is either below 2 standard deviations or above 2 standard deviations from the mean for a normal distribution? 95% 2.5%

25. Z-table What is the proportion less than the point with the Z-score -2,75?

26. Use Z-table

27. Quiz – height data

29. An intriguing fact

30. DISTRIBUTION, DISTRIBUTION, ARE YOU NORMAL?

31. Life expectancy data – histogram life expectancy frequency

32. Making conclusions from a histogram What can you tell about life expectancy data? how many modes? where is the mode? symmetric, left skewed or right skewed? outliers – yes or no? life expectancy frequency

33. Making conclusions from a histogram Where is the mode, the median, the mean? life expectancy frequency

34. Min. Q1 Median Q3 Max. 47.79 64.67 73.24 76.65 83.39 Five numbers summary

35. 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.

36. Lognormal distribution

37. 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)

38. 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.