1. Statistics

2. The usual course of events for conducting scientific work “The Scientific Method” Reformulate or extend hypothesis Develop a Working Hypothesis Observation Conduct an experiment or a series of controlled systematic observations Appropriate statistical tests Confirm or reject hypothesis

3. The usual course of events for conducting scientific work “The Scientific Method” Reformulate or extend hypothesis Develop a Working Hypothesis Observation Conduct an experiment or a series of controlled systematic observations Appropriate statistical tests Confirm or reject hypothesis In a group of crickets, small ones seem to avoid large ones There will be movement away from large cricket by small ones Record the number of times that small crickets move away from small and large crickets. Chi square test There is a significant difference in the number of times small crickets move away from large vs. small ones Avoidance may depend on previous experience

4. Imagine that you are collecting samples (i.e. a number of individuals) from a population of little ball creatures - Critterus sphericales Little ball creatures come in 3 sizes: Small = Medium = Large =

5. -sample 1 -sample 2 -sample 3 -sample 4 -sample 5 You end up with a total of five samples

6. The real population (all the little ball creatures that exist) Your samples

7. Each sample is a representation of the population BUT No single sample can be expected to accurately represent the whole population

8. To be statistically valid, each sample must be: 1) Random: Thrown quadrat?? Guppies netted from an aquarium?

9. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Assign numbers from a random number table

10. To be statistically valid, each sample must be: 2) Replicated:

11. But not - ‘Pseudoreplication’ Not pseudoreplication Pseudoreplication 10 samples from the same tree 10 samples from 10 different trees Sample size = 1Sample size = 10

12. TYPES OF DATA

13. RATIO DATA - constant size interval - a zero point with some reality e.g. Heights, rates, time, volumes, weights

14. INTERVAL DATA - constant size interval - no true zero point zero point depends on the scale used e.g. Temperature

15. Ordinal Scale - ranked data -grades, preference surveys

16. Nominal Scale Team numbers Drosophila eye colour

17. The kind of data you are dealing with is one determining factor in the kind of statistical test you will use.

18. Statistics and Parameters

19. Measures of: Central tendency - mean, median, mode Dispersion - range, mean deviation, variance, standard deviation, coefficient of variation

20. The real population (all the little ball creatures that exist) Central tendency - Mean

21. The real population (all the little ball creatures that exist) Your samples

22. The real population (all the little ball creatures that exist)  =  X i N Central Tendency 1) Arithmetic mean At Population level Measuring the diameters of all the little ball creatures that exist  population mean X i - every measurement in the population N - population size

23. Your samples X =  X i n n n n n

24. n Sample mean Sum of all measurements in the sample Sample size

25. If you have sampled in an unbiased fashion X =  X i n n n n n Each roughly equals 

26. Central tendency - Median Median - middle value of a population or sample e.g. Lengths of Mayfly (Ephemeroptera) nymphs 5 th value (middle of 9) 123456789

27. Median value Odd number of valuesEven number of values Median = middle value Median = + 2

28. Odd number of values (i.e. n is odd) Even number of values Or - to put it more formally Median = X (n+1) 2 Median = X (n/2) + X (n/2) + 1 2

29. Frequency (= number of times each measurement appears in the population Values (= measurements taken) c. Mode - the most frequently occurring measurement Mode Central tendency - Mode

30. Measures of Dispersion Why worry about this?? -because not all populations are created equal Distribution of values in the populations are clearly different BUT means and medians are the same Mean & median

31. Measures of Dispersion - 1. Range - difference between the highest and lowest values Remember little ball creatures and the five samples Range = -

32. Range - crude measure of dispersion Note - three samples do not include the highest value and - two samples do not include the lowest

33. Measures of Dispersion - 2. Mean Deviation X is a measure of central tendency Take difference between each measure and the mean X i - X BUT  X i - X = 0 So this is not useful as it stands

34. Measures of Dispersion - 2. Mean Deviation (cont’d) But if you take the absolute value -get a measure of disperson  |X i - X| and n = mean deviation

35. Measures of Dispersion - 3. Variance -eliminate the sign from deviation from mean Square the difference (X i - X) 2 And if you add up the squared differences - get the “sum of squares”  (X i - X) 2 (hint: you’ll be seeing this a lot!)

36. Measures of Dispersion - 3. Variance (cont’d) Sum of squares can be considered at both the population and sample level ss =  (X i - X) 2 SamplePopulation SS =  (X i -  ) 2

37. s 2 =  (X i - X) 2 SamplePopulation  2 =  (X i -  ) 2 Measures of Dispersion - 3. Variance (cont’d) If you divide by the population or sample size - get the mean squared deviation or VARIANCE N n-1 Population variance Sample variance

38. s 2 =  (X i - X) 2 Note something about the sample variance n-1 Measures of Dispersion - 3. Variance (cont’d) Degrees of freedom or df or

39. Measures of Dispersion - 4. Standard Deviation - just the square root of the variance  =  (X i -  ) 2 N Population Sample s =  (X i - X) 2 n-1

40. Standard Deviation - very useful Most data in any population are within one standard deviation of the mean

41. NORMAL DISTRIBUTION

42. Type of data Discrete Continuous Other distributions 2 categories & Bernoulli process > 2 categories Use a Binomial model to calculate expected frequencies Use a Poisson distribution to calculate expected frequencies From previous slide show

43. Type of data Discrete Continuous Other distributions 2 categories & Bernoulli process > 2 categories Use a Binomial model to calculate expected frequencies Use a Poisson distribution to calculate expected frequencies Now we’re dealing with:

44. Normal Distribution - bell curve

45. Central Limit Theorem Any continuous variable influenced by numerous random factors will show a normal distribution.

46. Normal curve is used for: 2) Continuous random data Weight, blood pressure weight, length, area, rates Data points that would be affected by a large number of random (=unpredictable) events Blood pressure age physical activity smokingdiet genes stress

47. Normal curves can come in different shapes So, for comparison between them, we need to standardize their presentation in some way

48. Standarize by calculating a Z-Score Z = value of a random variable - mean standard deviation Z = X - µ  or

49. Example of a z-score calculation The mean grade on the Biometrics midterm is 78.4 and the standard deviation is 6.8. You got a 59.7 on the exam. What is your z-score? Z = X - µ  Z = 59.7 - 78.4 = -2.75 6.8

50. If you look at the formula for z-scores: z = value of a random variable - mean standard deviation z is also the number of standard deviations a value is from the mean

51. Each standard deviation away from the mean defines a certain area of the normal curve