1. STATISTICS For Research

2. Why Statistics?

3. 1. Quantitatively describe and summarize data A Researcher Can:

4. 2. Draw conclusions about large sets of data by sampling only small portions of them

5. 3. Objectively measure differences and relationships between sets of data. A Researcher Can:

6. Samples should be taken at randomSamples should be taken at random Each measurement has an equal opportunity of being selectedEach measurement has an equal opportunity of being selected Otherwise, sampling procedures may be biasedOtherwise, sampling procedures may be biased Random Sampling

7. A characteristic CANNOT be estimated from a single data pointA characteristic CANNOT be estimated from a single data point Replicated measurements should be taken, at least 10.Replicated measurements should be taken, at least 10. Sampling Replication

8. Mechanics 1. Write down a formula 2. Substitute numbers into the formula 3. Solve for the unknown.

9. The Null Hypothesis H o = There is no difference between 2 or more sets of dataH o = There is no difference between 2 or more sets of data – any difference is due to chance alone – Commonly set at a probability of 95% (P .05)

10. The Alternative Hypothesis H A = There is a difference between 2 or more sets of dataH A = There is a difference between 2 or more sets of data –the difference is due to more than just chance –Commonly set at a probability of 95% (P .05)

11. Averages Population Average = mean ( x )Population Average = mean ( x ) a Population mean = (  )a Population mean = (  ) – take the mean of a random sample from the population ( n )

12. Population Means To find the population mean (  ), add up (  ) the valuesadd up (  ) the values (x = grasshopper mass, tree height) (x = grasshopper mass, tree height) divide by the number of values (n):  =  xdivide by the number of values (n):  =  x — — n

13. Measures of Variability Calculating a mean gives only a partial description of a set of dataCalculating a mean gives only a partial description of a set of data – Set A = 1, 6, 11, 16, 21 – Set B = 10, 11, 11, 11, 12 Means for A & B ??????Means for A & B ?????? Need a measure of how variable the data are.Need a measure of how variable the data are.

14. Range Difference between the largest and smallest valuesDifference between the largest and smallest values – Set A = 1, 6, 11, 16, 21 Range = ???Range = ??? – Set B = 10, 11, 11, 11, 12 Range = ???Range = ???

15. Standard Deviation

16. A measure of the deviation of data from their mean.A measure of the deviation of data from their mean.

17. The Formula __________ SD =  N ∑ X 2 - ( ∑ X) 2 ________ N (N-1) N (N-1)

18. SD Symbols SD = Standard Dev  = Square Root ∑ X 2 = Sum of x 2 ’d ∑ (X) 2 = Sum of x’s, then squared N = # of samples

19. The Formula __________ SD =  N ∑ X 2 - ( ∑ X) 2 ________ N (N-1) N (N-1)

20. X X 2 X X 2 297 88,209 301 90,601 306 93,636 312 97,344 314 98,596 317 100,489 325 105,625 329 108,241 334 111,556 350 122,500  X = 3,185  X 2 = 1,016,797

22. You can use your calculator to find SD! Once You’ve got the Idea:

23. The Normal Curve

25. SD & the Bell Curve

26. % Increments

28. Skewed Curves median

29. Critical Values Standard Deviations  2 SD above or below the mean = due to MORE THAN CHANCE ALONE.

30. Critical Values The data lies outside the 95% confidence limits for probability.

31. Chi-Square 2 Chi-Square  2

32. Chi-Square Test Requirements Quantitative dataQuantitative data Simple random sampleSimple random sample One or more categoriesOne or more categories Data in frequency (%) formData in frequency (%) form

33. Chi-Square Test Requirements Independent observationsIndependent observations All observations must be usedAll observations must be used Adequate sample size (  10)Adequate sample size (  10)

34. Example

35. Chi-Square Symbols 2 =  (O - E) 2  2 =  (O - E) 2 E O = Observed Frequency E = Expected Frequency  = sum of d f = degrees of freedom (n-1) 2 = Chi Square  2 = Chi Square

36. Chi-Square Worksheet

37. Chi-Square Analysis Table value for Chi Square = 9.49 4 d f 4 d f P=.05 level of significance P=.05 level of significance Is there a significant difference in car preference????

38. SD & the Bell Curve

39. T-Tests

40. T-Tests For populations that do follow a normal distribution

41. T-Tests Drawing conclusions about similarities or differences similarities or differences between population means (  )

42. T-Tests Is average plant biomass the same in two different geographical areas ???Is average plant biomass the same in two different geographical areas ??? Two different seasons ???Two different seasons ???

43. T-Tests COMPLETELY confident answer =COMPLETELY confident answer = – measure all plant biomass in each area Is this PRACTICAL?????Is this PRACTICAL?????

44. Instead: Take one sample from each populationTake one sample from each population Infer from the sample means and SD whether the populations have the same or different means.Infer from the sample means and SD whether the populations have the same or different means.

45. Analysis SMALL t values = high probability that the two population means are the sameSMALL t values = high probability that the two population means are the same LARGE t values = low probability (means are different)LARGE t values = low probability (means are different)

46. Analysis T calculated > t critical = reject H o T calculated > t critical = reject H o t critical t critical t critical t critical

47. We will be using computer analysis to perform the t- test

48. Simpson’s Diversity Index

49. Nonparametric Testing For populations that do NOT follow a normal distributionFor populations that do NOT follow a normal distribution – includes most wild populations

50. Answers the Question If 2 indiv are taken at RANDOM from a community, what is the probability that they will be the SAME species????If 2 indiv are taken at RANDOM from a community, what is the probability that they will be the SAME species????

51. The Formula D = 1 -  n i (n i - 1) D = 1 -  n i (n i - 1) ————— ————— N (N-1) N (N-1)

52. Example

53. Example D = 1- 50(49)+25(24)+10(9) D = 1- 50(49)+25(24)+10(9)———————————85(84) D = 0.56 D = 0.56

54. Analysis Closer to 1.0 = Closer to 1.0 = – more Homogeneous community Farther away from 1.0 = Farther away from 1.0 = – more Heterogeneous community

55. You can calculate by hand to find “D”You can calculate by hand to find “D” School Stats package MAY calculate it.School Stats package MAY calculate it.

56. © 2000 Anne F. Maben All rights reserved