1. STATISTICS For Research

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

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

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

5. 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

6. 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

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

8. 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)

9. 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)

10. 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 )

11. Population Means To find the population mean (  ), add up () the valuesadd up (Σ) 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

12. 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.

13. 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 = ???

14. Standard Deviation

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

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

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

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

19. 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

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

22. The Normal Curve

24. SD & the Bell Curve

25. % Increments

27. Skewed Curves median

28. Critical Values Standard Deviations  2 SD above or below the mean = “due to more than chance alone.” THIS MEANS: The data lies outside the 95% confidence limits for probability. Your research shows there is something significant going on...

29. Chi-Square 2 Chi-Square  2

30. 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 Independent observationsIndependent observations All observations must be usedAll observations must be used Adequate sample size (  10)Adequate sample size (  10)

31. Example

32. Chi-Square Symbols  2 = Σ (O - E) 2 E O = Observed Frequency E = Expected Frequency Σ = sum of df = degrees of freedom (n -1)  2 = Chi Square

33. Chi-Square Worksheet

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

35. SD & the Bell Curve

36. T-Tests

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

38. T-Tests To draw conclusions about similarities or differences between population means (  )To draw conclusions about similarities or differences between population means (  ) Is average plant biomass the same inIs average plant biomass the same in –two different geographical areas ??? –two different seasons ???

39. T-Tests To be COMPLETELY confident you would have to measure all plant biomass in each area.To be COMPLETELY confident you would have to measure all plant biomass in each area. –Is this PRACTICAL?????

40. Instead: Take one sample from each population.Take one sample from each population. Infer from the sample means and standard deviation (SD) whether the populations have the same or different means.Infer from the sample means and standard deviation (SD) whether the populations have the same or different means.

41. 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 (the means are different)LARGE t values = low probability (the means are different)

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

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

44. Simpson’s Diversity Index

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

46. 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????

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

48. Example

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

50. Analysis Closer to 1.0 = Closer to 1.0 = – more Homogeneous community (low diversity) Farther away from 1.0 = Farther away from 1.0 = – more Heterogeneous community (high diversity)

51. 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.

52. Designed by Anne F. Maben Former AP Science Coach, LACOE for the Los Angeles County Science Fair © 2013 All rights reserved This presentation is for viewing only and may not be published in any form