1. STATISTICS For Research

2. Why Statistics?

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

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

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

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

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

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

9. The Null Hypothesis
Ho = 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
HA = 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 ) a Population mean = (  )
take the mean of a random sample from the population ( n )

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

13. Measures of Variability
Calculating 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 ??????
Need a measure of how variable the data are.

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

15. Standard Deviation

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

17. SD = N ∑X2 - (∑X)2 ________ N (N-1)
The Formula
__________
SD = N ∑X2 - (∑X) ________
N (N-1)

18. ∑(X)2 = Sum of x’s, then squared N = # of samples
SD Symbols
SD = Standard Dev
 = Square Root
∑X = Sum of x2’d
∑(X)2 = Sum of x’s, then squared
N = # of samples

19. SD = N ∑X2 - (∑X)2 ________ N (N-1)
The Formula
__________
SD = N ∑X2 - (∑X) ________
N (N-1)

20. X X2
,209
,601
,636
,344
,596
,489
,625
,241
,556
,500
X = 3,  X2 = 1,016,797

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

23. The Normal Curve

24. 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

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

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

34. Example

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

36. Chi-Square Worksheet

37. Chi-Square Analysis 4 d f Table value for Chi Square = 9.49
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
between population means
(  )

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

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

44. Instead: Take one sample from each population
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 same
LARGE t values = low probability (means are different)

46. Analysis
Tcalculated > tcritical = reject Ho
tcritical tcritical

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 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????

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

52. Example

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

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

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

56. Former AP Science Coach, LACOE Los Angeles County Science Fair
Designed by
Anne F. Maben
Former AP Science Coach, LACOE
for the
Los Angeles County Science Fair
© 2012 All rights reserved
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