Basic Statistics

1. Basic Statistics Michael Hylin

2. Scientific Method • Start w/ a question • Gather information and resources (observe) • Form hypothesis • Perform experiment and collect data • Analyze data • Interpret data & draw conclusions • form new hypotheses • Retest (frequently done by other scientists) • i.e. replicate & extend

3. Example • Pavlov noticed that when the dogs saw the lab tech they salivated the same as when they saw meat powder (Observation) • Predicted that other stimuli could elicit this response when paired w/ meat powder (Hypothesis)

4. Example • Pavlov found that when a bell was paired w/ the presence of meat powder an association occurred (Experimentation) • Concluded that pairing of US w/ CS could lead to CR (Interpretation) • Research since Pavlov has demonstrated the mechanism of how CC works (e.g. Aplysia)

5. Basics of Experimental Design • Types of Variables • Types of Comparisons • Types of Groups

6. Types of Variables • Independent Variable • Manipulated by the experimenter • May have several • Dependent Variable • Dependent upon the IV • The data • IV → DV

7. Types of Comparisons • Between-subjects • Comparing one group to another • Within-subjects • Comparing a subject’s results at one point to another point • Usually referred to as repeated-measures

8. Types of Groups • Experimental Group • Receives experimental manipulation • Control Group • “controls” for the effect of manipulation

9. Example • A researcher has a new drug (M100) that improves semantic memory in normal individuals. • The researcher decides to test M100’s effectiveness by giving the drug to participants and testing their ability to memorize a list of words. Other participants are given a sugar pill and told to memorize the list as well.

10. Example • What is the IV? the DV? • Additional IVs & DVs • What was the control? • What type of comparison was being done? • Could it be different?

11. What about statistics? • Why do we need statistics? • Cannot rely solely upon anecdotal evidence • Make sense of raw data • Describe behavioral outcomes • Test hypotheses

12. Measures of Central Tendency • Mode • Frequency, most common ‘score’ • Median • Point at or below 50% of scores fall when the data is arranged in numerical order • Used typically w/ non-normal distributions • Mean (Often expressed ) • Sum of the scores divided by the number of scores

13. Mean

14. Example • Data for number of words recalled 8, 14, 17, 10, 8 • Mode = 8 • Median = 10 (8 , 8, 10, 14, 17) • Mean = 8+14+17+10+8 = 11.4 5

15. Measures of Variability • Range • Difference between highest and lowest scores • Variance (s2) • Standard Deviation (s) • Standard Error of the Mean (S.E.M.)

16. Variance • Equation for Variance Where:

17. Variance • Another Equation for Variance Where: &

18. Standard Deviation • Equation for Standard Deviation Or

19. Example • Data for number of words recalled 8, 14, 17, 10, 8 • Range = 17 – 8 = 9 • Variance = 15.8 • Standard Deviation = 3.97

20. Example Variance Standard Deviation

21. Mean & Standard Deviation

22. Null Hypothesis • Start w/ a research hypothesis • “Manipulation” has an effect • e.g. Students given study techniques have a higher GPA • Set up the null hypothesis • “Manipulation” has NO effect • e.g. Students w/ techniques are no diff. than those w/o techniques

23. Null Hypothesis • Does the manipulation have an effect • Use a critical value to test our hypothesis • Usually 0.05

24. Hypothesis Testing True State of the World Decision H0 True H0 False Reject H0 Type I Error p = α Correct decision p = 1 – β = Power Accept H0 Correct decision p = 1 - α Type II Error p = β

25. Hypothesis Testing • Not truly ‘proving’ our hypothesis • In reality we are setting up a situation where there is no relationship between the variables and then testing whether or not we can reject this (null hypothesis)

26. Independent T-Test • Test whether our samples come from the same population or different populations

27. Equation for Independent T-Test

28. å å = = X X 12 15 . . 94 76 1 2 Group 1 (study techniques) Group 2 (no techniques) GPA GPA 3.41 2.54 3.16 3.10 2.98 2.10 2.95 2.40 3.26 2.80

32. Since our observed t = 3.04 which is greater than 2.306 we can reject the null hypothesis • Therefore the probability of the difference we observed occurring when the null hypothesis is true is less than 0.05 (5%) • As a result our effect is likely due to the training

33. Degrees of Freedom • 6, 8, 10 • Mean = 8 • If we change two numbers the other is determine if we want to keep Mean = 8 • 67 & 1013 then the final number is 4

34. IV with more than two levels • Sometimes we want to compare more that just two groups • Cannot just due multiple t-tests • Increase alpha • Simple analysis of variance • 1-way ANOVA

35. Multiple IVs • Factoral ANOVA • Allow for comparison of more than one IV • IVs can be between or within • If both its called mixed ANOVA (repeated measures) • Interaction of IVs • E.g. 2x2 ANOVA • IV1 Study group (no study vs. study) • IV2 Time at testing (pre. vs. post.)

36. Example

37. ANOVA Table

38. Example

39. ANOVA Table

40. F-Score Equation

42. What about further group comparisons • Significant main effects with more than 2 levels • Post hoc comparisons • Significant interactions • Simple effects