1. Reasoning in Psychology Using Statistics Psychology 138 2015

2. Reasoning in Psychology Using Statistics Announcements Exam 2 in lecture and lab on Wednesday Be prepared to do calculations (including square roots) on calculator

3. Reasoning in Psychology Using Statistics Cautions with Correlations Mathematical cautions –Different scales: convert to z-scores –Restriction of range (e.g., age & height) –Outliers (especially in small samples) Interpretive caution –Causal claims

4. Reasoning in Psychology Using Statistics Pearson’s r, z transformation Y X 1 2 3 4 5 6 123 45 6 Both variables on same scale Correlation stays the same What happens to means? Change all scores to z-scores 3.6 z y z x -1.5 -0.5 0 0.5 -1.5 -1 -.5 0.5 1 1.5 Convert X and Y to z-scores 1.0 1.5

5. Reasoning in Psychology Using Statistics Restriction of range Total data for positive correlation between SAT and GPA. What correlation between SAT and GPA in only those with admitted and studied (400 < SAT < 700)? Get r = 0

6. Reasoning in Psychology Using Statistics Outliers One extreme score can change correlation (especially in small sample). On left, 5 observations, high X associated with high Y: good predictability. On right, same 5 observations plus 1 other, high X associated with high or low Y: poor predictability.

7. Reasoning in Psychology Using Statistics Causal claims We ’ d like to say: –X causes Y To be able to do this: 1.The causal variable must come first 2.There must be co-variation between the two variables 3.Need to eliminate plausible alternative explanations Careful: Do not make casual claims based on correlations Correlation procedures address point 2, but say nothing about points 1 and 3.

8. Reasoning in Psychology Using Statistics Causal claims –Happy people sleep well Directionality Problem Or is it that sleeping well makes you happy?

9. Reasoning in Psychology Using Statistics Causal claims –Happy people sleep well –Or does sleeping well make you happy? –OR something else makes people happy and sleep well! Regular exercise Minimal use of drugs & alcohol Being a conscientious person Being a good relationship Etc. Third Variable Problem:

10. Reasoning in Psychology Using Statistics Review for Exam 2: Descriptive statistics Statistical procedures to help organize, summarize & simplify large sets of data 1.One variable (frequency distribution) Display results in a frequency distribution table & histogram (or bar chart if categorical variable). Make a deviations table to get measures of central tendency ( mode, median, mean ) & variability ( range, standard deviation, variance ). 2.Two variables (bivariate distribution) Display results: Make a scatterplot. Make a bivariate deviations or z-table table to get Pearson’s r. 3.Z-scores & normal distribution

11. Reasoning in Psychology Using Statistics Example Are hours sleeping related to GPA? –You conduct a survey. Your sample of 10 gives these results for average hours per night sleeping: 7, 6, 7, 8, 8, 7, 9, 5, 9, 6 You also have respondents give their overall GPA: 2.4, 3.9, 3.5, 2.8, 3.0, 2.1, 3.9, 2.9, 3.6, 2.7 –We will focus on sleep results first and then both variables together. What kind of scales are they? To find standard deviation, will we use formula for population or sample?

12. Reasoning in Psychology Using Statistics Step 1: Frequency distribution & histogram Hrs. sleep n=10 7,6,7,8,8 7,9,5,9,6 Xfp%cfc% 9 8 7 6 5 ∑101.0100

13. Reasoning in Psychology Using Statistics Step 1: Frequency distribution & histogram Hrs. sleep n=10 7,6,7,8,8 7,9,5,9,6 Xfp%cfc% 92 82 73 62 51 ∑ 10 1.0100 Will enter first two columns as X and Y axes for frequency distribution

14. Reasoning in Psychology Using Statistics Step 1: Frequency distribution & histogram Hrs. sleep n=10 p = f/n Xfp%cfc% 92.220 82.220 73.330 62.220 51.110 ∑ 1.0 100

15. Reasoning in Psychology Using Statistics Step 1: Frequency distribution & histogram Xfp%cfc% 92.220 82.220 73.330 62.220 51.1101 ∑ 1.0 100

16. Reasoning in Psychology Using Statistics Step 1: Frequency distribution & histogram Xfp%cfc% 92.220 82.220 73.330 62.220330 51.1101 ∑ 1.0 100

17. Reasoning in Psychology Using Statistics Step 1: Frequency distribution & histogram Xfp%cfc% 92.220 82.220 73.330660 62.220330 51.1101 ∑ 1.0 100

18. Reasoning in Psychology Using Statistics Step 1: Frequency distribution & histogram Xfp%cfc% 92.220 82.220880 73.330660 62.220330 51.1101 ∑ 1.0 100

19. Reasoning in Psychology Using Statistics Step 1: Frequency distribution & histogram Xfp%cfc% 92.22010100 82.220880 73.330660 62.220330 51.1101 ∑ 1.0 100

20. Reasoning in Psychology Using Statistics Step 1: Frequency distribution & histogram Hrs. sleep 6 5 4 3 2 1 56789 SCORE FREQUENCYFREQUENCY Xf 92 82 73 62 51

21. Reasoning in Psychology Using Statistics A weighted mean Suppose that you combine two groups together. –How do you compute the new group mean? Group 1Group 2New Group 110 140

22. Reasoning in Psychology Using Statistics A weighted mean Suppose that you combine two groups together. –How do you compute the new group mean? Group 1Group 2New Group 110 140 Xf 92 82 73 62 51 Be careful computing the mean of this distribution, remember there are groups here 99887776659988777665

23. Reasoning in Psychology Using Statistics Characteristics of a mean & standard deviation The mean –Change/add/delete a given score, then the mean will change. –Add/subtract a constant to each score, then the mean will change by adding(subtracting) that constant. –Multiply (or divide) each score by a constant, then the mean will change by being multiplied by that constant. The standard deviation –Change/add/delete a given score, then the mean will change. –Add/subtract a constant to each score, then the standard deviation will NOT change. –Multiply (or divide) each score by a constant, then the standard deviation will change by being multiplied by that constant.

24. Reasoning in Psychology Using Statistics Step 2: Deviations table Hrs. sleep n = 10 9 9 8 8 7 7 7 6 6 5 X Create table, sorted in descending order

25. Reasoning in Psychology Using Statistics Step 2: Deviations table Hrs. sleep n = 10 9 9 8 8 7 7 7 6 6 5 X Mode = 7 (filled in) Median = 7 (arrow) Mean = (∑X)/n = 72/10 = 7.2 Range = 5 to 9 ∑ 72

26. Reasoning in Psychology Using Statistics Step 2: Deviations table Hrs. sleep n = 10 91.8 9 8.8 8 7-.2 7 7 6-1.2 6 5-2.2 X Mode = 7 Median = 7 Mean = (∑X)/n = 72/10 = 7.2 Range = 5 to 9 ∑ 72 7.2 ∑ 72 7.2 0 0 = 9-7.2

27. Reasoning in Psychology Using Statistics Step 2: Deviations table Hrs. sleep n = 10 91.83.24 91.83.24 8.8.64 8.8.64 7-.2.04 7-.2.04 7-.2.04 6-1.21.44 6-1.21.44 5-2.24.84 X Mode = 7 Median = 7 Mean = ∑X/n = 72/10 = 7.2 Range = 5 to 9 SD for sample ∑ 72 7.2 ∑ 72 7.2 0 0 15.6 = SS = √15.6/9 = √1.73 = 1.32 = 1.8 2

28. Reasoning in Psychology Using Statistics Step 3: Scatterplot 4.0 3.5 3.0 2.5 2.0 1.5 1.0 56789 PersonHrs.GPA A72.4 B63.9 C73.5 D82.8 E83.0 F72.1 G93.9 H52.9 I93.6 J62.7 Hours of sleep

29. Reasoning in Psychology Using Statistics Step 3: Scatterplot 4.0 BG 3.5 CI 3.0 HDE 2.5 JA 2.0 F 1.5 1.0 56789 PersonHrs.GPA A72.4 B63.9 C73.5 D82.8 E83.0 F72.1 G93.9 H52.9 I93.6 J62.7 What does shape of envelope indicate about correlation? low positive correlation Hours of sleep

30. Reasoning in Psychology Using Statistics Step 3: Scatterplot, Effect of outlier 4.0 BG 3.5 CI 3.0 HDE 2.5 JA 2.0 F 1.5 1.0 K 56789 PersonHrs.GPA A72.4 B63.9 C73.5 D82.8 E83.0 F72.1 G93.9 H52.9 I93.6 J62.7 K51.0 What does shape of envelope indicate about correlation? moderate positive correlation Hours of sleep

31. Reasoning in Psychology Using Statistics Step 3: Scatterplot, Effect of outlier 4.0 BG 3.5 CI 3.0 HDE 2.5 JA 2.0 F 1.5 1.0 K 56789 PersonHrs.GPA A72.4 B63.9 C73.5 D82.8 E83.0 F72.1 G93.9 H52.9 I93.6 J62.7 K91.0 What does shape of envelope indicate about correlation? low negative correlation Hours of sleep

32. Reasoning in Psychology Using Statistics Step 4: Bivariate Deviations Table X Y 91.83.243.90.820.671.476 91.83.243.60.520.270.936 80.80.643.0-0.080.01-0.064 80.80.642.8-0.280.86-0.224 7-0.20.403.50.420.18-0.084 7-0.20.042.4-0.680.460.136 7-0.20.042.1-0.980.960.196 6-1.21.443.90.820.67-0.984 6-1.21.442.7-0.380.140.456 5-2.24.842.9-0.180.030.396 72 0.015.630.8 0.03.472.24 7.2SS X 3.08SS Y SP Sum Mean n=10 Note signs! +r or – r?

33. Reasoning in Psychology Using Statistics Pearson’s r & summary statistics = ___2.24___ √ 15.6 * 3.47 = _2.24_ √54.132 = _2.24_ =.304 7.357 XY co-deviations X deviations, Y deviations

34. Reasoning in Psychology Using Statistics An example Based on normative data: Normal, μ = 50.0, σ = 10.0 μ 6040 –Write down what you know –Make a sketch of the distribution (make a note: population or sample) –Determine the shape –What is best measure of center? –What is best measure of variability? –Mark the mean (center) and standard deviation on your sketch Preparing for your analyses SRA (Scientific Reasoning Assessment) (fictional)

35. Reasoning in Psychology Using Statistics z-scores & Normal Distribution SRA (Scientific Reasoning Assessment) (fictional)  6040 If George got a 35 on the SRA, what is his percentile rank? Question 1 That’s 6.68% at or below this score (definition of percentile) Unit Normal Table Based on normative data: Normal distr.,  = 50.0, σ = 10.0 1.0 0.0668 Since a normal distribution, can use Unit Normal Table to infer percentile.

36. Reasoning in Psychology Using Statistics z-scores & Normal Distribution SRA (Scientific Reasoning Assessment) (fictional)  40 What proportion of people get between a 40 and 60 on the SRA? Question 2 Based on normative data: Normal distr., μ = 50.0, σ = 10.0 Unit Normal Table That leaves 68% between these two scores 0.1587 1.0 4060 That’s about 32% outside these two scores Since a normal distribution, can use Unit Normal Table to infer percentile.

37. Reasoning in Psychology Using Statistics z-scores & Normal Distribution SRA (Scientific Reasoning Assessment) (fictional) Suppose that Chandra took a different reasoning assessment (the RSE: Based on normative data, Normal distr., μ= 100, σ = 15). She received a 130 on the RSE. Assuming that they are highly positively correlated, what is the equivalent score on the SRA? Question 3a transformation Based on normative data: Normal distr., μ = 50.0, σ = 10.0

38. Reasoning in Psychology Using Statistics z-scores & Normal Distribution SRA (Scientific Reasoning Assessment) (fictional) Suppose that Chandra took a different reasoning assessment ( RSE: Based on normative data, Normal distr., μ= 100, σ = 15). She received a 130 on the RSE. Assuming that they are highly positively correlated, what is the equivalent score on the SRA? Question 3a Based on normative data: Normal distr., μ = 50.0, σ = 10.0 transformation (for RSE) (for SRA) Now know that predict equivalent only if r RSE,SRA = 1.0.

39. Reasoning in Psychology Using Statistics z-scores & Normal Distribution SRA (Scientific Reasoning Assessment) (fictional) Suppose that Chandra took a different reasoning assessment (the RSE: Based on normative data, Normal distr., μ= 100, σ = 15). She received a 130 on the RSE. Assuming that they are perfectly positively correlated, what is the equivalent score on the SRA? Question 3c Based on normative data: Normal distr., μ = 50.0, σ = 10.0 transformation (for RSE) (for SRA) What percent of those taking either test will score below Chandra? Know z = 2 From Unit Normal Table, p(z ≥ 2) =.0228 p(z < 2) = 1 -.0228 =.9772 = 98%

40. Reasoning in Psychology Using Statistics z-scores, Normal Distribution, & Correlation SRA (Scientific Reasoning Assessment) (fictional) Suppose that Chandra took a different reasoning assessment ( RSE: Based on normative data, Normal distr., μ= 100, σ = 15). She received a 130 on the RSE. Assuming that they are highly positively correlated, what is the equivalent score on the SRA? Question 3b Based on normative data: Normal distr., μ = 50.0, σ = 10.0 transformation (for RSE) = 2.0 (for SRA) If r RSE,SRA =.8, what is our best estimate of her actual score? More on this later in course z y = r z x so z SRA =.8 * 2 = 1.6

41. Reasoning in Psychology Using Statistics Wrap up In lab: continue to review, including SPSS Questions?