1. Reasoning in Psychology Using Statistics Psychology 138 2015

2. Reasoning in Psychology Using Statistics Exam 1(s) Lab Ex1, mean = 68.7/75 = 91.6%Lecture Ex1, mean = 57.9/75 = 77.2% Combined (Lab + Lecture) Ex1, mean = 126.6/150 = 84.4%

3. Reasoning in Psychology Using Statistics Descriptive statistics Summaries or pictures of the distribution Numeric descriptive statistics –Shape: modality, and skew (and kurtosis, not cover much) –Measures of Center: Mode, Median, Mean –Measures of Variability (Spread): Range, Inter-Quartile Range, Standard Deviation (& variance)

4. Reasoning in Psychology Using Statistics Measures of Center Useful to summarize or describe distribution with single numerical value. –Value most representative of the entire distribution, that is, of all of the individuals –Central Tendency: 3 main measures –Mean (M) –Median (Mdn) –Mode Note: “Average” may refer to each of these three measures, but it usually refers to Mean.

5. Reasoning in Psychology Using Statistics The Mean Most commonly used measure of center Arithmetic average –Computing the mean – Formula for population mean (a parameter): – Formula for sample mean (a statistic): Add up all of the X’s Divide by the total number in the population Divide by the total number in the sample M= –Note: Mean is mathematical result, not necessarily score on scale (e.g., average of 2.5 children)

6. Reasoning in Psychology Using Statistics The Mean –Conceptualizing the mean As the center of the distribution As the representative score in the distribution

7. Reasoning in Psychology Using Statistics The Mean –Conceptualizing the mean As center of distribution As representative score in distribution

8. Reasoning in Psychology Using Statistics The Mean –Conceptualizing the mean As center of distribution As representative score in distribution

9. Reasoning in Psychology Using Statistics The Mean –Conceptualizing the mean As center of distribution As representative score in distribution

10. Reasoning in Psychology Using Statistics The Mean –Conceptualizing the mean As center of distribution As representative score in distribution Balancing point

11. Reasoning in Psychology Using Statistics The Mean –Conceptualizing the mean As center of distribution As representative score in distribution 1 2 3 4 5 6 7 8 9 10 1+10 = 11 Mean = 11/2 = 5.5 Balancing point

12. Reasoning in Psychology Using Statistics The Mean –Conceptualizing the mean As center of distribution As representative score in distribution 1 2 3 4 5 6 7 8 9 10 Balancing points

13. Reasoning in Psychology Using Statistics The Mean –Conceptualizing the mean As center of distribution As representative score in distribution 1 2 3 4 5 6 7 8 9 10 What happens if we add an observation to our distribution?

14. Reasoning in Psychology Using Statistics The Mean –Conceptualizing the mean As center of distribution As representative score in distribution 1 2 3 4 5 6 7 8 9 10 What happens if we add an observation to our distribution?

15. Reasoning in Psychology Using Statistics The Mean –Conceptualizing the mean As center of distribution As representative score in distribution 1 2 3 4 5 6 7 8 9 10 What happens if we add an observation to our distribution?

16. Reasoning in Psychology Using Statistics The Mean –Conceptualizing the mean As center of distribution As representative score in distribution 1 2 3 4 5 6 7 8 9 10 What happens if we add an observation to our distribution?

17. Reasoning in Psychology Using Statistics The Mean –Conceptualizing the mean As center of distribution As representative score in distribution 1+10+7 = 18 Mean = 18/3 = 5.5 1 2 3 4 5 6 7 8 9 10 Balancing point What happens if we add an observation to our distribution?

18. Reasoning in Psychology Using Statistics 6.0 The Mean –Conceptualizing the mean As center of distribution As representative score in distribution 1 2 3 4 5 6 7 8 9 10 1+10+7 = 18 Mean = 18/3 = What happens if we add an observation to our distribution?

19. Reasoning in Psychology Using Statistics The Mean –Conceptualizing the mean As center of distribution As representative score in distribution 1+10+7 = 18 Mean = 18/3 = 6.0 1 2 3 4 5 6 7 8 9 10 What happens if we add an observation to our distribution?

20. Reasoning in Psychology Using Statistics The Mean –Conceptualizing the mean As center of distribution As representative score in distribution 1+10+7 = 18 Mean = 18/3 = 6.0 1 2 3 4 5 6 7 8 9 10 What happens if we add an observation to our distribution?

21. Reasoning in Psychology Using Statistics The Mean –Conceptualizing the mean As center of distribution As representative score in distribution 1 2 3 4 5 6 7 8 9 10 1+10+7 = 18 Mean = 18/3 = 6.0 What happens if we add an observation to our distribution? New Balancing point

22. Reasoning in Psychology Using Statistics The Mean –Conceptualizing the mean As center of distribution As the representative score in the distribution 1 2 3 4 5 6 7 8 9 10 Girl Scout bake sale for camping trip $12 $25$30$6 $18 $15 $13 12+25+30+6+18+15+13=119 119/7 = 17 To be fair, let’s give everybody the same amount.

23. Reasoning in Psychology Using Statistics The Mean –Conceptualizing the mean As center of distribution As representative score in distribution 1 2 3 4 5 6 7 8 9 10 Girl Scout bake sale for camping trip $17 So everybody is represented by same score, the mean is the “standard” 17+17+17+17+17+17+17=119 119/7 = 17 12+25+30+6+18+15+13=119 119/7 = 17

24. Reasoning in Psychology Using Statistics A weighted mean Suppose that you combine 2 groups together. –How do you compute new group mean? But it only works this way when the two groups have exactly the same number of scores Average the 2 averages

25. Reasoning in Psychology Using Statistics A weighted mean Suppose that you combine 2 groups together. –How do you compute new group mean? Group 1Group 2New Group $205!? I only have $191

26. Reasoning in Psychology Using Statistics A weighted mean Suppose that you combine 2 groups together. –How do you compute new group mean? Group 1 $12 $25 $30 $6 $18 $15 $13 Group 2 $25 $30 $17 New Group 12+25+30+6+18+15+13+25+17+30=191 Mean = 191/10 = 19.1 $12 $25 $30 $6 $18 $15 $13 $25 $30 $17

27. Reasoning in Psychology Using Statistics A weighted mean Suppose that you combine 2 groups together. –How do you compute new group mean? Group 1 $17 Group 2 $24 New Group The mean is the representative score in the distribution The mean is the representative score in the distribution

28. Reasoning in Psychology Using Statistics Characteristics of a mean –Suppose that one of the girl scouts discovered that she had really made $23 instead of $30. So now the total is 119-7=112 112/7 = $16 (instead of $17) Change/add/delete a given score, then the mean will change. 5 10 15 20 25 30 17

29. Reasoning in Psychology Using Statistics Characteristics of a mean –Suppose that one of the girl scouts discovered that she had really made $23 instead of $30. So now the total is 119-7=112 112/7 = $16 (instead of $17) Change/add/delete a given score, then the mean will change. 5 10 15 20 25 30 17

30. Reasoning in Psychology Using Statistics Characteristics of a mean –Suppose that one of the girl scouts discovered that she had really made $23 instead of $30. So now the total is 119-7=112 112/7 = $16 (instead of $17) Change/add/delete a given score, then the mean will change. 16 5 10 15 20 25 30

31. Reasoning in Psychology Using Statistics Characteristics of a 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. –Suppose that you want to factor out a $2 camping fee for each girl scout. Subtract 2 from each amount. Now the total is $105, so the mean is 105/7 = $15. –But notice you could have just subtracted $2 from the previous mean of $17 and arrived at the same answer.

32. Reasoning in Psychology Using Statistics Multiply (or divide) each score by a constant, then the mean will change by being multiplied by that constant. 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. –Suppose that the troop sponsor agreed to match the money made by each girl scout (they give each girl scout an additional amount of money equal to however much each made on the sale). So now the total is $238, and the mean for each girl is 238/7 = $34 –Which is 2 times the original mean Characteristics of a mean

33. Reasoning in Psychology Using Statistics The median Median divides distribution in half: 50% of individuals in distribution have scores at or below the median. –Case1: Odd number of scores $12 $25 $30 $6 $18 $15 $13 Step1: put scores in order

34. Reasoning in Psychology Using Statistics The median Median divides distribution in half: 50% of individuals in distribution have scores at or below the median. –Case1: Odd number of scores $12 $25$30$6 $18 $15 $13 Step1: put scores in order Step2: find middle score That’s the median, a score on scale

35. Reasoning in Psychology Using Statistics The median Median divides distribution in half: 50% of individuals in distribution have scores at or below the median. Step1: put scores in order Step2: find middle 2 scores That’s the median Note: mathematical result not a score on scale Step3: find arithmetic average of 2 middle scores $12$25$30$18$15$13$6$18 –Case2: Even number of scores

36. Reasoning in Psychology Using Statistics The mode Mode: score or category with greatest frequency. –Pick variable in frequency table or graph with highest frequency (mode always a score on scale). Mode =Modes =52, 8 Mode =Medium

37. Reasoning in Psychology Using Statistics Which center when? Depends on a number of factors, like scale of measurement and shape. –The mean is the most preferred measure and it is closely related to measures of variability –However, there are times when the mean is not the appropriate measure.

38. Reasoning in Psychology Using Statistics Which center when? If data on nominal scale: Mode only –Unranked categories (e.g. eye color) –Not a numeric scale –Can not do arithmetic operations on values –Can not calculate cumulative percentages Eye color Green Mode = Brown Median =

39. Reasoning in Psychology Using Statistics Which center when? If data on ordinal scale: Median (plus Mode) –Not a numeric scale (e.g., T-shirt size) –Can not do arithmetic operations on values –Can calculate cumulative percentages on frequencies (median is score at 50 th percentile) Median of T-shirt size = Medium Mode of T-shirt size = Medium

40. Reasoning in Psychology Using Statistics Which center when? If data on interval or ratio scale BUT: –Distributions open-ended Response category like “5 or more” Extreme values unknown, so can not calculate mean –Distributions skewed with long tails Extreme values over influence mean E.g., income sample of 50 –47 middle income ($60,000-$100,000) and 3 millionaires or billionaires –Median = $80,000 –Mean = $135,000 or $60,000,000 Median (plus Mode)

41. Reasoning in Psychology Using Statistics Which center when? If data on interval or ratio scale AND no exclusionary conditions: Mean (plus Median) (plus Mode) –Numeric scale –Can do arithmetic calculations on values –Have benefit of other statistics using the mean, such as standard deviation

42. Reasoning in Psychology Using Statistics Which center when? = mean = median mode = mean > median mode > mean < median mode < mean median, 2 modes Impact of shape on center (interval or ratio scale) Positively skewed distribution Mean & median pulled toward tail Negatively skewed distribution

43. Reasoning in Psychology Using Statistics Chicago distributions Mode 0-10,000 175-200,000 Median 45,734 261,600 Mean ? 325,212 Check out your hometown: http://www.city-data.com/

44. Reasoning in Psychology Using Statistics Buyer beware: Know your distribution Mode 0-10,000 175-200,000 Median 45,734 261,600 Mean ? 325,212 buying selling The average price of houses in this neighborhood is … When you say “average” are you talking about the median or the mean?

45. Reasoning in Psychology Using Statistics Wrap up Today’s lab –Compute mean, median, & mode both by hand & using SPSS Questions?