Psych 230 Psychological Measurement and Statistics

1. Psych 230Psychological Measurement and Statistics Pedro Wolf September 2, 2009

2. Previously on “let’s learn statistics in five weeks” • the logic of research • samples, populations, and variables • descriptive and inferential statistics • statistics and parameters • understanding experiments • experimental and correlational studies • independent and dependent variables • characteristics of scores • nominal, ordinal, interval, and ratio scales • continuous and discrete

3. Which Scale? Does the variable have an intrinsic value? NO YES Nominal Does the variable have equal values between scores? NO YES Ordinal Does the variable have a real zero point? NO YES Interval Ratio

4. Continuous • A continuous scale allows for fractional amounts • it ‘continues’ between the whole-number amount • decimals make sense • Examples: • Height • Weight • IQ

5. Discrete • In a discrete scale, only whole-number amounts can be measured • decimals do not make sense • usually, nominal and ordinal scales are discrete • some interval and ratio variables are also discrete • number of children in a family • Special type of discrete variable: dichotomous • only two amounts or categories • pass/fail; living/dead; male/female

6. Today…. • Why graphical representations of data? • Stem and leaf plots. • Box plots. • Frequency • what is it • how a frequency distribution is created • Graphing frequency distributions • bar graphs, histograms, polygons • Types of distribution • normal, skewed, bimodal • Relative frequency and the normal curve • percentiles, area under the normal curve

7. “… look at the data” (Robert Bolles, 1998) • Raw data is often messy, overwhelming, and un-interpretable. • Many data sets can have thousands of measurements and hundreds of variables. • Graphical representations of data can make data interpretable • Looking at the data can inspire ideas.

8. What in the world could these data mean?Imagine over 30,000 observations

9. After plotting those data • By plotting the data and • superimposing it on map • data, suddenly the previous • slide’s data can tell a story • Of course not all data can • tell such a story • People have developed • various ways to visualize • their data graphically

10. Stem and Leaf Plots 5 | 4 6 7 9 9 5 6 | 3 4 6 8 8 5 7 | 2 2 5 6 4 8 | 1 4 8 3 9 | 0 10 | 6 1 N = 18 • data - 54, 56, 57, 59, 59, 63, 64, 66, 68, 72 … • preserves the data in tact. • is a way to see the distribution • numbers on the left of the line are called • the stems and represent the leading edge of • each of the numbers • numbers on the right of the line are called • the leaves and represent the individual • numbers • indicate their value by completing the • stem.

11. Box Plots • Each of the lines in a box plot • represents either quartiles or the • range of the data. • In this particular plot the dots • represent outliers.

12. Frequency distributions - why? • Standard method for graphing data • easy way of visualizing group data • Introduction to the Normal Distribution • underlies all of the statistical tests we will be studying this semester • understanding the concepts behind statistical testing will make life a lot easier later on

13. Frequency

14. Frequency - some definitions • Raw scores are the scores we initially measure in a study • The number of times a score occurs in a set is the score’s frequency • A distribution is the general name for any organized set of data • A frequency distribution organizes the scores based on each score’s frequency • N is the total number of scores in the data

15. Understanding Frequency Distributions • A frequency distribution table shows the number of times each score occurs in a set of data • The symbol for a score’s frequency is simply f • N = ∑f

16. Raw Scores • The following is a data set of raw scores. We will use these raw scores to construct a frequency distribution table.

17. Frequency Distribution Table

18. Frequency Distribution Table - Example • Make a frequency distribution table for the following scores: 5, 7, 4, 5, 6, 5, 4

19. Frequency Distribution Table - Example • Make a frequency distribution table for the following scores: 5, 7, 4, 5, 6, 5, 4 ValueFrequency 7 1

20. Frequency Distribution Table - Example • Make a frequency distribution table for the following scores: 5, 7, 4, 5, 6, 5, 4 ValueFrequency 7 1 6 1

21. Frequency Distribution Table - Example • Make a frequency distribution table for the following scores: 5, 7, 4, 5, 6, 5, 4 ValueFrequency 7 1 6 1 5 3

22. Frequency Distribution Table - Example • Make a frequency distribution table for the following scores: 5, 7, 4, 5, 6, 5, 4 ValueFrequency 7 1 6 1 5 3 4 2

23. Frequency Distribution Table - Example • Make a frequency distribution table for the following scores: 5, 7, 4, 5, 6, 5, 4 Xf 7 1 6 1 5 3 4 2

24. Learning more about our data • What are the values for N and ∑X for the scores below?

25. Results via Frequency Distribution Table What is N? N = ∑f

26. Results via Frequency Distribution Table What is ∑X?

27. Results via Frequency Distribution Table What is ∑X? (17 * 1) = 17 (16 * 0) = 0 (15 * 4) = 60 (14 * 6) = 84 (13 * 4) = 52 (12 * 1) = 12 (11 * 1) = 11 (10 * 1) = 10 __________ Total = 246

28. Graphing Frequency Distributions

29. Graphing Frequency Distributions • A frequency distribution graph shows the scores on the X axis and their frequency on the Y axis

30. Graphing Frequency Distributions • A frequency distribution graph shows the scores on the X axis and their frequency on the Y axis • Why? • Because it’s not easy to make sense of this:

31. Graphing Frequency Distributions • A frequency distribution graph shows the scores on the X axis and their frequency on the Y axis • Why? • Because it’s not easy to make sense of this: • On a scale of 0-10, how excited are you about this class: __________ 0=absolutely dreading it 10=extremely excited/highlight of my semester • Data (raw scores) 5 7 2 3 5 5 5 8 7 7 4 5 10 7 5 4 5 5 7 3 6 2 6 3 5 5 7 2 4 6 3 7 5 5 7 3 5 6 5 5 8 6 7 5 3 5 7 2 3 5 4 5 4 8 3 6 5 5 5 1 2 4 7 5 5 4 3 3 7 5 8 6 3 5 10 0 6 6 3 8 5 4 3 2 4 6 3 7 5 5 7 5 7 5 10 7 5 4 5 5 7 6 3 8 1 5 5 6 4 9 8 5 8 5 7 5 10 7 5 4 5 5 7 4 8 4 5 8 5 5 7 5 5 5 2 4 6 3 7 5 2 4 6 3 7 5 8 6 3 5 10 0 6 7 2 8 8 5 5 8 6 3 6 2 6 3 5 5 7 2 5 10 7 5 4 5 5 7 5 7 5 10 7 5 4 5 5 5 7 2 3 3 7 5 8 6 3 5 10 0 6

32. Graphing Frequency Distributions • Xf • 10 4 • 9 7 • 8 35 • 7 40 6 33 • 5 43 • 4 11 • 3 11 • 2 3 • 1 6 • 0 4 0 1 2 3 4 5 6 7 8 9 10

33. Graphing Frequency Distributions • A frequency distribution graph shows the scores on the X axis and their frequency on the Y axis • The type of measurement scale (nominal, ordinal, interval, or ratio) determines whether we use: • a bar graph • a histogram • a frequency polygon

34. Graphs - bar graph • A frequency bar graph is used for nominal and ordinal data

35. Graphs - bar graph • A frequency bar graph is used for nominal and ordinal data Values on the x-axis

36. Graphs - bar graph • A frequency bar graph is used for nominal and ordinal data Frequencies on the y-axis

37. Graphs - bar graph • A frequency bar graph is used for nominal and ordinal data In a bar graph, bars do not touch

38. Graphs - histogram • A histogram is used for a small range of different interval or ratio scores

39. Graphs - histogram • A histogram is used for a small range of different interval or ratio scores Values on the x-axis

40. Graphs - histogram • A histogram is used for a small range of different interval or ratio scores Frequencies on the y-axis

41. Graphs - histogram • A histogram is used for a small range of different interval or ratio scores In a histogram, adjacent bars touch

42. Graphs - frequency polygon • A frequency polygon is used for a large range of different scores

43. Graphs - frequency polygon • A frequency polygon is used for a large range of different scores In a freq. polygon, there are many scores on the x-axis

44. Constructing a Frequency Distribution • Step 1: make a frequency table • Step 2: put values along x-axis (bottom of page) • Step 3: put a scale of frequencies along y-axis (left edge of page) • Step 4 (bar graphs and histograms) • make a bar for each value • Step 4 (frequency polygons) • mark a point above each value with a height for the frequency of that value • connect the points with lines

45. Graphing - example • A researcher observes driving behavior on a road, noting the gender of drivers, type of vehicle driven, and the speed at which they are traveling. Which type of graph should be used for each variable? • Gender? • nominal: bar graph • Vehicle Type? • nominal: bar graph • Speed? • ratio: frequency polygon

46. Use and Misuse of Graphs -2

47. Use and Misuse of Graphs • Which graph is correct? • Neither does a very good job at summarizing the data • Beware of graphing tricks

48. Types of Distributions

49. Distributions • Frequency tables, bar-graphs, histograms and frequency polygons describe frequency distributions

50. Distributions - Why? • Describing the shape of this frequency distribution is important for both descriptive and inferential statistics • The benefit of descriptive statistics is being able to understand a set of data without examining every score