1. Research Methods & Design in Psychology Lecture 3 Descriptives & Graphing Lecturer: James Neill

2. Overview Univariate descriptives & graphs Non-parametric vs. parametric Non-normal distributions Properties of normal distributions Graphing relations b/w 2 and 3 variables

3. Empirical Approach to Research A positivistic approach ASSUMES: the world is made up of bits of data which can be ‘measured’, ‘recorded’, & ‘analysed’ Interpretation of data can lead to valid insights about how people think, feel and behave

4. What do we want to Describe? Distributional properties of variables: Central tendency(ies) Shape Spread / Dispersion

5. Basic Univariate Descriptive Statistics Central tendency Mode Median Mean Spread Interquartile Range Range Standard Deviation Variance Shape Skewness Kurtosis

6. Basic Univariate Graphs Bar Graph – Pie Chart Stem & Leaf Plot Boxplot Histogram

7. Measures of Central Tendency Statistics to represent the ‘centre’ of a distribution – Mode (most frequent) – Median (50 th percentile) – Mean (average) Choice of measure dependent on – Type of data – Shape of distribution (esp. skewness)

8. Measures of Central Tendency XXX?Ratio XXXInterval XXOrdinal XNominal MeanMedianMode

9. Measures of Dispersion Measures of deviation from the central tendency Non-parametric / non-normal: range, percentiles, min, max Parametric: SD & properties of the normal distribution

10. Measures of Dispersion XXXRatio X?XXInterval XOrdinal Nominal SDPercentile s Range, Min/Max

11. Describing Nominal Data Frequencies – Most frequent? – Least frequent? – Percentages? Bar graphs – Examine comparative heights of bars – shape is arbitrary Consider whether to use freqs or %s

12. Frequencies Number of individuals obtaining each score on a variable Frequency tables graphically (bar chart, pie chart) Can also present as %

13. Frequency table for sex

14. Bar chart for frequency by sex

15. Pie chart for frequency by sex

16. Bar chart: Do you believe in God?

17. Bar chart for cost by state

18. Bar chart vs. Radar Chart

20. Mode Most common score - highest point in a distribution Suitable for all types of data including nominal (may not be useful for ratio) Before using, check frequencies and bar graph to see whether it is an accurate and useful statistic.

21. Describing Ordinal Data Conveys order but not distance (e.g., ranks) Descriptives as for nominal (i.e., frequencies, mode) Also maybe median – if accurate/useful Maybe IQR, min. & max. Bar graphs, pie charts, & stem-&-leaf plots

22. Stem & Leaf Plot Useful for ordinal, interval and ratio data Alternative to histogram

23. Box & whisker Useful for interval and ratio data Represents min. max, median and quartiles

24. Describing Interval Data Conveys order and distance, but no true zero (0 pt is arbitrary). Interval data is discrete, but is often treated as ratio/continuous (especially for > 5 intervals) Distribution (shape) Central tendency (mode, median) Dispersion (min, max, range) Can also use M & SD if treating as continuous

25. Describing Ratio Data Numbers convey order and distance, true zero point - can talk meaningfully about ratios. Continuous Distribution (shape – skewness, kurtosis) Central tendency (median, mean) Dispersion (min, max, range, SD)

26. Univariate data plot for a ratio variable

27. The Four Moments of a Normal Distribution Mean <-SkewSkew->

28. The Four Moments of a Normal Distribution Four mathematical qualities (parameters) allow one to describe a continuous distribution which as least roughly follows a bell curve shape: 1 st = mean (central tendency) 2 nd = SD (dispersion) 3 rd = skewness (lean / tail) 4 th = kurtosis (peakedness / flattness)

29. Mean (1 st moment ) Average score Mean =  X / N Use for ratio data or interval (if treating it as continuous). Influenced by extreme scores (outliers)

30. Standard Deviation (2 nd moment ) SD = square root of Variance =  (X - X) 2 N – 1 Standard Error (SE) = SD / square root of N

31. Skewness (3 rd moment ) Lean of distribution +ve = tail to right -ve = tail to left Can be caused by an outlier Can be caused by ceiling or floor effects Can be accurate (e.g., the number of cars owned per person)

32. Skewness (3 rd moment ) Negative skew Positive skew

33. Ceiling Effect

34. Floor Effect

35. Kurtosis (4 th moment ) Flatness or peakedness of distribution +ve = peaked -ve = flattened Be aware that by altering the X and Y axis, any distribution can be made to look more peaked or more flat – so add a normal curve to the histogram to help judge kurtosis

36. Kurtosis (4 th moment ) Red = Positive (leptokurtic) Blue = negative (platykurtic)

37. Key Areas under the Curve for Normal Distributions For normal distributions, approx. +/- 1 SD = 68% +/- 2 SD ~ 95% +/- 3 SD ~ 99.9%

38. Areas under the normal curve

39. Types of Non-normal Distribution Bi-modal Multi-modal Positively skewed Negatively skewed Flat (platykurtic) Peaked (leptokurtic)

40. Non-normal distributions

42. Rules of Thumb in Judging Severity of Skewness & Kurtosis View histogram with normal curve Deal with outliers Skewness / kurtosis 1 Skewness / kurtosis significance tests

43. Histogram of weight

44. Histogram of daily calorie intake

45. Histogram of fertility

46. Example ‘normal’ distribution 1

47. Example ‘normal’ distribution 2

48. Example ‘normal’ distribution 3

49. Example ‘normal’ distribution 4

50. Example ‘normal’ distribution 5

51. Skewed Distributions & the Mode, Median & Mean +vely skewed mode < median < mean Symmetrical (normal) mean = median = mode -vely skewed mean < median < mode

52. Effects of skew on measures of central tendency

53. More on Graphing (Visualising Data)

54. Edward Tufte Graphs:  Reveal data  Communicate complex ideas with clarity, precision, and efficiency

55. Tufte's Guidelines 1 Show the data Substance rather than method Avoid distortion Present many numbers in a small space Make large data sets coherent

56. Tufte's Guidelines 2 Encourage eye to make comparisons Reveal data at several levels Purpose: Description, exploration, tabulation, decoration Closely integrated with statistical and verbal descriptions

57. Tufte’s Graphical Integrity 1 Some lapses intentional, some not Lie Factor = size of effect in graph size of effect in data Misleading uses of area Misleading uses of perspective Leaving out important context Lack of taste and aesthetics

58. Tufte's Graphical Integrity 2 Trade-off between amount of information, simplicity, and accuracy “It is often hard to judge what users will find intuitive and how [a visualization] will support a particular task” (Tweedie et al)

59. Chart scale

62. Types of Graphs

63. Cleveland’s Hierarchy

64. Volume

65. Food Aid Received by Developing Countries

66. Percentage of Doctors Devoted Solely to Family Practice in California 1964-1990

67. Distortive Variations in Scale

69. Restricted Scales

71. Example Graphs Depicting the Relationship between Two Variables (Bivariate)

72. People Histogram

73. Separate Graphs

74. Example Graphs Depicting the Relationship between Three Variables (Multivariate)

75. Clustered bar chart

76. 19 th vs. 20 th century causes of death

77. Demographic distribution of age

78. Where partners first met

79. Line graph

81. Causes of Mortality

82. Bivariate Normality

83. Exampes of More Complex Graphs

84. Sea Temperature

86. Inferential Statistical Analaysis Decision Making Tree

88. Links Presenting Data – Statistics Glossary v1.1 - http://www.cas.lancs.ac.uk/glossary_v1.1/presdata.html http://www.cas.lancs.ac.uk/glossary_v1.1/presdata.html A Periodic Table of Visualisation Methods - http://www.visual- literacy.org/periodic_table/periodic_table.htmlhttp://www.visual- literacy.org/periodic_table/periodic_table.html Gallery of Data Visualization Univariate Data Analysis – The Best & Worst of Statistical Graphs - http://www.csulb.edu/~msaintg/ppa696/696uni.htmhttp://www.csulb.edu/~msaintg/ppa696/696uni.htm Pitfalls of Data Analysis – http://www.vims.edu/~david/pitfalls/pitfalls.htm http://www.vims.edu/~david/pitfalls/pitfalls.htm Statistics for the Life Sciences – http://www.math.sfu.ca/~cschwarz/Stat- 301/Handouts/Handouts.html http://www.math.sfu.ca/~cschwarz/Stat- 301/Handouts/Handouts.html