1. BASIC NOTATION

2. Summation (  ) X i = The number of meals I have on day “i” X= 1,2,3,2,1  X i = ???  X i 2 = ??? (  X i ) 2 = ???

3. Qualitative Variables Nominal Political affiliation Republican Democrat Independent Gender Female Male

4. Quantitative Variables Ordinal Categories have relative value/order Example Very Depressed Depressed Slightly depressed Not depressed

5. Quantitative Variables Interval Categories have relative value/order Difference in measurement = Difference in characteristic Example Temperature Fahrenheit, 83,84,85 … Difference from 83 to 84 = Difference from 84 to 85

6. Quantitative Variables Ratio Categories have relative value/order Difference in measurement = Difference in characteristic True zero (0) point exists Example Temperature Kelvin, 0,1,2,…343,345,346 … Height 0 inches, 1 inch, …. 86 inches (Shaq)

7. Statistical Analyses DesignInterval/RatioNominal One SampleOne sample z or t testGoodness of fit χ2 Two Indepenedent samplesIndependent t testχ2 test of independence Two dependent SamplesDependent t test McNemar test for significance of change K independent samplesANOVAχ2 test of independence CorrelationPearson R PredictionRegression

8. Frequency Distributions Tables Ungrouped (list of scores) Grouped (grouped by ranges) Graphs histograms frequency polygons

9. Table Distributions The variable: Time (in minutes) between getting out of bed this morning and eating your first bite of food. Time (min) Ungrouped : (6, 28, 27, 7, 7, 24, 39, 55, 13, 17, 13, 13, 3, 23, 18, 37, 2, 8, 11, 18, 22, 2, 21, 31, 12) Bad Grouped Frequency Distribution X f 0-10 7 11-20 8 21-30 6 31-40 3 41-50 0 51-60 1 25 Good Grouped Frequency Distribution X f 1-10 7 11-20 8 21-30 6 31-40 3 41-50 0 51-60 1 25

11. Distribution Characteristics Modality - Peaks Symmetry – Mirror Reflection Asymptoticness – Extreme Values on both Sides

13. Normal Distributions USA Unimodal Symmetric Asymptotic

14. Inflection points Where curve changes from convex to concave or concave to convex Also = 1 standard deviation from the mean

15. CENTRAL TENDENCY WHAT IS A TYPICAL SCORE LIKE? Mode: Most common value; number of peaks; always an observed value Median: Middle of distribution; not affected much by outliers Mean: Average; greatly affected by outliers

16. -Most common score(s) 1,2,2,2,3,4,5,6,7Unimodal Mode=2 1,3,3,4,4,5,6,7,8BimodalModes=3,4 1,3,3,4,4,5,6,6,8TrimodalModes=3,4,6 1,2,3,4,5,6,7,8,9Amodal CENTRAL TENDENCY Modes

17. Modes in Populations -Unimodal -Bimodal -Trimodal -Amodal ?

18. -Middle score in distribution -Odd number of scores 5-point data set: 2,3,5,9,12 Median=5 1,2,5,5,7,9,500,700,999 Median=? -Even number of scores 4-point data set: 3,5,8,9 Median=(5+8)/2=6.5 1,2,5,5,7,9,500,700,999,1122 Median=? CENTRAL TENDENCY Medians

19. Medians in Populations

20. CENTRAL TENDENCY Means

21. More modes, medians and means MeasureDefinitionLevel of measurementDisadvantage Mode most frequent valuenom., ord., int./rat.Crude Medianmiddle valueord., int./rat. Only two points contribute Meanbalancing pointint./rat.Affected by skew

22. The Spread of Distributions -How different are scores from central tendency? -Range -Standard Deviation

23. Measure of Spread RANGE -Highest value – Lowest Value -Affected only by end points -Data set 1 -1,1,1,50,99,99,99 -Data set 2 -1,50,50,50,50,50,99

24. Why ‘range’ is weak

25. The Spread of Distributions -How different are scores from central tendency? -Always, by definition of the mean

26. Population Standard Deviation

27. Sample Variance and Standard deviation Also known as “Estimated Population Standard Deviation”

28. Sample Variance and Standard deviation Why do we use N-1 for sample? Because sample means are closer to sample mean than to population mean, which underestimates the estimate Population 2,4,6,and 8, σ = (2+4+6+8)/4 = 5 Scores 2 and 6 σ 2 = (2-5) 2 +(6-5) 2 = 9 + 1 = 10 Scores 2 and 6, = (2+6)/2 = 4 S 2 = (2-4) 2 +(6-4) 2 = 4 + 4 = 8 N-1 adjusts for bias

29. Sample Variance  SUM OF SQUARED DEVIATIONS  DEGREES OF FREEDOM STANDARD DEVIATION

30. Differences Between Sample and Population Standard Deviation 1) Sigma vs. S 2) Population mean versus Sample mean 3) N vs. N-1

31. Super Important Relationship Standard Deviation is square root of variance SAMPLE STANDARD DEVIATION = SQUARE ROOT OF THE SAMPLE VARIANCE POPULATION STANDARD DEVIATION = SQUARE ROOT OF THE POPULATION VARIANCE

32. Population Standard Deviation

33. Sample Standard Deviation