1. Review Data: {2, 5, 6, 8, 5, 6, 4, 3, 2, 1, 4, 9} What is F(5)? A.2 B.4 C.6 D.8

2. Review Find the normal distribution A.C. B.D. Score Density Score Density Score Density Score Density

3. Review Estimate the probability of a score between 1 and 2. A.0.33 B.0.5 C.1.0 D.1.5 Score Density

4. Central Tendency & Scale Types 9/9

5. Outline Central Tendency –Mean –Median –Mode Scale Types –Nominal –Ordinal –Interval –Ratio Different statistics for different variables

6. Central Tendency Statistics – simplify a large set of data to a single (meaningful) number Central Tendency –One useful kind of summary information –Intuitively: typical, average, normal value Three statistics for central tendency –Mean –Median –Mode

7. Mean Sum of scores divided by number of scores Sample mean: IQ: X = [94, 108, 145, 121, 88, 133]  X = 94 + 108 + 145 + 121 + 88 + 133 = 689 M = 689/n = 689/6 = 114.83 Equal apportionment –If everyone had mean score, total would be the same Balance point, seesaw analogy (Fig 3.3) Equal upward and downward distances M8894108121133145

8. Population Mean Finite population Infinite population X = [1,1,2,2,2,2,3,3,3,3,3,4,4,4,5,5,6] x = 1 f(x) = 2  X = 1*2 x = 2 f(x) = 4  X = 2*4 x = 3 f(x) = 5  X = 3*5 Probability p(x) = fraction of population with value x

9. Mean of Infinite Population Half of all leprechauns have 1 pot of gold. The other half have 2. Mean? 92% of dogs have 4 legs. 5% have 3 legs. 2% have 2 legs. 1% have 5 legs. 100 dogs. 92 with 4 legs, 5 with 3 legs, 2 with 2 legs, 1 with 5 legs. 4+…+4+3+3+3+3+3+2+2+5 = 392 392/100 = 3.92

10. Median Middle value –Higher than half the scores, lower than other half Not average of minimum and maximum Sorting approach X = [4,7,5,8,6,2,1,4,3,5,6,8,7,4,3,6,9] X = [1,2,3,3,4,4,4,5,5,6,6,6,7,7,8,8,9] Same as 50 th percentile

11. Household Income Frequency (.5 M households) Mean vs. Median Both based on a notion of balance Mean sensitive to each datum's distance from middle Median better for irregular distributions –Skew –Outliers Mean MedianMean Median ($46k) Mean ($63k) Mean excluding outlierMean

12. Mode Most common value Peak in the distribution for continuous variables Simple and insensitive Most useful when mean, median not definable –College majors, sex, favorite color

13. Scale types We usually use numbers to represent values of variables –Numbers are just a model or analogy for real world –Some properties relevant, some superfluous Sex –Males = 1; females = 2 –Females not twice males Analogy still limited for more “numerical” variables –Height, reaction time –Can’t multiply together Numbers have many properties –Which are relevant for a given variable? –Determines what kinds of statistics make sense Scale of a variable –Summarizes what numerical properties are meaningful –4 types of scales: Nominal, Ordinal, Interval, Ratio

14. Nominal Scale Values are just labels –Sex: {male, female} –Color: {red, green, blue, …} No structure or relationships between values Essentially non-numeric –Can use numbers for “coding” but just as placeholders –Red = 1; green = 2; blue = 3 Only mathematical notion is equality (=) –Two scores are equal, or they’re not Few meaningful statistics –Frequencies: Number of scores of a given value –Mode: Value with greatest frequency

15. Ordinal Scale Values are ordered, but differences aren’t meaningful –Preferences, contest placings, years of education –1 st - 2 nd  2 nd - 3 rd Mathematical notion of greater-than (>, =) Additional meaningful statistics –Median, quantiles –Range, interquartile range

16. Interval Scale Differences between scores are meaningful –Today 4° warmer than yesterday Ratios of scores not meaningful –2° not twice as hot as 1° –No real zero point –E.g. Fahrenheit vs. Celcius; IQ Mathematical notion of subtraction (–, >, =) Additional meaningful statistics –Mean –Variance, standard deviation

17. Ratio Scale Zero is meaningful –Weight, time, etc. Ratios between scores make sense –Twice as heavy, twice as long Mathematical notion of division (/, –, >, =) No notable new statistics 0

18. Summary of Scale Types Scale Meaningful OperationsModeMedianMean Nominal= Ordinal> = Interval– > = Ratio/ – > = 0

19. Review Data: {1, 4, 7, 8, 2, 12, 8} What is the mean? A.4 B.5 C.6 D.7 E.8

20. Review Data: {1, 4, 7, 8, 2, 12, 8} What is the mode? A.4 B.5 C.6 D.7 E.8

21. Review What scale type is this variable? A person’s first name A.Nominal B.Ordinal C.Interval D.Ratio