Central Tendency & Scale Types

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

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

Mean Sum of scores divided by number of scores Sample mean: IQ: X = [94, 108, 145, 121, 88, 133] SX = 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 88 94 108 121 133 145 M

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 SX = 1*2 x = 2 f(x) = 4 SX = 2*4 x = 3 f(x) = 5 SX = 3*5 Probability p(x) = fraction of population with value x

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

Median Middle value Not average of minimum and maximum 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 50th percentile

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 Median Mean Mean Household Income Frequency (.5 M households) Median ($46k) Mean ($63k) Mean excluding outlier Mean

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

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

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

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

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

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

Meaningful Operations Summary of Scale Types Scale Meaningful Operations Mode Median Mean Nominal = Ordinal > = Interval – > = Ratio / – > =