HUMBEHV 3HB3 Measures of Central Tendency & Variability Week 2
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1 Describig Data Distributios HUMBEHV 3HB3 Measures of Cetral Tedecy & Variability Week 2 Prof. Patrick Beett Ofte we wish to summarize distributios of data, rather tha showig histograms Two basic descriptios of a distributio iclude its middle (cetral tedecy) ad how spread out it is (variability) Part 1 - Cetral Tedecy Example Biocular Rivalry Times Ofte wat to describe typical score i data Biocular Rivalry (Youg Adults) How to extract oe typical score? Commo measures: - mode (most commo) - media (middle score; 50th percetile) - mea (average) Mode = Mode is most commo score percept duratio (sec)
2 Mode - Most Commo Score Mode may be poorly defied The mode is the most commo/frequet score If 2 adjacet scores occur with equal frequecy: - mode = average of those two scores If 2 o-adjacet scores occur with equal frequecy: - distributio is bi-modal - report both umbers If rage of umber occur with early equal frequecy: - mode is ill-defied - report as: the mode fell withi the rage of X to Y Data Set Score Data Set Score Data Set Score I some cases, small fluctuatios i frequecies ca produce BIG chages i mode Media = Middle Score Odd umber of scores (N=11) > scores: > sorted scores: Media = Middle Score Eve umber of scores (N=10) > scores: > sorted scores: > media(scores): 100 Media Locatio = (N+1)/2 = (11+1)/2 = 6 > media(scores): (99+100)/2 = 99.5 Media Locatio = (N+1)/2 = (10+1)/2 = 5.5
3 Media = Middle Score Cumulative Plot Biocular Rivalry (Youg Adults) > scores: , > sorted scores: ,999 > media(scores): 100 Odd umber of scores (N=11) Media Locatio = (N+1)/2 = (11+1)/2 = 6 Cumulative Media = % lie Media is the middle score (50th percetile) Duratio (s) Mode vs. Media Biocular Rivalry (Youg Adults) Iterlude: Summatio Notatio Suppose you have a set of scores, X: Mode = Media = X1, X2, X3,, X - represet a arbitrary score as Xi Followig otatio represets the summatio of all scores: Sigma = X i = (X 1 + X 2 + X X ) i=1 percept duratio (sec)
4 = Summatio (cotiued) Ofte use a compact form to represet summig over the etire sample: Summatio (cotiued) Operatios to the right of Sigma are performed o idividual Y s before summatio = X equivalet! = X 2 = (X X X X 2 ) = X i = (X 1 + X 2 + X X ) i=1 = Summatio (cotiued) Brackets idicate operatios performed after summatio = X 2 = (X 1 + X 2 + X X ) 2 Mea most commo idex of cetral tedecy the average score: - (sum of all scores) divided by (umber of scores) X = < X i i=1 = (X 1 + X 2 + X X )
5 early illustratio of calculatig a mea Mode, Media, & Mea Biocular Rivalry (Youg Adults) Mode = Media = 1.23 Mea = 1.71 The Seve Pillars of Statistical Wisdom, Stephe M Stigler percept duratio (sec) Mode Advatages & Disadvatages Advatages - robust to outliers (extreme scores) - value actually appears i the data - represets the greatest probability of subjects havig a score - ca be foud for omial data e.g., mode of household pet type dog ; o aalogous mea or media Disadvatages - depeds o how we bi scores - ca be poorly defied & ustable for flat distributios Media Advatages & Disadvatages Advatages - robust to outliers (extreme scores) - ca be calculated eve with flat distributios - good idex of typical score i skewed distributios Disadvatages - o mathematical formula for the media difficult to use media i mathematical derivatios/equatios - i some situatios, betwee-sample variability is greater for media tha mea (i.e., media less stable tha mea)
6 Mea Advatages & Disadvatages Advatages: - i some situatios, betwee-sample variatio is less for sample meas tha sample modes & medias i.e., meas are more stable - eters readily ito algebraic equatios Disadvatages: - less robust to extreme values - value may ot actually exist i the data - iterpretatio depeds more o iterval properties of scores Statistics i the ews Hamilto Spectator Sept 14, 2017 Trimmed Meas To make the mea less sesitive to extreme values, we ca trim a certai percetage of values off of the tails - Example: scores = {2, 2, 3, 5, 6, 7, 8, 8, 9, 501} mea = Now trim 10% off tails at both eds of sorted scores: - trimmed scores = {2, 2, 3, 5, 6, 7, 8, 8, 9, 501} 10% trimmed mea = 6 Amout of trimmig varies across applicatios/situatios: - e.g., 20% trimmed mea removes upper & lower 20% of scores sorted data: % trimmig % trimmed mea % trimmig % trimmed mea % trimmig % trimmed mea
7 Part 1 - Cetral Tedecy (summary) Mode, Media, Mea - methods of calculatio - advatages & disadvatages Summatio Notatio Trimmed Meas
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