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 )

early illustratio of calculatig a mea Mode, Media, & Mea Biocular Rivalry (Youg Adults) 0 200 400 600 800 Mode = 0.875 Media = 1.23 Mea = 1.

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

Median and IQR The median is the value which divides the ordered data values in half.

Median and IQR The median is the value which divides the ordered data values in half. STA 666 Fall 2007 Web-based Course Notes 4: Describig Distributios Numerically Numerical summaries for quatitative variables media ad iterquartile rage (IQR) 5-umber summary mea ad stadard deviatio Media