Measures of Dispersion, Skew, & Kurtosis (based on Kirk, Ch. 4) {to be used in conjunction with Measures of Dispersion Chart }

Transcription

1 Percetles Psych 54, 9/8/05 p. /6 Measures of Dsperso, kew, & Kurtoss (based o Krk, Ch. 4) {to be used cojucto wth Measures of Dsperso Chart } percetle (P % ): a score below whch a specfed percetage of scores falls (start wth a percet, ad covert to a score) vs. percetle rak (P R ): percetage of scores that falls below a gve score (start wth a score, ad covert to a percet)...we wll retur to ths topc later whe computg the meda, we are terested computg the 50 th percetle (P 50 ) / fb recall (from revew): Md = P 50 = X ll + ( ), where: f X ll s the real lower lmt of the terval cotag the meda (the logc s that you are startg at the bottom of the terval ad coutg up to t) s the sze of the class terval ( f you are workg wth ugrouped dstrbutos) s the umber of scores the etre dstrbuto (thus, / s the locato of the mdpot) Σf b s the umber of scores below X ll (ths tells you how may scores up from the bottom that you ve already come...so, takg ths away from / tells you how may more scores you have to go) f s the umber of scores there are the terval cotag the meda (ths tells you how may peces to dvde that terval to) Example: X = {4, 4, 4, 5, 5, 5, 5, 6, 7, 8, 8} Md (X) = Whe computg the sem-terquartle rage (Q), we eed the 5 th ad 75 th percetles (P 5 ad P 75 ). The formula for the meda ca be geeralzed to apply to ay percetle, cludg the 5 th ad 75 th :

2 Psych 54, 9/8/05 p. /6 ( PR /00) f b P % = X ll + ( ), where: f X ll s the real lower lmt of the terval cotag the percetle of terest (the logc s that you are startg at the bottom of the terval ad coutg up to t); cosder the sample sze to determe what terval (whe workg wth raw, ugrouped data, ths s just a score) to work wth (e.g., e.g. f you have 0 cases, the 5 th percetle s assocated wth the.5(0) =.5 th case...use the real lower lmt of the terval cotag the.5 th case) s the sze of the class terval ( f you are workg wth ugrouped dstrbutos) s the umber of scores the etre dstrbuto P R s the percetle rak you are workg wth (ths s 50 for the meda/q, 5 for Q, 75 for Q 3, etc.); thus, (P R /00) tells you how may cases to the dstrbuto (from the bottom) you must come to get to the case of terest (you actually use ths to get X ll ) Σf b s the umber of scores below X ll (ths tells you how may scores up from the bottom that you ve already come...so, takg ths away from (P R /00) tells you how may more scores you have to go) f s the umber of scores there are the terval cotag the percetle of terest (ths tells you how may peces to dvde that terval to) Example: X = {4, 4, 4, 5, 5, 5, 5, 6, 7, 8, 8} Q 3 = Q = Q =

3 Psych 54, 9/8/05 p. 3/6 problem: statstcal packages compute percetles dfferetly from each other (ad from the above method) * see quartlesmore.html f you have further terest ths topc * ote that P teds to gve more extreme values (.e., percetles that are further away from the meda), whle Excel teds to gve less extreme values (.e., percetles that are closer to the meda); ether program uses the algorthm preseted above (ote, however, that the above algorthm s thought to be the best for ormal dstrbutos) percetle rak: aga, here we are gog from a score to a percet, so we just solve for P R the above equato: P R 00 f ( P X = ( f ( % b + ll ) )) Example: Usg the above data set, fd the percetle rak for the score of 8. More o tadard Devato Calculatg tadard Devato: Recall that the D approxmates the average dstace of scores from the mea (devatos). Why ot just take the true average of these devatos? The mea s the balacg pot of all scores...there are equal postve devatos ad egatve devatos, so the sum of all devatos s equal to zero...always!...so ths wo t gve us ay terestg formato. Why ot take the absolute value of the devatos? Ths s the rght dea, but mathematcas have foud that the resultg values do ot work well wth more advaced calculatos (they are ot mathematcally tractable) What s the soluto? Frst square all the devatos ad sum them (ths gves us the sums of squares, ). The, take the average ad covert back to orgal uts by takg the square root. Varatos o the tadard Devato Formula: The formula gve the chart s for fdg the stadard devato for a sample. It s purely a descrptve statstc:

4 Psych 54, 9/8/05 p. 4/6 = = We may also be terested fdg the stadard devato for a kow populato. I ths case, the formula s the same, but ow we re comparg dvdual scores to a populato mea stead of a sample mea: σ = = μ) If our ultmate terest s drawg a ferece about a populato based o a sample, we must use a slghtly dfferet formula. tatstcas (who make up populatos ad draw samples from these populatos) have foud that the formula for wll uderestmate the true populato stadard devato. They have foud that dvdg by - (stead of ) compesates for ths bas. = ˆ σ = For ay of the above formulas for stadard devato, t s oteworthy that a rawscore formula may also be appled. Below s the case for the sample D: = = X ( = X ) Ths formula makes calculatos easer (especally whe there are may cases), but t s ot as coceptually meagful as the devato formula. Effects of Lear Trasformatos of cores o tadard Devato What happes whe we add a costat to all scores?

5 Psych 54, 9/8/05 p. 5/6 What happes whe we multply each score by a costat? tadard Devato ad the Normal Curve (see Krk, p. 5) Varace Varace s aother measure of dsperso that s ofte used feretal statstcs. It s smply the D squared ad s represeted by, σ, or σˆ. For example: = = kew kew refers to the asymmetry of a dstrbuto ad ca be computed as:

6 Psych 54, 9/8/05 p. 6/6 k = = 3 3 Note that the devatos the umerator of the formula are cubed. Ths meas that each score ca cotrbute a egatve (scores to the left of the mea) or postve (scores to the rght of the mea) value to k. Also ote that the farther out a score s (.e., the tal), the greater pull t wll have o the value of k. For example, f the mea s 50, a score of 5 wll add to the sum of the cubed devatos, but a score of 5 wll add -5 3, or -5,65 to ths sum. If there are a lot of relatvely extreme scores oe tal, the skew formula wll reflect ths. Negatve kew (-) ymmetrcal (0) Postve kew (+) Kurtoss Kurtoss reflects how peaked or flat a dstrbuto s ad s calculated as: Kur = = platykurtc (-) mesokurtk (0) leptokurtc (+) Examples (use data from chart): k= Kur =