JUST THE MATHS UNIT NUMBER STATISTICS 3 (Measures of dispersion (or scatter)) A.J.Hobson

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1 JUST THE MATHS UNIT NUMBER 8.3 STATISTICS 3 (Measures of dispersio (or scatter)) by A.J.Hobso 8.3. Itroductio The mea deviatio Practica cacuatio of the mea deviatio The root mea square (or stadard) deviatio Practica cacuatio of the stadard deviatio Other measures of dispersio Exercises Aswers to exercises

2 UNIT STATISTICS 3 MEASURES OF DISPERSION (OR SCATTER) 8.3. INTRODUCTION Averages typify a whoe coectio of vaues, but they give itte iformatio about how the vaues are distributed withi the whoe coectio. For exampe, 99.9, 00.0, 00. is a coectio which has a arithmetic mea of 00.0 ad so is 99.0,00.0,0.0; but the secod coectio is more widey dispersed tha the first. It is the purpose of this Uit to examie two types of quatity which typify the distace of a the vaues i a coectio from their arithmetic mea. They are kow as measures of dispersio (or scatter) ad the smaer these quatities are, the more custered are the vaues aroud the arithmetic mea THE MEAN DEVIATION If the vaues x, x 2, x 3,...,x have a arithmetic mea of x, the x x, x 2 x, x 3 x,...,x x are caed the deviatios of x, x 2, x 3,..., x from the arithmetic mea. Note: The deviatios add up to zero sice (x i x) = x i x = x x = 0 DEFINITI0N The mea deviatio (or, more accuratey, the mea absoute deviatio) is defied by the formua M.D. = x i x

3 8.3.3 PRACTICAL CALCULATION OF MEAN DEVIATION I cacuatig a mea deviatio, the foowig short-cuts usuay tur out to be usefu, especiay for arger coectios of vaues: (a) If a costat, k, is subtracted from each of the vaues x i (i =,2,3...), ad aso we use the fictitious arithmetic mea, x k, i the formua, the the mea deviatio is uaffected. x i x = (x i k) (x k). (b) If we divide each of the vaues x i (i =,2,3,...) by a positive costat,, ad aso we use the fictitious arithmetic mea x, the the mea deviatio wi be divided by. x i x = x i x. Summary If we code the data usig both a subtractio by k ad a divisio by, the vaue obtaied from the mea deviatio formua eeds to mutipied by to give the correct vaue THE ROOT MEAN SQUARE (OR STANDARD) DEVIATION A more commo method of measurig dispersio, which esures that egative deviatios from the arithmetic mea do ot ted to cace out positive deviatios, is to determie the arithmetic mea of their squares, ad the take the square root. DEFINITION The root mea square deviatio (or stadard deviato ) is defied by the formua R.M.S.D. = (x i x) 2. 2

4 Notes: (i) The root mea square deviatio is usuay deoted by the symbo, σ. (ii) The quatity σ 2 is caed the variace PRACTICAL CALCULATION OF THE STANDARD DEVIATION I cacuatig a stadard deviatio, the foowig short-cuts usuay tur out to be usefu, especiay for arger coectios of vaues: (a) If a costat, k, is subtracted from each of the vaues x i (i =,2,3...), ad aso we use the fictitious arithmetic mea, x k, i the formua, the σ is uaffected. (x i x) 2 = [(x i k) (x k)] 2. (b) If we divide each of the vaues x i (i =,2,3,...) by a costat,, ad aso we use the fictitious arithmetic mea x, the σ wi be divided by. (x i x) 2 = ( xi x ) 2. Summary If we code the data usig both a subtractio by k ad a divisio by, the vaue obtaied from the stadard deviatio formua eeds to mutipied by to give the correct vaue, σ. (c) For the cacuatio of the stadard deviatio, whether by codig or ot, a more coveiet formua may be obtaied by expadig out the expressio (x i x) 2 as foows: σ 2 = [ x 2 i 2x x i + ] x 2. 3

5 That is, σ 2 = x 2 i 2x 2 + x 2, which gives the formua σ = ( x 2 i ) x 2. Note: I advaced statistica work, the above formuae for stadard deviatio are used oy for descriptive probems i which we kow every member of a coectio of observatios. For iferece probems, it may be show that the stadard deviatio of a sampe is aways smaer tha that of a tota popuatio; ad the basic formua used for a sampe is σ = (x i x) OTHER MEASURES OF DISPERSION We metio here, briefy, two other measures of dispersio: (i) The Rage This is the differece betwee the highest ad the smaest members of a coectio of vaues. (ii) The Coefficiet of Variatio This is a quatity which expresses the stadard deviatio as a percetage of the arithmetic mea. It is give by the formua C.V. = σ x 00. 4

6 EXAMPLE The foowig grouped frequecy distributio tabe shows the diameter of 98 rivets: Cass Cs. Mid Freq. Cum. (x i 6.6) 0.02 f i x i 2 x i 2 f i x i f i x i x Itv. Pt. x i f i Freq. = x i Totas Estimate the arithmetic mea, the stadard deviatio ad the mea (absoute) deviatio of these diameters. Soutio Fictitious arithmetic mea = Actua arithmetic mea = Fictitious stadard deviatio = Actua stadard deviatio = Fictitious mea deviatio =

7 Actua mea deviatio EXERCISES. For the coectio of umbers 6.5, 8.3, 4.7, 9.2,.3, 8.5, 9.5, 9.2 cacuate (correct to two paces of decimas) the arithmetic mea, the stadard deviatio ad the mea (absoute) deviatio. 2. Estimate the arithmetic mea, the stadard deviatio ad the mea (absoute) deviatio for the foowig grouped frequecy distributio tabe: Cass Iterva Frequecy The foowig tabe shows the registered speeds of 00 speedometers at 30m.p.h. Regd. Speed Frequecy Estimate the arithmetic mea, the stadard deviatio ad the coefficiet of variatio ANSWERS TO EXERCISES. Arithmetic mea 8.40, stadard deviatio.88, mea deviatio Arithmetic mea 65, stadard deviatio 24.66, mea deviatio Arithmetic mea 3.4, stadard deviatio.44, coefficiet of variatio