StatiStical MethodS for GeoGraphy

Transcription

1 Peter A. Rogerso StatStcal MethodS for GeoGraphy A Studet S Gude F o u r t h e d t o 00_Rogerso_4e_BAB1405B0092_Prelms.dd 3 10/13/2014 5:30:29 PM

2 2 DESCRIPTIVE STATISTICS Learg Objectves Types of Data 26 Vsual Descrptve Methods 27 Measures of Cetral Tedecy 31 Measures of Varablty 33 Other Numercal Measures for Descrbg Data 35 Descrptve Spatal Statstcs 38 Descrptve Statstcs SPSS 21 for Wdows 49 I Chapter 1, a fudametal dstcto was draw betwee descrptve ad feretal statstcs. We saw that descrbg data costtutes a mportat early phase of the scetfc method. I ths chapter, we wll focus upo vsual ad umercal descrptve summares of data. We wll beg by descrbg dfferet types of data ad by coverg some of the vsual approaches that are commoly used to explore ad descrbe data. Followg ths, umercal measures of descrpto are revewed. Fally, descrpto s dscussed for the specal cotext of spatal data. 2.1 TYPES OF DATA Data may be classfed as omal, ordal, terval, or rato. Nomal data are observatos that have bee placed to a set of mutually exclusve ad collectvely exhaustve categores. Examples of omal data clude sol type ad vegetato type. Ordal data cosst of observatos that are raked. Thus, t s possble to say that oe observato s greater tha (or less tha) aother, but wth ths much formato t s ot possble to say by how much a observato s greater or less tha aother. It s ot ucommo to fd ordal data almaacs ad statstcal abstracts; a example s data o the sze of ctes, by rak. 02_Rogerso_4e_BAB1405B0092_Ch-02.dd 26 10/3/ :40:35 AM

3 Descrptve Statstcs 27 Whe t s possble to say by how much oe observato s greater or less tha aother, data are ether terval or rato. Wth terval data, dffereces values are detfable. For example, o the Fahrehet temperature scale, 44 degrees s 12 degrees warmer tha 32 degrees. However, the zero s ot meagful o the terval scale, ad cosequetly rato terpretatos are ot possble. Thus, 44 degrees s ot twce as warm as 22 degrees. Rato data, o the other had, does have a meagful zero. Thus, 100 degrees Kelv s twce as warm as 50 degrees Kelv. Most umercal data are rato data deed, t s dffcult to thk of examples for terval data other tha the Fahrehet ad Celsus scales. Data may cosst of values that are ether dscrete or cotuous. Dscrete varables take o oly a fte set of values examples clude the umber of suy days a year, the aual umber of vsts by a famly to a local publc faclty, ad the mothly umber of collsos betwee automobles ad deer a rego. Cotuous varables take o a fte umber of values; examples clude temperature ad elevato. 2.2 VISUAL DESCRIPTIVE METHODS Suppose that we wsh to lear somethg about the commutg behavor of resdets a commuty. Perhaps we are o a commttee that s vestgatg the potetal mplemetato of a publc trast alteratve, ad we eed to kow how may mutes, o average, t takes people to get to work by car. We do ot have the resources to ask everyoe, ad so we decde to take a sample of automoble commuters. Let s say we survey = 30 resdets, askg them to record ther average tme t takes to get to work. We receve the resposes show pael (a) of Table 2.1. We may summarze our data vsually by costructg a hstogram, whch s a vertcal bar graph. To costruct a hstogram, the data are frst grouped to categores. The hstogram cotas oe vertcal bar for each category. The heght of the bar represets the umber of observatos the category (.e., the absolute frequecy), ad t s commo to ote the mdpot of the category o the horzotal axs. Fgure 2.1 s a hstogram for the hypothetcal commutg data Table 2.1, produced by SPSS for Wdows. A alteratve to the hstogram s the frequecy polygo; t may be draw by coectg the pots formed at the mddle of the top of each vertcal bar. Data may also be summarzed va box plots. Fgure 2.2 depcts a box plot for the commutg data. The horzotal le rug through the rectagle deotes the meda (21), ad the lower ad upper eds of the rectagle (sometmes called the hges ) represet the 25th ad 75th percetles, respectvely. Vellema ad Hoagl (1981) ote that there are two commo ways to draw the whskers, whch exted upward ad dowward from the hges. Oe way s to sed the whskers out to the mmum ad maxmum values. I ths case, the box plot represets a graphcal summary of what s sometmes called a fve-umber summary of the dstrbuto (the mmum, maxmum, 25th ad 75th percetles, ad the meda). 02_Rogerso_4e_BAB1405B0092_Ch-02.dd 27 10/3/ :40:35 AM

4 28 Statstcal Methods for Geography Table 2.1 Commutg data (a) Data o dvduals Idvdual o. Commutg tme (m.) Idvdual o. Commutg tme (m.) (b) raked commutg tmes 5, 5, 6, 9, 10, 11, 11, 12, 12, 14, 16, 17, 19, 21, 21, 21, 21, 21, 22, 23, 24, 24, 26, 26, 31, 31, 36, 42, 44, Frequecy Std. Dev = Mea = 21.9 N = Fgure 2.1 Hstogram for commutg data 02_Rogerso_4e_BAB1405B0092_Ch-02.dd 28 10/3/ :40:36 AM

5 Descrptve Statstcs 29 Commutg tme Fgure 2.2 Boxplot for commutg data There are ofte extreme outlers the data that are far from the mea, ad ths case t s ot preferable to sed whskers out to these extreme values. Istead, whskers are set out to the outermost observatos that are stll wth a dstace from the hge that s equal to 1.5 tmes the terquartle rage (where the terquartle rage s determed as the dfferece betwee the 75th ad 25th percetles, equvalet to the vertcal legth of the rectagle). All other observatos beyod ths are cosdered outlers, ad are show dvdually. I the commutg data, 1.5 tmes the terquartle rage s equal to 1.5(14.25) = The whsker extedg dowward from the lower hge exteds to the mmum value of 5, sce ths s greater tha the lower hge (11.75) mus The whsker extedg upward from the upper hge stops at 44, whch s the hghest observato less tha (whch tur s equal to the upper hge (26) plus ). Note that there s a sgle outler observato 9 ad t has a value of 77 mutes. A stem-ad-leaf plot s a alteratve way to dsplay the frequeces of observatos. It s smlar to a hstogram tlted oto ts sde, wth the actual dgts of each observato s value used place of bars. The leadg dgts costtute the stem, ad the tralg dgts make up the leaf. Each stem has oe or more leaves, wth each leaf correspodg to a observato. The vsual depcto of the frequecy of leaves coveys to the reader a mpresso of the frequecy of observatos that fall wth gve rages. Joh Tukey, the desger of the stem-ad-leaf plot, has sad, If we are gog to make a mark, t may as well be a meagful oe. The smplest ad most useful meagful mark s a dgt (Tukey 1972, p. 269). For the commutg data, whch have at most two-dgt values, the frst dgt s the stem, ad the secod s the leaf (see Fgure 2.3). Note that for each tem the stem, the fal dgts are arraged umercal order, from lowest to hghest. 02_Rogerso_4e_BAB1405B0092_Ch-02.dd 29 10/3/ :40:36 AM

6 30 Statstcal Methods for Geography Frequecy Stem & Leaf Extremes > =77) Stem wdth: Each leaf: 1 case(s) Fgure 2.3 Stem-ad-leaf plot for commutg data To gve aother example that makes use of vsual descrpto, cosder school dstrct admstrators, who ofte take cesuses of the umber of school-age chldre ther dstrct, so that they may form hopefully accurate estmates of future erollmet. Table 2.2 gves the hypothetcal resposes of 750 households whe asked how may school-age chldre are co-resdets. Table 2.2 Frequecy of chldre households Number of chldre Absolute frequecy Total 750 The absolute frequeces may be traslated to relatve frequeces by dvdg by the total umber of observatos ( ths case, 750). Table 2.3 reveals, for example, that 26.7% of all households surveyed had oe chld. Note that the sum of the relatve frequeces s equal to oe. Note also that we ca easly costruct a hstogram usg the relatve frequeces stead of the absolute frequeces (see Fgure 2.4); the hstogram pael (b) has precsely 02_Rogerso_4e_BAB1405B0092_Ch-02.dd 30 10/3/ :40:36 AM

7 Descrptve Statstcs 31 Table 2.3 Absolute ad relatve frequeces Number of chldre Absolute frequecy Relatve frequecy /750 = /750 = /750 = /750 = /750 =.067 Total (a) (b) Fgure 2.4 Number of chldre households: (a) absolute frequecy; (b) relatve frequecy the same shape as that pael (a); the vertcal scale has just bee chaged by a factor equal to the sample sze of MEASURES OF CENTRAL TENDENCY We may cotue our descrptve aalyss of the data Table 2.1 by summarzg the formato umercally. The sample mea commutg tme s smply the average of our observatos; t s foud by addg all of the dvdual resposes ad dvdg by the umber of observatos. The sample mea s tradtoally deoted by x ; our example, we have x = 658/30 = mutes. I practce, ths could sesbly be rouded to 22 mutes. We ca use otato to state more formally that the mea s the sum of the observatos, dvded by the umber of observatos: x x = =Σ 1 (2.1) 02_Rogerso_4e_BAB1405B0092_Ch-02.dd 31 10/3/ :40:37 AM

8 32 Statstcal Methods for Geography where x deotes the value of observato, ad where there are observatos. (A revew of mathematcal covetos ad mathematcal otato may lkely be order for may readers; see Appedx B.) The meda s defed as the observato that splts the raked lst of observatos (arraged from lowest to hghest, or hghest to lowest) half. Whe the umber of observatos s odd, the meda s smply equal to the mddle value o a raked lst of the observatos. Whe the umber of observatos s eve, we take the meda to be the average of the two values the mddle of the raked lst. Half of all respodets our example have commutes that are loger tha the meda, ad half have commutes that are shorter. Whe the resposes are raked as pael (b) of Table 2.1, the two the mddle are 21 ad 21. The meda ths case s therefore equal to 21 mutes. The mode s defed as the most frequetly occurrg value; here the mode s also 21 mutes, sce that value occurs more frequetly tha ay other outcome. May varables have dstrbutos where a small umber of hgh values cause the mea to be much larger tha the meda; ths s typcally true for come dstrbutos ad dstace dstrbutos. For example, Rogerso et al. (1993) used the US Natoal Survey of Famles ad Households to study the dstace that adult chldre lved from ther parets. For adult chldre wth both parets alve ad lvg together, the mea dstace to parets s over 200 mles, ad yet the meda dstace s oly 25 mles! Because the mea s ot represetatve of the data crcumstaces such as these, t s commo to use the meda as a measure of cetral tedecy. Whe data are avalable oly for categores, grouped meas may be calculated. Ths s acheved by assumg that all of the data wth a partcular category take o the mdpot value of the category. For example, Table 2.4 portrays some hypothetcal data o come (uts have bee delberately omtted to keep the example locato-free!). Table 2.4 Absolute frequeces assocated wth observatos o come Icome Frequecy (Number of dvduals) < 15, ,000 34, ,000 54, ,000 99, The grouped mea s foud by assumg that the te dvduals the frst category have a come of 7500 (the mdpot of the category), the 20 dvduals the secod category each have a come of 25,000, the 30 dvduals the ext category each have a come of 45,000, ad those the fal category each have a come of 77,500. All 02_Rogerso_4e_BAB1405B0092_Ch-02.dd 32 10/3/ :40:37 AM

9 Descrptve Statstcs 33 of these dvdual values are added, ad the result s dvded by the umber of dvduals. Thus, the grouped mea for ths example s: More formally, 10(7,500) + 20(25,000) + 30(45,000) + 15(77,500) x G Σ fx md g = =, 1 G Σ f = 1 = 41, 167 (2.2) (2.3) where x g deotes the grouped mea, G s the umber of groups, f s the umber of observatos group, ad x,md deotes the value of the mdpot of the group. The grouped mea may be see as a weghted average, where the mdpot of each group s weghted by the frequecy of observatos that group. After all of these weghted quattes have bee calculated ad summed, the last step the calculato of a weghted average s to dvde by the sum of the weghts. A smlar way to look at the cocept of the grouped mea s to mage that we eed to create a specfc value for each observato, eve though we do t have ths formato. I the example above, we oly kow that there are te observatos that are less tha 15,000 we do t kow ther dvdual values. Image startg a lst we wll guess that the frst observato s 7,500, the secod observato s 7,500, ad so o. The begg of our lst thus has te values of 7,500. The we move o to the secod category there are 20 observatos the ext category ad so we add 20 staces of 25,000 to the lst, sce ths would be our best guess for each of these observatos. Whe we get doe wth the costructo of ths lst, we smply add the umbers (whch gves us the umerator Equatos 2.2 ad 2.3), ad we dvde by the umber of tems o the lst (whch s the deomator Equatos 2.2 ad 2.3). It s ot ucommo to fd that the last category s ope-eded; stead of the 55,000 99,999 category, t mght be more commo for data to be reported a category labeled 55,000 ad above. I ths case, a educated estmate of the average salary for those ths group should be made. It would also be useful to make a umber of such estmates ad repeat the calculato of the grouped mea, to see how sestve the result s to dfferet choces for the estmate. 2.4 MEASURES OF VARIABILITY We may also summarze hstograms ad datasets by characterzg ther varablty. The commutg data Table 2.1 rage from a low of 5 mutes to a hgh of 77 mutes. The rage s the dfferece betwee the two values: here t s equal to 77 5 = 72 mutes. 02_Rogerso_4e_BAB1405B0092_Ch-02.dd 33 10/3/ :40:38 AM

10 34 Statstcal Methods for Geography The terquartle rage s the dfferece betwee the 25th ad 75th percetles. Wth observatos, the 25th percetle s represeted by observato ( + 1)/4, whe the data have bee raked from lowest to hghest. The 75th percetle s represeted by observato 3( + 1)/4. These wll ofte ot be tegers, ad that case terpolato s used, just as t s for the meda whe there s a eve umber of observatos. For the commutg data, the 25th percetle s represeted by observato (30 + 1)/4 = Iterpolato betwee the 7th ad 8th lowest observatos requres that we go 3/4 of the way from the 7th lowest observato (whch s 11) to the 8th lowest observato (whch s 12). Ths mples that the 25th percetle s (sce s 3/4 of the way from 11 to 12; more formally, ths ca be foud by (a) multplyg 3/4 by the dfferece the two observatos (3/4 (12 11) = 3/4, ad the (b) addg the result to the smaller of the observatos (11 + 3/4 = 11.75). Smlarly, the 75th percetle s represeted by observato 3(30 + 1)/4 = Sce both the 23rd ad 24th observatos are equal to 26, the 75th percetle s equal to 26. The terquartle rage s the dfferece betwee these two values, or = The sample varace of the data (deoted s 2 ) may be thought of as the average squared devato of the observatos from the mea. To esure that the sample varace gves a ubased estmate of the true, ukow varace of the populato from whch the sample was draw (deoted σ 2 ), s 2 s computed by takg the sum of the squared devatos, ad the dvdg the result by 1, stead of by. Here the term ubased mples that f we were to repeat ths samplg may tmes, we would fd that the average or mea of our may sample varaces would be equal to the true varace. Thus, the sample varace s foud by takg the sum of squared devatos from the mea, ad the dvdg by 1: s 2 = 1 2 Σ( x x) = 1 (2.4) A approxmate terpretato of the varace s that t represets the average squared devato of a observato from the mea (t s a approxmate terpretato because 1, stead of, s used the deomator). I our example, s 2 = mutes 2. The sample stadard devato s equal to the square root of the sample varace; here we have s = = mutes. Note that the uts for the stadard devato are the same as those for the varable tself here, for example, the stadard devato s gve mutes. Sce the sample varace characterzes the average squared devato from the mea, by takg the square root to calculate the stadard devato, we are puttg the measure of varablty back o a scale closer to that used for the mea ad the orgal data. Although the stadard devato s ot equal to the average absolute devato of a observato from the mea, t s usually close. Varaces for grouped data are foud by assumg that all observatos are at the mdpot of ther category, ad the calculato s based o the sum of squared devatos of these mdpot values from the grouped mea: 02_Rogerso_4e_BAB1405B0092_Ch-02.dd 34 10/3/ :40:38 AM

11 Descrptve Statstcs 35 s G 2 = 1 g = 2, md g Σ f ( x x ) G ( Σ f ) 1 = 1 (2.5) For the data Table 2.4, the grouped varace s (10(7,500 41, 167) 2 +20(25,000 41,167) (45,000 41,167) + 15(77,500 41,167) ) = ( ) 1 8 (2.6) The square root of ths, 22,301, s the grouped stadard devato. 2.5 OTHER NUMERICAL MEASURES FOR DESCRIBING DATA Coeffcet of Varato Cosder the sellg prce of homes two commutes. I commuty A, the mea prce s 150,000 (uts are aga delberately omtted, so that the llustrato may apply to more tha oe ut of currecy!). The stadard devato s 75,000. I commuty B, the mea sellg prce s 80,000, ad the stadard devato s 60,000. The stadard devato s a absolute measure of varablty; ths example, such varablty s clearly lower commuty B. However, t s also useful to thk terms of relatve varablty. Relatve to ts mea, the varablty commuty B s greater tha that commuty A. More specfcally, the coeffcet of varato s defed as the rato of the stadard devato to the mea. Here, the coeffcet of varato commuty A s 75,000/150,000 = 0.5; commuty B, t s 60,000/80,000 = Skewess Skewess measures the degree of asymmetry exhbted by the data ad hstogram. Fgure 2.5 s clearly asymmetrc, ad t reveals that there are more observatos below the mea tha above t ths s kow as postve skewess. Postve skewess ca also be detected by comparg the mea ad meda. Whe the mea s greater tha the meda as t s here, the dstrbuto s postvely skewed. I cotrast, whe there are a small umber of low observatos ad a large umber of hgh oes, the mea s less tha the meda, ad the data exhbt egatve skewess (see Fgure 2.6). The sample skewess s computed by frst addg together the cubed devatos from the mea ad the dvdg by the product of the cubed stadard devato ad the umber of observatos: 02_Rogerso_4e_BAB1405B0092_Ch-02.dd 35 10/3/ :40:39 AM

12 36 Statstcal Methods for Geography Relatve frequecy x Fgure 2.5 A postvely skewed dstrbuto Relatve frequecy x Fgure 2.6 A egatvely skewed dstrbuto skewess = Σ ( x x) = 1 3 s 3 (2.7) The 30 commutg tmes Table 2.1 have a postve skewess of If skewess equals zero, the hstogram s symmetrc about the mea Kurtoss Kurtoss measures how peaked the hstogram s. Its defto s smlar to that for skewess, wth the excepto that the fourth power s used stead of the thrd: kurtoss = Σ( x x) = 1 s 4 4. (2.8) 02_Rogerso_4e_BAB1405B0092_Ch-02.dd 36 10/3/ :40:40 AM

13 Descrptve Statstcs 37 (a) Relatve frequecy Fgure 2.7(a) Leptokurtc dstrbuto (b) Relatve frequecy Fgure 2.7(b) Platykurtc dstrbuto Data wth a hgh degree of peakedess are sad to be leptokurtc, ad have values of kurtoss over 3.0 (see Fgure 2.7a). Flat hstograms are platykurtc, ad have kurtoss values less tha 3.0 (Fgure 2.7b). The kurtoss of the commutg tmes s equal to 7.68, ad hece the dstrbuto of commutg tmes s relatvely peaked Stadard Scores Sce data come from dstrbutos wth dfferet meas ad dfferet degrees of varablty, t s commo to stadardze observatos. Oe way to do ths s to trasform each observato to a z-score by frst subtractg the mea of all observatos, ad the dvdg the result by the stadard devato: 02_Rogerso_4e_BAB1405B0092_Ch-02.dd 37 10/3/ :40:40 AM

14 38 Statstcal Methods for Geography x x z =. (2.9) s z-scores may be terpreted as the umber of stadard devatos a observato s away from the mea. Data below the mea have egatve z-scores, whle data above the mea have postve z-scores. For the commutg data Table 2.1, the z-score for dvdual 1 s ( )/14.3 = Ths dvdual has a commutg tme that s 1.18 stadard devatos below the mea. 2.6 DESCRIPTIVE SPATIAL STATISTICS To ths pot, our dscusso of descrptve statstcs has bee geeral, the sese that the cocepts ad methods covered apply to a wde rage of data types. I ths secto, we revew a umber of descrptve statstcs that are useful provdg umercal summares of spatal data. Descrptve measures of spatal data are mportat uderstadg ad evaluatg such fudametal geographc cocepts as accessblty ad dsperso. For example, t s mportat to locate publc facltes so that they are accessble to defed populatos. Spatal measures of cetralty appled to the locato of dvduals the populato wll result geographc locatos that are some sese optmal wth respect to accessblty to the faclty. Smlarly, t s mportat to characterze the dsperso of evets aroud a pot. It s useful to summarze the spatal dsperso of dvduals aroud a hazardous waste ste. Are dvduals wth a partcular dsease less dspersed aroud the ste tha are people wthout the dsease? If so, ths could dcate that there s creased rsk of dsease at locatos ear the ste The Measuremet of Dstace The measuremet of dstace plays a key role the descrpto of spatal data. The shortest dstace betwee two pots o a global scale s called the great crcle dstace. Ths s represeted by a arc that, f exteded, would form a crcle whose plae would pass through the ceter of the earth. Ar travel over log dstaces s plaed usg these arcs. The great crcle dstace betwee ay two pots o a sphere may be calculated as follows. Let (a 1, b 1 ) ad (a 2, b 2 ) be the (lattude, logtude) pars for pots 1 ad 2, respectvely. The great crcle dstace betwee pots 1 ad 2 s d = r{arccos[sa s a + cosa cosa cos( b b )]} (2.10) where the sphere has radus r ad where the otato arccos[] sgfes the agle whose cose s equal to the term wth the square brackets. (A equvalet alteratve otato for arccos[x] s cos -1 [x].) 02_Rogerso_4e_BAB1405B0092_Ch-02.dd 38 10/3/ :40:41 AM

15 Descrptve Statstcs 39 Example Let s fd the great crcle dstace betwee Lodo (lattude a 1 = 51.51N, b 1 = logtude 0.13W) ad New York Cty (lattude a 2 = 40.67N, logtude b 2 = 73.94W). The earth has a radus of approxmately 3,963 mles. We must assume that t s a sphere (although all grade school chldre kow that t s actually a oblate spherod). Usg Equato 2.10, ths results d = 3,963 {arcos[(0.7827)(0.6517)+(0.6224)(0.7585)(0.2788)]} = 3,963{arccos[0.6417]} = 3,963 (.8741) = 3,464 mles The agle whose cose s s degrees. Oe fal pot about Equato 2.10 t requres the use of agles radas, ad ot degrees. To covert a agle from degrees to radas, multply by 3.14/180. Here 50.08(3.14)/180 = radas. The fal step s to multply ths result by r: (0.8741)(3,963) = 3,464 mles. For the calculato of dstaces o a smaller scale (say, wth a cty) we ca make the much more drastc assumpto that the world s flat. The we ca supermpose a grd o a map of the area, ad use the grd coordates for our calculatos. The two most commo measures of dstace o such a flat plae are Eucldea ad Mahatta dstaces. Let (x 1, y 1 ) ad (x 2, y 2 ) represet the x- ad y-coordates of pots 1 ad 2, respectvely. The Eucldea dstace s the shortest dstace betwee two pots ad s calculated as d = ( x x ) + ( y y ) (2.11) You may recogze ths s as the Pythagorea formula for the legth of the hypoteuse (the sold le Fgure 2.8) of a rght tragle. The Mahatta dstace s gve by d = ( x2 x1) + ( y2 y1 ) (2.12) where the vertcal les Equato 2.12 dcate that the absolute value s to be take. Ths s the dstace betwee two pots whe travel s restrcted to occur oly horzotal ad vertcal drectos. It s equal to the sum of the legth of the two dotted les Fgure 2.8. For example, suppose (x 1, y 1 ) = (8,4) ad (x 2, y 2 ) = (7,10). The the Eucldea dstace s 2 2 d = ( 7 8) + ( 10 4) = 37 = 6.08 ad the Mahatta dstace s d = ( 7 8) + ( 10 4 ) = 1+6 = 7. 02_Rogerso_4e_BAB1405B0092_Ch-02.dd 39 10/3/ :40:42 AM

16 40 Statstcal Methods for Geography Y X 2, Y 2 X 1, Y 1 X Fgure 2.8 Mahatta ad Eucldea dstaces Mea Ceter The most commoly used spatal measure of cetral tedecy s the mea ceter. For pot data, the x- ad y-coordates of the mea ceter are foud by smply fdg the mea of the x-coordates ad the mea of the y-coordates, respectvely. For areal data, the mea ceter ca be foud by assumg that we kow the x- ad y-coordates assocated wth the cetrods of each subrego. Geographc cetrods are pot locatos represetg the balace pot of the rego. Thus f you were holdg up a physcal map of the rego (say made out of metal, plastc, wood, or some other materal), the geographc cetrod s that locato where you could balace the map o oe fger. A populato cetrod s the same as the populato-weghted ceter of populato the very quatty we are dscussg ths secto! Gve the x- ad y-coordates of the cetrods for each surego, t s ofte useful to attach weghts to them. To fd the mea ceter of populato for stace, the weghts are take as the umber of people lvg each subrego. The weghted mea of the x-coordates ad y-coordates the provdes the locato of the mea ceter of populato. More specfcally, whe there are subregos, Σ wx Σ wy = 1 = 1 x = ; y = (2.13) Σ w Σ w where the w are the weghts (e.g., populato rego ) ad x ad y are the coordates of the cetrod rego. Coceptually, ths s detcal to assumg that all dvduals 02_Rogerso_4e_BAB1405B0092_Ch-02.dd 40 10/3/ :40:43 AM

17 Descrptve Statstcs 41 lvg a partcular subrego lve at a prespecfed pot ( ths case, the cetrod) of that subrego. The mea ceter s detcal to the locato that would be foud f the x- ad y-coordates for each dvdual were frst wrtte out, ad the the mea of all the x s ad all the y s were foud. Equato 2.13, through ts use of a weghted mea, merely provdes a qucker way to arrve at the soluto. The mea ceter of populato the USA has mgrated west ad south over tme (see Fgure 2.9). The weghted mea ceter has the property that t mmzes the sum of squared dstaces that dvduals must travel (assumg that each perso travels to the cetralzed faclty located at the mea ceter). Although t s easy to calculate, ths terpretato s a lttle usatsfyg t would be cer to be able to fd a cetral locato that mmzes the sum of dstaces, rather tha the sum of squared dstaces Meda Ceter The locato that mmzes the sum of dstaces traveled s kow as the meda ceter. Although ts terpretato s more straghtforward tha that of the mea ceter, ts calculato s more complex. Calculato of the meda ceter s teratve, ad oe begs by guessg a tal locato (a coveet tal locato s the weghted mea ceter). The the ew x- ad y-coordates (deoted by x ad y, respectvely) are updated usg the followg: wx Σ = 1 d x = ; y = w Σ = 1 d wy Σ d w Σ = 1 d = 1 (2.14) where d s the dstace from pot to the specfed tal locato of the meda ceter. Ths process s the carred out aga ew x- ad y-coordates are aga foud usg these same equatos, wth the oly dfferece beg that d s redefed as the dstace from pot to the most recetly calculated locato for the meda ceter. Ths teratve process s termated whe the ewly computed locato of the meda ceter does ot dffer sgfcatly from the prevously computed locato. I the applcato of socal physcs to spatal teracto, populato dvded by dstace s cosdered a measure of populato potetal or accessblty. If the w s are defed as populatos, the each terato fds a updated locato based upo weghtg each pot or areal cetrod by ts accessblty to the curret meda ceter. The meda ceter s the fxed pot that s mapped to tself whe weghted by accessblty. Alteratvely stated, the meda ceter s a accessblty-weghted mea ceter, where accessblty s defed terms of the dstaces from each pot or areal cetrod to the meda ceter. 02_Rogerso_4e_BAB1405B0092_Ch-02.dd 41 10/3/ :40:44 AM

18 Fgure 2.9 Mea Ceter of Populato for the Uted States: 1790 to 2010 Source: US Bureau of the Cesus (2011) 02_Rogerso_4e_BAB1405B0092_Ch-02.dd 42 10/3/ :40:44 AM

19 Descrptve Statstcs Stadard Dstace Aspatal measures of varablty, such as the varace ad stadard devato, characterze the amout of dsperso of data pots aroud the mea. Smlarly, the spatal varablty of locatos aroud a fxed cetral locato may be summarzed. The stadard dstace (Bach 1963) s defed as the square root of the average squared dstace of pots to the mea ceter: s d Σ d = 1 = 2 c (2.15) where d c s the dstace from pot to the mea ceter ad s the umber of locatos aroud the cetral locato. Although Bach s measure of stadard dstace s coceptually appealg as a spatal verso of the stadard devato, t s ot really ecessary to mata the strct aalogy wth the stadard devato by takg the square root of the average squared dstace. Wth the aspatal verso (.e., the stadard devato), loosely speakg, the square root udoes the squarg ad thus the stadard devato may be roughly terpreted as a quatty that s o the same approxmate scale as the average absolute devato of observatos from the mea. Squarg ad takg square roots s carred out part because devatos from the mea may be ether postve or egatve. But, the spatal verso, dstaces are always postve, ad so a more terpretable ad atural defto of stadard dstace would be to smply use the average dstace of observatos from the mea ceter: d s Σ dc = =1 (2.16) I practce, the results foud usg Equatos 2.15 ad 2.16 wll be qute smlar. Equatos 2.15 ad 2.16 represet deftos of stadard dstace whe each pot or locato s gve equal weght the calculato. More commoly, each locato wll have a assocated weght, w. I ths case, the weghted stadard dstaces are calculated usg 2 Σ wd c = 1 sd = Σ w = 1 (2.17) s d Σ wd c = 1 = Σ w = 1 02_Rogerso_4e_BAB1405B0092_Ch-02.dd 43 10/3/ :40:45 AM

20 44 Statstcal Methods for Geography The uts for all of these deftos of stadard dstace are the uts of the dstace varable (typcally mles, klometers, etc.) Relatve Dstace Oe drawback to the stadard dstace measure descrbed above s that t s a measure of absolute dsperso; t retas the uts whch dstace s measured. Furthermore, t s affected by the sze of the study area. The two paels of Fgure 2.10 show stuatos where the stadard dstace s detcal, but clearly the amout of dsperso about the cetral locato, relatve to the study area, s lower pael (b). Relatve dstace s a measure of relatve dsperso ad t may be derved by dvdg the stadard dstace by the radus of a crcle wth area equal to the sze of the study area (McGrew ad Moroe 2000). Ths makes the measure of dsperso utless ad stadardzes for the sze of the study area, thereby facltatg comparso of dsperso study areas of dfferet szes. For a crcular study area, the relatve dstace s s d,rel = s d /r ad for a square study area, 2 sdrel, = sd π / s (where r ad s represet the radus of the crcle ad the sde of the square, respectvely). Note that the maxmum relatve dstace for a crcle s 1; ths occurs whe all pots are located o the crcumferece of the crcle. For a square, the maxmum relatve dstace s π / 2 = , ad ths occurs whe all pots are located at corers of the square Illustrato of Spatal Measures of Cetral Tedecy ad Dsperso The descrptve spatal statstcs outled above are ow llustrated usg the data Table 2.5. Ths s a smple set of te locatos; a smple ad small dataset has bee chose X X (a) (b) Fgure 2.10 Illustrato of stadard dstace. Note that dsperso relatve to the study area s lower pael (b) 02_Rogerso_4e_BAB1405B0092_Ch-02.dd 44 10/3/ :40:46 AM

21 Descrptve Statstcs 45 delberately, to facltate the dervato of the quattes by had, f desred. The study area s assumed to be a square wth the legth of each sde equal to oe. I ths example, we assume mplctly that there are equal weghts at each locato (or, equvaletly, that oe dvdual s at each locato). The mea ceter s at (0.4513, ), ad t s foud smply by takg the mea of each colum. The locato of the meda ceter s at (0.4611, ). Accuracy to three dgts s acheved after 33 teratos. The frst few teratos of Equato 2.14 are show Table 2.6. Table 2.5 x y coordate pars x y Table 2.6 Covergece of teratos toward the meda ceter Iterato x-coordate y-coordate _Rogerso_4e_BAB1405B0092_Ch-02.dd 45 10/3/ :40:46 AM

22 46 Statstcal Methods for Geography It s terestg to ote that the approach to the y-coordate of the meda ceter s mootoc, whle the approach to the x-coordate s a damped harmoc. The sum of squared dstaces to the mea ceter s ; ote that ths s lower tha the sum of squared dstaces to the meda ceter (0.9328). Smlarly, the sum of dstaces to the meda ceter s 2.655, ad ths s lower tha the sum of dstaces to the mea ceter (2.712). The stadard dstace s (whch s the square root of /10); ote that ths s smlar to the average dstace of a pot from the mea ceter (2.712/10 = ) Agular Data Agular data arse a umber of geographcal applcatos; the aalyss of wd drecto, ad the study of the algmet of crystals bedrock provde two examples. The latter example has bee partcularly mportat the study of cotetal drft, ad establshg the tmg of reversals earth s magetc feld. Specal cosderatos arse the vsual ad umercal descrpto of agular data. Cosder the 146 observatos o wd drecto gve Table 2.7. A hstogram Table 2.7 Hypothetcal data o wd drecto Drecto Agular drecto Frequecy North 0 10 Northeast 45 8 East 90 5 Southeast South Southwest West Northwest Total 146 could be costructed, but t s ot clear how the horzotal axs should be labeled. A hstogram could arbtrarly start wth North o the left, as Fgure 2.11a; aother possblty s to arrage for the mode to be ear the mddle of the hstogram, as Fgure 2.11b. Nether of the optos depcted Fgure 2.11 for costructg a hstogram usg agular data s deal, sce the observatos at the far left of the horzotal axs are smlar 02_Rogerso_4e_BAB1405B0092_Ch-02.dd 46 10/3/ :40:46 AM

23 Descrptve Statstcs 47 (a) Frequecy N NE E SE S SW W NW Drecto (b) Frequecy SE S SW W NW Drecto N NE E Fgure 2.11 Absolute frequeces for drectoal data drecto to the observatos o the far rght of the horzotal axs. I partcular, there s o provso for wrappg the hstogram aroud o tself. A alteratve s the crcular hstogram (Fgure 2.12a). Here bars exted outward all drectos, reflectg the ature of the data. As s the case wth more typcal hstograms, the legths of the bars are proportoal to the frequecy. A slght varato of ths s the more commo rose dagram (Fgure 2.12b); here the rectagular bars have bee replaced wth pe- or wedge-shapes. The cocetrc rgs show Fgure 2.12b are ofte ot dsplayed; they are show here to emphasze how the relatve frequeces are used to costruct the dagram. The rose dagram s effectve portrayg vsually the ature of agular data. There are also specal cosderatos that are ecessary whe cosderg umercal summares of agular data. Cosder the very smple case where we have two observatos oe observato s 1 ad the other s 359. If 0 s take to be orth, both of these 02_Rogerso_4e_BAB1405B0092_Ch-02.dd 47 10/3/ :40:46 AM

24 48 Statstcal Methods for Geography (a) N (b) N W E W E S S Fgure 2.12 Absolute frequeces for drectoal data: (a) crcular hstogram; (b) rose dagram observatos are very close to orth. However, f we take the smple average or mea of 1 ad 359, we get ( )/2 = 180 due south! Clearly, some other approach s eeded, sce the average of two observatos that are very close to orth should ot be south. Here we descrbe how to fd the mea ad varace for agular data. Mea: 1. Fd the se ad cose of each agular observato. 2. Fd the mea of the ses ( S ) ad the mea of the coses ( C ) Fd R = S + C 4. The mea agle (say α ) s the agle whose cose s equal to C / R ad whose se s equal to S / R. Thus, α = arccos (C / R) ad α = arcs ( S / R ). Varace: A measure of varace for agular data (termed the crcular varace) provdes a dcato of how much varablty there s the data. For example, f all observatos cossted of the same agle, the varablty, ad hece the crcular varace, should be zero. The crcular varace, desgated by S 0, s smply equal to 1 R. It vares from zero to oe. A hgh value ear oe dcates that the agular data are dspersed, ad come from may dfferet drectos. A value ear zero mples that the observatos are clustered aroud partcular drectos. Readers terested more detal regardg agular data may fd more extesve coverage Marda ad Jupp (1999). 02_Rogerso_4e_BAB1405B0092_Ch-02.dd 48 10/3/ :40:48 AM

25 Descrptve Statstcs 49 Example 2.1 Three observatos of wd drecto yeld measuremets of 43, 88, ad 279. Fd the agular mea ad the crcular varace. Soluto. We start by costructg the followg table: Observato Cose Se ( ) = The mea of the coses s equal to C = / The mea of the ses s equal to S = ( )/3 = The 2 2 R = = The mea agle, α, s the agle whose cose s equal to C / R = / = , ad whose se s equal to S / R = / = Usg ether a calculator or table, we fd that the mea agle s arccos(0.7994) = arcs(0.6008) = 37. The crcular varace s equal to 1 R = = Ths value s closer to oe tha to zero, dcatg a tedecy for hgh varablty that s, the agles are relatvely dspersed ad are comg from dfferet drectos. 2.7 DESCRIPTIVE STATISTICS IN SPSS 21 FOR WINDOWS Data Iput After startg SPSS, data are put for the varable or varables of terest. Each colum represets a varable. For the commutg example set out Table 2.1, the 30 observatos were etered to the frst colum of the spreadsheet. Alteratvely, respodet ID could have bee etered to the frst colum (.e., the sequece of tegers, from 1 to 30), ad the commutg tmes would the have bee etered the secod colum. The order that the data are etered to a colum s umportat Descrptve Aalyss Smple descrptve statstcs Oce the data are etered, clck o Aalyze (or Statstcs, older versos of SPSS for Wdows). The clck o Descrptve Statstcs. The clck o Explore. A splt 02_Rogerso_4e_BAB1405B0092_Ch-02.dd 49 10/3/ :40:50 AM

26 50 Statstcal Methods for Geography box wll appear o the scree; move the varable or varables of terest from the left box to the box o the rght that s headed Depedet Lst by hghlghtg the varable(s) ad clckg o the arrow. The clck o OK. Table 2.8 Descrptve summary of commutg data usg SPSS Case Processg Summary Cases Vald Mssg Total N Percet N Percet N Percet VAR % 0.0% % Descrptves Statstc Std. Error VAR00001 Mea % Cofdece Lower Boud Iterval for Mea Upper Boud % Trmmed Mea Meda Varace Std. Devato Mmum 5.00 Maxmum Rage Iterquartle Rage Skewess Kurtoss Other optos Optos for producg other related statstcs ad graphs are avalable. To produce a hstogram, for stace, before clckg OK above, clck o Plots, ad you ca the check a box to produce a hstogram. The clck o Cotue ad OK Results Table 2.8 dsplays results of the output. I addto to ths table, box plots (Fgure 2.2), stem ad leaf dsplays (Fgure 2.3) ad, optoally, hstograms (Fgure 2.1) are also produced. 02_Rogerso_4e_BAB1405B0092_Ch-02.dd 50 10/3/ :40:50 AM

27 Descrptve Statstcs 51 SOLVED EXERCISES 1. A ew park s plaed for a commuty, ad plaers wsh to cosder the mea ceter of four resdetal areas as a possble locato. Usg the coordates ad resdetal populatos lsted below, fd the weghted mea ceter. x-coordate y-coordate resdetal populato Soluto. We wll frst fd the x-coordate of the weghted mea ceter. Coceptually, ths s the average x-coordate amog all resdets. There are 200 people wth a x-coordate of 2, 100 wth a x-coordate of 4, ad so o. If we totaled the x-coordates amog all resdets, the result would be 200(2) + 100(4) + 50(4) (8) = = There are = 550 resdets, ad so the average x-coordate amog all resdets s 2600/550 = Smlarly, the y-coordate of the mea ceter s 200(1) + 100(4) + 50(8) + 200(2) 550 Thus, the weghted mea ceter s at (4.73, 2.55) = = Usg the followg data, fd the mea, meda, ad rage: Data values: 29, 35, 17, 30, 231, 6, 27, 35, 23, 29, 13 Soluto. The mea (or average) s equal to the sum of the observatos ( = 475), dvded by the umber of observatos ( = 11); thus the mea s 475/11 = To fd the meda, frst order the observatos: 6, 13, 17, 23, 27, 29, 29, 30, 35, 35, 231. The meda s observato umber ( + 1)/2 o ths ordered lst. Sce = 11, ( + 1)/2 = 6; the 6th observato o the lst (29) s the meda. Fally, the rage s equal to the largest observato (231), mus the smallest (6), ad s therefore equal to = Fd the skewess ad kurtoss for the followg data: 4, 7, 8, 13. (Cotued) 02_Rogerso_4e_BAB1405B0092_Ch-02.dd 51 10/3/ :40:50 AM

28 52 Statstcal Methods for Geography (Cotued) Soluto. For the skewess, we use Equato 2.7. The equato mples that we wll eed the mea ad the stadard devato. The mea s equal to ( )/4 = 32/4 = 8. The stadard devato s the square root of the varace. The varace s foud by fdg the sum of the squared devatos from the mea, ad the dvdg the result by 1, where s the umber of observatos. The squared devatos from the mea are (4 8) 2, (7 8) 2, (8 8) 2, ad (13 8) 2 ; summg these yelds ( 4) 2 + ( 1) = = 42. The varace s thus equal to 42/(4 1) = 14, ad the stadard devato s s = 14 = The umerator of the skewess s equal to the sum of the cubed devatos from the mea. The cubed devatos from the mea are (4 8) 3, (7 8) 3, (8 8) 3, ad (13 8) 3, ad the sum of these quattes s ( 4) 3 + ( 1) = ( 64) + ( 1) = 60. The deomator s equal to the umber of observatos, multpled by the cube of the stadard devato: 4( ) = The skewess s equal to 60/ = The data exhbt a small amout of postve skewess. We use Equato 2.8 to fd the kurtoss. The devatos from the mea are frst rased to the fourth power: (4 8) 4, (7 8) 4, (8 8) 4, ad (13 8) 4 ; these are summed to fd the umerator of the kurtoss, ad ths result s ( 4) 4 + ( 1) = = 882. The deomator of the kurtoss s equal to the umber of observatos, multpled by the stadard devato rased to the fourth power (ad ote that ths latter quatty s equal to the square of the varace). The deomator s therefore 4( ) = 4(14 2 ) = 784. The kurtoss s equal to 882/784 = Note that sce ths s less tha 3, the dstrbuto ca be descrbed as flat, or platykurtc. 4. Fd the grouped mea for the followg data: Category Frequecy Soluto. To fd the grouped mea, we assume that all observatos are at the mdpot of the category they are. There are fve observatos that are less tha 20. All we kow s that they are betwee 0 ad 20; we do ot kow ther exact values. We assume that they are all mdway betwee 0 ad 20 ad therefore assg them values of 10. Smlarly, the 15 observatos the ext category are all assged a value of 30, whch s the mdpot of the category they are ( ). The mea of all of these observatos, oce they are assged the mdpot of ther categores, s foud by addg all of ther assged values, ad the dvdg the result by the total umber of observatos. 02_Rogerso_4e_BAB1405B0092_Ch-02.dd 52 10/3/ :40:50 AM

29 Descrptve Statstcs 53 Thus, we have 5 10 = 50 (whch s the total of the fve observatos the frst category), plus = 450 (whch s the total of the 15 observatos the secod category), plus 10 50, plus The soluto s therefore equal to ths total of = 1840, dvded by the total umber of observatos ( = 42); ths s equal to 1840/42 = EXERCISES 1. The 236 values that appear below are the 1990 meda household comes ( dollars) for the 236 cesus tracts of Buffalo, New York. (a) For the frst 19 tracts, fd the mea, meda, rage, terquartle rage, stadard devato, varace, skewess, ad kurtoss, usg oly a calculator (though you may wat to check your results usg a statstcal software package). I addto, costruct a stem-ad-leaf plot, a box plot, ad a hstogram for these 19 observatos. (b) Use a statstcal software package to repeat part (a), ths tme usg all 236 observatos. (c) Commet o your results. I partcular, what does t mea to fd the mea of a set of medas? How do the observatos that have a value of 0 affect the results? Should they be cluded? How mght the results dffer f a dfferet geographc scale was chose? 22342, 19919, 8187, 15875, 17994, 30765, 31347, 27282, 29310, 23720, 22033, 11706, 15625, 6173, 15694, 7924, 10433, 13274, 17803, 20583, 21897, 14531, 19048, 19850, 19734, 18205, 13984, 8738, 10299, 10678, 8685, 13455, 14821, 23722, 8740, 12325, 10717, 21447, 11250, 16016, 11509, 11395, 19721, 23231, 21293, 24375, 19510, 14926, 22490, 21383, 25060, 22664, 8671, 31566, 26931, 0, 24965, 34656, 24493, 21764, 25843, 32708, 22188, 19909, 33675, 15608, 15857, 18649, 21880, 17250, 16569, 14991, 0, 8643, 22801, 39708, 17096, 20647, 30712, 19304, 24116, 17500, 19106, 17517, 12525, 13936, 7495, 10232, 6891, 16888, 42274, 43033, 43500, 22257, 22931, 31918, 29072, 31948, 36229, 33860, 32586, 32606, 31453, 32939, 30072, 32185, 35664, 27578, 23861, 18374, 26563, 30726, 33614, 30373, 28347, 37786, 48987, 56318, 49641, 85742, 43229, 53116, 44335, 30184, 36744, 39698, 0, 21987, 66358, 46587, 26934, 27292, 31558, 36944, 43750, 49408, 37354, 31010, 35709, 32913, 25594, 25612, 28980, 28800, 28634, 18958, 26515, 24779, 21667, 24660, 29375, 29063, 30996, 45645, 39312, 34287, 35533, 27647, 24342, 22402, 28967, 39083, 28649, 23881, 31071, (Cotued) 02_Rogerso_4e_BAB1405B0092_Ch-02.dd 53 10/3/ :40:50 AM

30 54 Statstcal Methods for Geography (Cotued) 27412, 27943, 34500, 19792, 41447, 35833, 41957, 14333, 12778, 20000, 19656, 22302, 33475, 26580, 0, 24588, 31496, 30179, 33694, 36193, 41921, 35819, 39304, 38844, 37443, 47873, 41410, 34186, 36798, 38508, 38382, 37029, 48472, 38837, 40548, 35165, 39404, 34281, 24615, 34904, 21964, 42617, 58682, 41875, 40370, 24511, 31008, 16250, 29600, 38205, 35536, 35386, 36250, 31341, 33790, 31987, 42113, 37500, 33841, 37877, 35650, 28556, 27048, 27736, 30269, 32699, 28988, 22083, 27446, 76306, Te mgrato dstaces correspodg to the dstaces moved by recet mgrats are observed ( mles): 43, 6, 7, 11, 122, 41, 21, 17, 1, 3. Fd the mea ad stadard devato, ad the covert all observatos to z-scores. 3. By had, fd the mea, meda, ad stadard devato of the umber of bedrooms, for the frst te observatos of the Mlwaukee dataset. 4. Usg SPSS or Excel, aswer the followg questos usg the full Mlwaukee house sale dataset: (a) Make hstograms of () house sale values for those observatos that have more tha two bedrooms, ad () those observatos that have fewer tha three bedrooms. Commet o the dffereces. (b) Fd the mea, meda, ad stadard devato for sales prce, umber of bedrooms, age of house, ad lot sze. 5. Usg SPSS or Excel, aswer the followg questos usg the full Hypothetcal UK Housg Prces dataset: (a) Provde descrptve formato o house prces, umber of bedrooms, umber of bathrooms, floor area, ad date bult. For each, provde the mea, meda, stadard devato, ad skewess. Also for each, provde a box plot. (b) What percetage of homes have garages? (c) What percetage of homes were bult durg each of the followg tme perods: pre-war, ter-war, ad post-war? 6. For the frst te observatos of the Mlwaukee dataset: (a) fd the mea ceter; (b) fd Bach s stadard dstace. Exercses 7 11 are questos pertag to otato; see Appedx B for a revew. 7. Gve a = 3, b = 4, ad 02_Rogerso_4e_BAB1405B0092_Ch-02.dd 54 10/3/ :40:51 AM

31 Descrptve Statstcs 55 Observato x y Fd the followg: (a) Σ y (b) Σ y 2 (c) Σax (d) Πx (e) Σ 3 =2 y + by (f) 3x 2 + y 4 (g) 32!/( 30!) xkyk (h) Σ k 8. Let a = 5, x 1 = 6, x 2 = 7, x 3 = 8, x 4 = 10, x 5 = 11, y 1 = 3, y 2 = 5, y 3 = 6, y 4 = 14, ad y 5 = 12. Fd the followg: (a) Σx (b) Σx y (c) Σ x + ay 3 (d) Σ y 2 =1 (e) ( ) = = 1 Σ a (f) Σ k 2( y k 3) = 5 (g) Σ x x = 1 ( ) 9. Fd 8!/3! (Cotued) 02_Rogerso_4e_BAB1405B0092_Ch-02.dd 55 10/3/ :40:55 AM

32 56 Statstcal Methods for Geography (Cotued) 10. Fd Use the followg table of commutg flows to determe the total umber of commuters leavg each zoe ad the total umber eterg each zoe. Also fd the total umber of commuters. For each aswer, also gve the correct otato, assumg y j deotes the umber of commuters who leave org to go to destato zoe j. Destato zoe Org zoe The followg data represet stream lk legths a rver etwork: 100, 426, 322, 466, 112, 155, 388, 1155, 234, 324, 556, 221, 18, 133, 177, 441. Fd the mea ad stadard devato of the lk legths. 13. For the followg aual rafall data, fd the grouped mea ad the grouped varace. Rafall Number of years observed < Notce that assumptos must be made about the mdpots of the ope-eded age groups. I ths example, use 15 ad 65 as the mdpots of the frst ad last rafall groups, respectvely. 14. A square grd s placed over a cty map. What s the Eucldea dstace betwee two places located at (1,3) ad (3,6)? 15. I the example above, what s the Mahatta dstace? 02_Rogerso_4e_BAB1405B0092_Ch-02.dd 56 10/3/ :40:55 AM

33 Descrptve Statstcs Use a rose dagram to portray the followg agular data: Drecto Frequecy Drecto Frequecy N 43 S 60 NE 12 SW 70 E 23 W 75 SE 45 NW Draw frequecy dstrbutos that have (a) postve skewess, (b) egatve skewess, (c) low kurtoss ad o skewess, ad (d) hgh kurtoss ad o skewess. 18. What s the coeffcet of varato amog the followg commutg tmes: 23, 43, 42, 7, 23, 11, 23, 55? 19. A square grd s placed over a cty map. There are resdetal areas at the coordates (0,1), (2,3) ad (5,6). The respectve populatos of the three areas are 2500, 2000, ad A cetralzed faclty s beg cosdered for ether the pot (4,4) or the pot (4,5). Whch of the two pots s the better locato for a cetralzed faclty, gve that we wsh to mmze the total Mahatta dstace traveled by the populato to the faclty? Justfy your aswer by gvg the total Mahatta dstace traveled by the populato to each of the two possble locatos. 20. Is the followg data o comes postvely or egatvely skewed? You do ot eed to calculate skewess, but you should justfy your aswer. Data thousads: 45, 43, 32, 23, 45, 43, 47, 39, 21, 90, (a) Fd the weghted mea ceter of populato, where ctes populatos ad coordates are gve as follows: Cty x y Populato A ,000 B ,500 C ,000 D ,000 E ,500 (b) Fd the uweghted mea ceter, ad commet o the dffereces betwee your two aswers. (c) Fd the Eucldea dstaces of each cty to the weghted mea ceter of populato. (Cotued) 02_Rogerso_4e_BAB1405B0092_Ch-02.dd 57 10/3/ :40:55 AM