On the graphical presentation of statistics

384

[[aupt ." ON THE G R A P H I C A L PRESENTATION STATISTICS.

[J. F. I.,

OF

BY" LEWIS M. HAUPT, Member of the Institute.

W h i l e statistics are said to be u n i n t e r e s t i n g , t h e y are, nevertheless, i n v a l u a b l e as indices to the p r o g r e s s i v e man of affairs w h o w o u l d keep in t o u c h w i t h the industrial dev e l o p m e n t of his own or o t h e r countries. B u t m a s s e s of figures a r r a n g e d in t a b u l a r form are c o n f u s i n g and often misleading, w i t h o u t g r e a t l a b o r in classifying or in reducing t h e m to units of comparison. F r e q u e n t l y only total quantities or v a l u e s are g i v e n of imports, exports, o u t p u t , p r o d u c t or prices, r e n d e r i n g a large a m o u n t of calculation necessary b e f o r e unit v a l u e s or p e r c e n t a g e s for v a r i o u s years, periods or c o m m o d i t i e s m a y be secured, and w i t h o u t w h i c h no comparison is possible. T o s h o w the g r o w t h of a n y p a r t i c u l a r industry, it is evid e n t t h a t at least two factors m u s t be r e p r e s e n t e d , viz., time and q u a n t i t y , or time and value. W h e n all t h r e e are known, and the totals only are given, t h e y m u s t be r e d u c e d to the v a l u e of the unit for each time stated, to d e t e r m i n e t h e actual returns. Since q u a n t i t y m a y be e x p r e s s e d in m a n y different denominations, as in pounds, tons, yards, bushels, bales, barrels, stones, cords, etc., t h e y m u s t also be b r o u g h t to a c o m m o n basis for comparison. T o avoid m a n y of t h e s e difficulties, the practice of repres e n t i n g statistics b y the aid of diagrams, plain, colored or s h a d e d in c o n v e n t i o n a l symbols, is r a p i d l y g r o w i n g in favor, b u t the e x i s t i n g c o n v e n t i o n s possess defects w h i c h render t h e m almost useless as b a s e s of c o m p a r i s o n for commercial purposes, unless a c c o m p a n i e d b y a b s t r a c t figures, in which case the f o r m e r o b j e c t i o n s are not eliminated. THE EXISTING I)EFECTS. F o r example, in c o m p a r i n g q u a n t i t i e s of a n y p a r t i c u l a r kind, as of population, a' column is b u i l t up of colored blocks h a v i n g three dimensions, r e q u i r i n g the eye to

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Graphical Presentaliolz of Statistics.

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c o m p a r e t h e s e v o l u m e s w i t h one a n o t h e r . Often they are n o t e v e n s i m i l a r vol um e s , so t h a t t h e c u b e s of t h e i r e d g e s c a n n o t be u s e d for t h e p u r p o s e i n t e n d e d , and unless t h e figures be b r a n d e d on t h e blocks, no i dea can be o b t a i n e d of e v e n t h e i r r e l a t i v e v o l u m e s . In m a n y o t h e r cases areas of v a r i o u s f o r m s and sizes are used, especially r e c t a n g l e s and circles, b u t f r e q u e n t l y sectors are e m p l o y e d in circles of d i f f e r e n t d i a m e t e r s , to i n d i c a t e g r o w t h from y e a r to year. H e r e , again, figures of two d i m e n s i o n s are t h e r e f o r e e r r o n e o u s l y us e d to r e p r e s e n t q u a n t i t i e s of only o n e d i m e n s i o n . It is t r u e t h a t t he c i r c u m f e r e n c e of the circle m a y be s u b d i v i d e d into arcs, r e p r e s e n t i n g prop o r t i o n a l p a r t s or p e r c e n t a g e s of traffic or product , b u t unless th e r atio s are i n d i c a t e d in the d e g r e e s of t he cent ral angle, or th e r e l a t i v e l e n g t h s of arc, t h e sectors possess b u t little a c t u a l v a l u e for c o m p a r i s o n . E v e n w h e r e st at i s t i c s are r e p r e s e n t e d in c o l u m n a r form, laid off on b o t h sides of a c e n t r a l axis, t he curve of increase is bisected, one-half of it b e i n g on e i t h e r side of the axis, and, t h e r e f o r e , t he c o r r e c t c u r v e is n o t s h o w n on t he diag r a m , b u t m u s t be d e d u c e d f r o m it by a m e n t a l operation. Ag ain , it f r e q u e n t l y h a p p e n s t h a t t h e i n t e r v a l s of t i m e w h e n th e r e c o r d is m a d e are q u i t e variable, so t h a t w i t h o u t i n t e r p o l a t i o n a c o r r e c t idea of the f o r m of t he c u r v e c a n n o t be o b t a i n e a e v e n w i t h t he a b s t r a c t figures stated. Cert a i n l y t h e r e c t a n g l e s d r a w n to scale, unless taken in conn e c t i o n with t h e i r t i m e i n t e r v a l s , b e c o m e q u i t e m i sl eadi ng. T h e a b s e n c e of u n i f o r m scales, and t he omission of any r e f e r e n c e t h e r e t o , r e s ul t s in r e p r e s e n t i n g t he part s as somet i m e s g r e a t e r t h a n t h e whole, a m a n i f e s t a b s u r d i t y . T h e s e and m a n y o t h e r i n c o n g r u i t i e s arise from the prom i s c u o u s p icto ri a l m e t h o d s in v o g u e i n t e n d e d to e d u c a t e t h e m i n d b y unscientific, p o p u l a r m e t h o d s , r e s u l t i n g in " c o n f u s i o n worse c o n f o u n d e d . " SUGGESTIONS FOR GENERAl, USE. All q u a n t i t i e s , of w h a t e v e r d e n o m i n a t i o n , kind or e x t e n t , w h i c h are c a p a b l e of b e i n g m e a s u r e d , m a y be e x p r e s s e d in a b s t r a c t figures, and such figures m a y al w ays be repreVoL. CXLVIII. No. 887. 25

386

t/az~pt :

[J. F. 1.,

s e n t e d b y linear d i s t a n c e s d r a w n to scale b y t a k i n g any n u m b e r of such parts as a unit. As the n u m b e r repres e n t i n g the q u a n t i t y or price m a y v a r y from time to time, so the l e n g t h s of the lines will v a r y for the c o r r e s p o n d i n g times, and t h e s e lines may, therefore, be compared, and the ratio of any one to a n y o t h e r be d e t e r m i n e d b y p r o j e c t i n g t h e m upon a s h e e t of cross-section paper, s t a r t i n g always from a s t r a i g h t line or edge. Thus, if the law of the i n c r e m e n t of any variable quantity is to be represented, the times at w h i c h the variations occur m a y be laid off along a s t r a i g h t line, while the actual v a l u e of t h e v a r i a t i o n m a y be p l a t t e d in a series of lines at r i g h t angles to the first, c o n s t i t u t i n g a s y s t e m of rectilinear co6rdinates, of w h i c h the times m a y be the abscissa (x) and the q u a n t i t i e s the o r d i n a t e s (y). By d r a w i n g a line t h r o u g h the e x t r e m i t y of the ordinates, the curve s h o w i n g the c h a n g e in v a l u e of the q u a n t i t y is at once revealed. If the i n c r e m e n t s for equal times are equal, the curve will b e c o m e a right line, and a p r e d i c t i o n can safely be m a d e as to f u t u r e probabilities. If it b e n d u p w a r d , the i n c r e m e n t s are i n c r e a s i n g ; if d o w n w a r d , decreasing. T h e y m a y oscillate violently, i n d i c a t i n g lack of s t a b i l i t y or g r e a t fluetuations in value, g e n e r a l l y traced to s o m e external cause affecting the v a l u e of the c o m m o d i t y u s e d in exchanges, as a plethoric or s t r i n g e n t m o n e y market, abundance or p a u c i t y of crops, etc. TO

ILLUSTRATE

DEVELOPMENTS.

T h e r e s u l t i n g diagrams, if p l a t t e d as indicated above, will give a simple, a c c u r a t e e x h i b i t of the conditions existing at any particular date, w h e n c e t h e y m a y readily be c o n n e c t e d w i t h correlative events, and the relation b e t w e e n e a u s e and effect be largely d e t e r m i n e d with accuracy. A s p o p u l a t i o n is the basis for, and principal e l e m e n t in, industrial development, and as it is i m p o r t a n t to d e t e r m i n e the p r o b a b l e increase as affecting q u e s t i o n s of c o n s u m p t i o n or traffic, it m a y be i n s t r u c t i v e to illustrate the a b o v e m e t h o d b y platting the curve of the g r o w t h of population d u r i n g the p a s t c e n t u r y in the U n i t e d States. Thus, on the diagram, it vgill

Nov., 1899.l

Graphical Presentation o f Stalz)tics.

387

be seen t h a t t h e axis of A" is horizontal, h a v i n g a scale of ten y e a r s to t inch, w hi l e t h a t of Yis vertical, h a v i n g a scale of Io,ooo,ooo to an inch. (See Plotc ]i) T a k i n g th e origin of c o g r d i n a t e s at O, and the d a t a from the c e n s u s r e p o r t s for each decade, b e g i n n i n g w i t h I79o, it will be seen t h a t t h e c u r v e at first d e p a r t s b u t sl i ght l y f r o m the h o r i z o n t a l axis, b u t g r a d u a l l y i n c r e a s e s its flexure upwards an d mu s t , in c o u r s e of time, a p p r o a c h a t a n g e n t to the vertical, b e c o m i n g , in short, an a s y m p t o t e - - t h i s , too, n o t w i t h s t a n d i n g t h e fact t h a t t h e ?ercenta2-c of increase deereases f r o m th e b e g i n n i n g to t he end of t he c e n t u r y . I t is t h u s shown at once t h a t t h e i n c r e m e n t is not a simple ratio, as in an a r i t h m e t i c a l p r o g r e s s i o n , b u t is b e i n g c o m p o u n d e d e a c h decade. H e n c e it is t h a t the p o p u l a t i o n u n d e r n o r m a l c o n d i t i o n s i n c r e a s e s m o r e r a p i d l y each decade, a l t h o u g h t h e p e r c e n t a g e m a y decrease. B e t w e e n i79 ° and I89O, for instance, t h e p o p u l a t i o n i n c r e a s e d 1,491 per cent., or I4"9t p e r cent. on an a v e r a g e each year, if c o m p u t e d as simple i n t e r e s t u p o n t he o r i g i n a l principal. Y e t t he r a t e e a ch d e c a d e b e f o r e t h e civil w a r was v a r i e d from 33 to 36 p e r cent., or o n l y 3½ to 3.~ p e r cent. per a n n u m . D u r i n g t h e w a r d e c a d e it was o n l y 23 per cent., increasi n g to 3o, an d a g a i n d e c r e a s i n g d u r i n g t he last d e c a d e to 25 p e r cent. T h e a c t u a l i n c r e m e n t s of p o p u l a t i o n , h o w e v e r , w ere n e a r l y I2,5oo,ooo in I88o-9o, as c o m p a r e d w i t h less t h a n 2,ooo,ooo in 18oo-Io, w h e n it was 36 p e r cent. of t h e whole. W h a t this r a p i d or c o m p o u n d i n c r e m e n t p o r t e n d s in t he n e a r f u t u r e for t he t r a d e p r e s t i g e a nd influence of this n a t i o n m a y be m o r e c l e a r l y u n d e r s t o o d by e x t e n d i n g t h e c u r v e at the c o n s t a n t r a t i o of t h a t of t he past decade, n a m e l y , 25 p er cent., f r o m w h i c h it a p p e a r s t h a t the population of I89O will h a v e d o u b l e d itself b y I92o, or in t h i r t y years, an d b y I95o, it will h a v e a g a i n n e a r l y d o u b l e d if t h e r a t io r e m a i n c o n s t a n t and t he g o v e r n m e n t stable and liberal. I t is n o t to be e x p e c t e d , h o w e v e r , t h a t the ratio will rem a i n c o n s t a n t for so l ong a period, b u t it will not be mat:y y e a r s b e f o r e o u r p o p u l a t i o n will r e a c h 15o,ooo,ooo souls.

388

[-[alt2t.:

[J. F. I.,

TO REPRESENT VARIATIONS. F l u c t u a t i o n s in values m a y also be i l l u s t r a t e d most forcibly by the use of the g r a p h i c a l m e t h o d . Take, for example, the relative v a l u e s of gold and silver d u r i n g the past c e n t u r y , a n d plat the curve of t h e i r ratios as per the d i a g r a m (see Plate I[), and it will reveal at a .glance the relation b e t w e e n the values of those commodities and legislation. T h e s t a b i l i t y of t h e i r relative values, from the close of the last c e n t u r y , t h r o u g h three wars and several financial crises, up to i873 , the y e a r of the passage of the d e m o n e t i z a t i o n act, is clearly shown, as well as the effects on the ratio due to s u b s e q u e n t acts. T h i s diagram, therefore, is a sharply-defined r e p r e s e n t a t i o n of the fact t h a t the v a l u e of silver as a m e d i u m of trade has depreciated rapidly, while t h a t of gold has as r a p i d l y appreciated, n e i t h e r b e i n g stable nor a " s t a n d a r d , " and, b u t for the p r o v i d e n t i a l d e v e l o p m e n t of large deposits and the r e d u c t i o n in the cost of e x t r a c t i o n of gold, the discrepancies in values would h a v e been m u c h greater. TO DETERMINE I"'RANCt[ISI~2S. C o l u m n s of figures are also c o n f u s i n g w h e n it is desired to d e t e r m i n e the values of a franchise from the t a b u l a t e d a n n u a l r e v e n u e s of a n y enterprise, and an a v e r a g e of the n e t r e t u r n s is also m i s l e a d i n g , since t h e profits will generally be less d u r i n g the earlier s t a g e s of d e v e l o p m e n t than in the later year. Here, again, the s u p e r i o r i t y of the graphical curves plays an i m p o r t a n t p a r t in r e v e a l i n g at once t h e actual condition of affairs. To i l l u s t r a t e : take the case of the M o n o n g a h e l a River I m p r o v e m e n t from 184o to 1896, and plat the receipts and expenses, as shown in Plate IlL T h e s e curves indicate r e a d i l y the f l u c t u a t i o n s due to i n t e r n a l and external causes far b e t t e r t h a n can be done by an array of figures. If a d a m breaks, the expense a c c o u n t goes up and the revenues d o w n : if e n l a r g e m e n t s or b e t t e r m e n t s are added, the opera t i n g expenses reach a h i g h e r plane, while the revenues are c o r r e s p o n d i n g l y increased. But p r o b a b l y the most practical use of this d i a g r a m is

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to d e t e r m i n e the value of the c o m p a n y ' s franchise by the e s t a b l i s h m e n t of the average financial gradient. T h i s is done, a p p r o x i m a t e l y at first, by a trial line with an a s s u m e d a n n u a l i n c r e m e n t , and by t h e n t a k i n g the algebraic s u m of the ordinates of t h e r e v e n u e above and below this line. It / / /

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this s u m is not equal to zero, it should be f u r t h e r a d j u s t e d until it is so, w h e n the true line becomes known. By capitalizing this i n c r e m e n t at the c u r r e n t rate per cent., the value of the franchise is readily found. T h e s e few i l l u s t r a t i o n s will serve, it is hoped, to estab-

39 o

Report c~f t/ze Cammisszbn on Water Suppl3,.

EJ. F. I.,

lish t h e g r e a t s u p e r i o r i t y of t h e l i n e a r m e t h o d s of r e p r e s e n t i n g s t a t i s t i c s as c o m p a r e d w i t h t h e p i c t o r i a l v o l u m e s a n d a r e a s , w h i c h a r e a l m o s t u s e l e s s to t h e s t a t i s t i c i a n as w e l l as to t h e p u b l i c . I f t h i s b r i e f a r t i c l e will a i d in elimin a t i n g so u n s a t i s f a c t o r y a m e t h o d , a n d in s u b s t i t u t i n g t h e r e f o r t h e g e n e r a l u s e of t h e m o r e scient'ific a n d a c c e p t e d p r o j e c t i o n s , it will h a v e s e r v e d its p u r p o s e .

SUMMARY o~' THE R E P O R T OF THE C O M M I S S I O N ON TIlE E X T E N S I O N AND I M P R O V E M E N T oF THE WATER SUPPLY o v THF C I T Y OF P H I L A D E L PHIA. Following is a s u m m a r y of t h e conclusions presented in the report of Messrs. Rudolph Hering, Samuel M. Gray and J o s e p h M. Wilson, the experts n a m e d by the Mayor of the city of Pfiiladelphia, w i t h t h e consent of the Councils, to examine and report upon the questions involved in providing for t h e u r g e n t requirements of the city in connection with its present and future water supply : " T h e deplorable condition of t h e city's water supply, which it is sought to remedy, is due to the pollution of its sources, to t h e lack of effective p u m p i n g machinery, and to the insufficient capacity of the d i s t r i b u t i n g system. " T h e question of first importance is the source of supply, a n d to this nearly all of our t h o u g h t and time has been devoted. " Most of the water is now obtained from t h e S c h u y l k i l l River, w i t h i n the city limits. Five p u m p i n g stations take from it about 2oo,ooo,ooo gallons daily. One p u m p i n g station is located on the tidal estuary of t h e Delaware River at L a r d n e r ' s Point, a n d supplies about I5,OOO,OOO gallons daily. " The Schuylkill water is being polluted at m a n y points from its source down to the city line. B e g i n n i n g with t h e mine waters, t h e coal dust and some sewage from t h e upper parts of t h e water shed, the pollution is increased below by t h e sewage of cities and villages situated a l o n g t h e river and its chief tributaries, by the m a n u f a c t u r i n g refuse a n d b y t h e surface water from agricultural districts, all of w h i c h r e n d e r t h e water sometimes turbid, unpalatable, impure and dangerous to health. " The Delaware water at L a r d n e r ' s P o i n t is less turbid after rains t h a n the S c h u y l k i l l water ; it is also softer and less polluted. Its flow is m a n y times larger. W h i l e this water is, therefore, now somewhat b e t t e r t h a n t h e Schuylkill water, the growth of the city, t h e newly-built or projected sewers above a n d below t h e intake, and the tidal oscillation of t h e water, t e n d to a continually increasing pollution also of t h e w a t e r t a k e n from t h e Delaware River. " It, therefore, becomes imperative e i t h e r to select a new source of supply or to improve the present one, so t h a t it will become t h o r o u g h l y satisfactory