A QUICK INTRODUCTION TO SOME STATISTICAL CONCEPTS

Transcription

1 A QUICK INTRODUCTION TO SOME STATISTICAL CONCEPTS Mean (average) page 2 Varance page 7 Standard devaton page Bvarate data page 3 Covarance page 4 Correlaton page 9 Regresson page 2 Resduals page 24 Addtonal formulas page 25 Regresson effect page 27 Ths document was prepared by Professor Gary Smon wth the advce (and consent) of Professor Wllam Slber, Stern School of Busness. If you have comments or suggestons, please send them to ether gsmon@stern.nyu.edu or wslber@stern.nyu.edu. Release date 26 AUGUST 2002

2 THE MEAN, OR AVERAGE Ths document wll ntroduce a number of statstcal concepts, and perhaps some of these may be very new to you. Statstcal topcs can be confusng because dentcal subject matter can be descrbed n very dfferent terms. The very same statstcal concept can be descrbed n several ways:. Data, as numbers. 2. Data, represented algebracally. 2m. Data, represented algebracally, allowng multple values. 3. Conceptually as random varables wth probabltes. A lst of data, as n ponts through 3, mght be descrbed as a varable. Each column of a data spreadsheet could be called a varable. In tem 3, we do not necessarly have data, and we conceptualze the dea as a random varable. Let s llustrate the notons frst for the concept of the average of a set of values. The average s also called the mean.. Consder the lst of values 48, 46, 54, 5, 53. The average (or mean) of ths lst s found as = 252 = Let n be the number of tems n a lst. Represent the lst as x, x 2,, x n. The three dots smply ndcate that we re omttng some of the values. Note that we re usng x as the symbol for the lst tems. You ll frequently see x as the generc th element of the lst. You can thnk of the symbol as a counter or as an ndex. Some people wll descrbe the lst as { x ; =,, n }. You can read ths as x-sub-, as runs from to n. x + x xn The average of ths lst s n n n. = ( x x x ) 2 Snce we ll be addng lsts of numbers rather frequently, t helps to create a smple notaton for ths concept. We use the summaton notaton n x to represent the sum of the x s, usng the ndex to = enumerate from the startng value = to the endng value = n. Then we have n x = x + x x n and the average can be = 2

3 x n = wrtten as x =. The symbol s nothng but a n = n n countng convenence. You should note that n x j = j= n xu. u= n x = = In nearly every case you ll encounter, the entre lst of n values wll be added, and t s burdensome to keep the notaton above and below the Σ sgn. You can then use x as a smpler notaton for x + x x n. Agan, the symbol s a mere countng convenence, and so x = x j = xu. The symbol x, whch we read as x bar, s the most common notaton for the average of the x s. Thus x = x. n 2m. It can happen that the lst of value x, x 2,, x n wll nvolve duplcatons. Suppose that there are k dfferent values and that we name them as v, v 2,, v k. Let s say that v occurs n tmes, v 2 occurs n 2 tmes, and so on. The data could then be reorganzed to look lke ths: Value Multplcty v n v 2 n 2 v 3 n v k TOTAL n k n Now x = nv. You wll also see ths as x = n nv n. The formulas n tem 2 above are stll correct, even f the lst nvolves duplcatons. 3. There are tmes n whch we consder problems hypothetcally, rather than wth numbers (as n tem ) or wth algebra symbols (as n tems 2 and 2m). In the hypothetcal form, we ll consder X as the phenomenon under dscusson, and we ll gve X the techncal name random varable. In ths style of thnkng, X s endowed wth randomness. We should wrte 3

4 X n upper case. We may have no data yet, but we can stll dscuss the possble values for X. Let s suppose that x s a possble value, and that ts assocated probablty s p. In such a case, the average of X s p x. When dealng wth random varables (rather than wth numbers or wth algebra symbols), the mean s generally denoted µ or perhaps µ X. We also say that µ s the expected value of X. The table below concerns the numbers of employees of an offce callng n sck on any partcular mornng. x : p : The probabltes are lkely based on past observatons, but they could as well be someone s hypothetcal conjectures. We can apply these probabltes to tomorrow s sck calls, for whch we certanly do not yet have data. We can use X to represent the phenomenon, and n ths case we d calculate the expected value of X as µ = = =.02 That s, we expect.02 employees to call n sck tomorrow. Below, bordered by ***, s a techncal note whch you can skp at frst readng. ************************************************************************ Ths descrpton works perfectly for random varables whose possble values can be dentfed and solated; such random varables are called dscrete. Other random varables are obtaned by measurement processes and have uncountably many values. These random varables are called contnuous and a specal mathematcal framework s needed to deal wth them. ************************************************************************ We wll not dscuss every sngle dea n all four forms noted above. However you should be aware that t s possble to descrbe statstcal notons n dfferent styles. 4

5 EXAMPLE: Over ten busness days, the number of shares traded of Mraco were these: Mraco s, of course, a very small company. The average number of shares per day s x = 430. Ths uses = 4,300. There are some duplcatons of values n ths lst. The value 200 occurs three tmes an 400 occurs twce. You can use these duplcatons to rearrange your computaton f you wsh. Ths rearranged form would be = 4,300 For ths stuaton, t s smple enough to add the orgnal lst of ten values wthout searchng for duplcates. EXAMPLE: The number of medcal emergency calls comng per day to the Eastsde Ambulance Servce s random, and we know (ether from past experence or as a hypothetcal suggeston) that The probablty of 0 calls s 0.0. The probablty of call s 0.5. The probablty of 2 calls s The probablty of 3 calls s The probablty of 4 calls s 0.0. The probablty of 5 calls s The mean number of calls per day s = = 2.35 We can conceptualze the number of calls on any sngle day as a random varable X. Ths calculaton shows that µ X =

6 EXAMPLE: A sample was taken of 20 suburban famles, and each sampled famly was asked how many cars t owned. The data were these: You can get x by smply addng these numbers and then dvdng by 20. However, the work s easer f you organze t lke so: famly owned no cars. 5 famles owned one car. 3 famles owned two cars. famly owned three cars. The total value s then = = 34 Then x = =.7. 6

7 THE VARIANCE Another useful statstcal summary s the varance. We wll use the varance as an ntermedary to get to the calculaton known as the standard devaton. The standard devaton s smply the square root of the varance, so the connecton between the varance and the standard devaton s very smple. If you are dealng wth the varance concept for the frst tme, you ll be dstressed by the fact that there are several dfferent formulas that lead to the same result. If you are usng a computer to do your arthmetc, you wll not generally be concerned about the detals. We strongly recommend the use of a computer, because arthmetc wth a calculator s error-prone. On the other hand, just to make sure the computer s dong t rght, as well as to make sure you understand the formula, you should do a sample calculaton by hand. Wth a modest amount of data, the varance can be calculated wthout a computer. However, the choce of formula requres a judgment call based on the appearance of the nformaton. A good choce wll lead to the answer more quckly, and wth lower probablty of error, than a bad computatonal choce. In formula terms, we can descrbe the process of fndng the varance of the lst x, x 2,, x n.. Start by fndng x, the average. 2. Fnd the n dfferences x - x, x 2 - x, x 3 - x,, x n - x. The dfferences are also called devatons. As a computatonal check, ths lst must sum to zero. 3. Square these n dfferences to produce the values (x - x ) 2, (x 2 - x ) 2, (x 3 - x ) 2,, (x n - x ) Sum these values to get (x - x ) 2 + (x 2 - x ) 2 + (x 3 - x ) (x n - x ) Dvde ths sum by n -, the sample sze, less. The usual symbol for the varance s s 2 when the computaton starts from data. (When we work from random varables, the symbol s σ 2.) The varance can be done through the formula whch summarzes the steps above. n s 2 = ( x x ) 2 It s a lttle puzzlng that ths s dvded by n -, rather than by n. There s a perfectly good explanaton for ths choce, but we re not yet ready for t. 7

8 EXAMPLE: Consder the lst of values 48, 46, 54, 5, 53. Ths was consdered prevously, and we found the average x = The lst of dfferences from the average s ths: -2.4, -4.4, 3.6, 0.6, 2.6 The value -2.4 was obtaned as Ths lst of fve values sums to zero, as t must. Next we square the values to produce the followng: 5.76, 9.36, 2.96, 0.36, 6.76 The value 5.76 was obtaned as (-2.4) 2 = (-2.4) (-2.4). Next we sum the lst of squares to get the total Fnally, we compute s 2 = = = You mght fnd t easer to lay out the arthmetc n a table. Values x Devatons x - x Squares of Devatons TOTAL Ths example should make t clear why we recommend that varances be calculated wth a computer! There are other computatonal formulas for the varance, but the technque gven here wll suffce for hand calculatons wth short lsts. Varances can also be found for random varables wth probabltes. For example, f the value x has assocated probablty p, then the expected value s p x, and t s usually denoted by µ. Ths was dscussed above. The varance would then be defned as ( ) 2 σ 2 = p x µ = p x µ 2 2 8

9 You mght note that a dfferent symbol, σ 2 rather than s 2, s used for ths stuaton Also, the forms p ( x µ ) and p x µ represent two dfferent computatonal strateges. As an example of the varance of a random varable, consder ths stuaton regardng values for the prce at whch a certan home wll sell. The prces are n thousands of dollars. The expected value here s Prce Probablty µ = = = Ths example llustrates well what we mean by a random varable. There s no data (yet). The home wll sell only one tme, and the entre process wll come down to a sngle number. Before the home sells, however, we conceptualze the random process whch creates the prce. Ths has been done by suggestng possble prces and correspondng probabltes. (Ths s of course a hypothetcal probablty mechansm, perhaps based on a suggeston from a real estate expert.) The random varable wll get some symbol, lkely X, and we makng statements of the form the probablty that X wll take the value 200 s The mean of ths random varable s µ = Ths s not among the lsted possble values (200, 225, 250, 275), but the statstcal analyst s not concerned. We can next fnd the varance of the random varable X though ths calculaton: σ 2 = 0.20 ( ) ( ) ( ) ( ) 2 = =

10 We could also work through p x 2, fndng 2 p x = = 8, , , , = 54, Ths would allow us to compute σ 2 2 = p x µ 2 = 54, = 54, , = The formulas p ( x µ wll produce the same answer ) and p x µ 0

11 THE STANDARD DEVIATION If the data values have unts of measurements, perhaps dollars, then the varance wll have unts that are the squares of the unts of the data. Thus, f x, x 2,, x n gves a lst of values n dollars, then the varance s 2 wll have unts of dollars 2. Ths would be read as dollars, squared or as square dollars. Ths s all very reasonable from a mathematcal or physcal pont of vew, but most users are not comfortable wth the concept of square dollars. Accordngly, t s common to take the square root to convert back to the orgnal unts. The standard devaton s the square root of the varance, and t wll have the same unts of measurements as the orgnal values. Thus, the standard devaton of a lst of dollar values wll also be n dollar unts. If you ve calculated a varance, then you get the standard devaton smply as the square root. There are no other added computatonal complextes assocated wth the standard devaton. For data the standard devaton s usually wrtten as s, and for random varables the standard devaton s usually wrtten as σ. It s farly easy to make quck judgments about standard devatons because of the followng emprcal rule that frequently appled to observed data. We ll state t twce, once for data and once for random varables. If x, x 2,, x n s a set of data values (such as annual rates of return on n mutual funds), and f x s the average and s the standard devaton, then About 2 3 of the values are n the nterval from x - s to x + s. About 95% of the values are n the nterval from x - 2s to x + 2s. If X denotes a random varable wth mean µ and wth standard devaton σ, then The probablty s about 2 3 that X wll take a value n the nterval from µ - σ to µ + σ. The probablty s about 0.95 that X wll take a value n the nterval from µ - 2σ to µ + 2σ.

12 EXAMPLE: Consder all Amercan males between the ages of 2 and 30. What would be the standard devaton of ther weghts? We certanly cannot answer ths queston precsely wthout data, but t s easy to make a plausble guess at the standard devaton s. Perhaps these men have an average weght of 65 pounds. It would be belevable that about 2 3 of these men have weghts between = 40 pounds and = 90 pounds, so that 25 pounds s a plausble guess at the standard devaton. We d also be reasonably happy wth the statement that about 95% have weghts between = 5 pounds and = 25 pounds. EXAMPLE: Consder data on daly orders for bagged refned flour from Carlborg Mlls n Branerd, Mnnesota. What would be a reasonable value for the standard devaton of the daly amounts ordered? The ablty to guess standard devatons depends on some famlarty wth the concept. It s easy to formulate a guess for the weghts of Amercan males. Snce we have no nformaton about ths flour mll and no experence wth flour data, we should probably not make a guess here. 2

13 BIVARIATE DATA The mean (or average) and the standard devaton are very common statstcal summares. These two smple quanttes can tell us qute a lot about a set of data or about a random varable. The notons get even more nterestng when we descrbe two sets of data (or two random varables) at the same tme. The word bvarate suggests that we are dealng wth two varables. Consder ths set of data on ten recently-sold homes: Area Prce All homes were located n the same suburban subdvson. The frst column gves the ndoor area n square feet, and the second column gves the prce n dollars. Spreadsheets are now the preferred way to exhbt data lke ths. You mght however also see ths n coordnate form: (800, 82400), (362, 72800), (89, 90000), (594, 67600), (605, 92500), (274, ), (290, 84400), (2393, ), (654, 7600), (2209, 22300) We can fnd the followng nformaton: Varable Average Standard devaton Area, Prce 95,870 26,90 Ths small table gves us qute a good mpresson about the data. We mght, however, lke to see somethng that tells us how the varables act together. Do the larger homes have hgher prces? 3

14 A smple frst step conssts of makng a scatterplot: Ths pcture has ten ponts, one for each of the homes. Yes, there s defntely the mpresson that the larger homes have hgher prces. COVARIANCE There are several quanttes that quantfy the relatonshp between the two varables. Let s use y as a symbol for Prce and let s use x as a symbol for Area. The quantty known as the sample covarance between x and y s gven as n s xy = ( x x)( y y) n = For the data on home prces and floor areas, ths calculaton s structured as s xy = { (,800,937 )( 82, ,870 ) 0 ( )( +,362,937 72,800 95, ( )( + 2, 209, ,00 95,870 } ) ) 4

15 = { 37-3, ( ) ( ) ( 575) ( 23, 070) +... ( ) ( ) ,230 } Although you should do a sample calculaton by hand to renforce your understandng of the concept, actual calculatons of ths type should certanly be done by computer! The value above s 0,257,646. Ths calculaton has the form sample sze - s xy = ( x-dfference from mean)( y-dfference from mean) so that the unts must come from the products. Snce x s measured n ft 2 (square feet) and y s measured n dollars, the covarance wll have unts of ft 2 -$. We cannot even provde good advce about how to pronounce such a thng, but you could try square-foot dollars or dollars-square-foot or maybe foot-foot-dollars. The unts are annoyng, and fortunately there s somethng we wll do about ths soon. Each summand of the covarance s xy s a product of two factors. Ether or both of the factors can be negatve, so some of the summands are postve, and some are negatve. A pcture can show what ths means. 5

16 Frst of all, let s note the pont of averages (937, 95870) and put t on our plot. It s shown here as the open crcle Prce Area

17 The contrbuton to the covarance of any sngle data pont s the area of a rectangle wth one corner at the pont of averages and the dagonally opposte corner at the data pont. The llustraton here s of a postve contrbuton to the covarance, as both the area and the prce are above average Prce Area

18 Here s another postve contrbuton to the covarance, as both the area and the prce are below average: Prce Area

19 Contrbutons to the covarance are negatve when one of the varables s above average whle the other s below average. For the home shown here, the area s above average, but the prce s below average Prce Area 2500 CORRELATION Data analysts frequently need to deal wth the relatonshp between two varables. In the example above, the covarance served ths purpose, but t had unts that are hard to nterpret. We can make ths much easer to understand be scalng the covarance by dvdng by the standard devatons of the two varables. The resultng calculaton s called the correlaton, or sometmes the correlaton coeffcent. The data-based correlaton s noted as r, and t s calculated as r = s x s xy s y Covarance of xand y = ( Standard devaton of x ) ( Standard devaton of y ) When a correlaton s used to descrbe two random varables (as opposed to ts calculaton from data), t s denoted as ρ, the Greek letter rho. 9

20 For our data on house szes and prces, ths s r = 0,257,646 26, The calculaton of r must produce a value between - and +; values outsde ths nterval are arthmetc errors. ************************************************************************ Wth some algebrac work, we can show that r = x x y y n sx sy so that r s (almost) an average of the product of scaled dfferences from the means. These correspond to the scaled areas of the rectangles assocated wth the covarances. ************************************************************************ Correlatons that are close to + represent strong postve relatonshps. The correlaton found above as tells us that large homes tend to have hgh prces, and small homes tend to have low prces. A correlaton wll be + exactly f and only f the data ponts all le on a lne wth postve slope. Postve slope refers to a lne that rses from the left end of the scatter plot to the rght end. Ths can happen only wth a perfect accountng relatonshp between x and y, and t thus rarely encountered wth real data. Correlatons that are close to - represent strong negatve relatonshps. For nstance, the prces of used 998 Toyota Corollas have a strong negatve relatonshp wth the mleage on the odometer. That s, hgh prces are generally found on cars wth low mleage, and low prces are found on cars wth hgh mleage. A correlaton wll be - exactly f and only f the data ponts all le on a lne wth negatve slope, referrng to a lne that drops from the left end of the scatter plot to the rght end. Ths requres a perfect accountng relatonshp between x and y. Correlatons that are close to zero represent weak relatonshps. We ve used the words close to, strong, and weak very loosely. Our feelngs about correlatons wll vary accordng to the problem. Sometmes we work wth real estate prces, sometmes wth bond nterest rates, and sometmes wth corporate profts. 20

21 REGRESSION In most statstcal work wth two varables, we move beyond the correlaton concept to explore the related noton of regresson. In the regresson context we thnk of one of the varables as a possble nfluence on the other. The correlaton concept treats the two varables symmetrcally, but regresson does not. In the regresson context, the varable that s dong the (potental) nfluencng s called the ndependent varable. The varable that s (potentally) nfluenced s the dependent varable. Thus, the ndependent varable s a (potental) nfluence on the dependent varable. We are beng very, very careful wth the word potental. Consder a data base of manufacturng frms n whch we are concerned wth year 2000 R&D (research and development) spendng and year 200 profts. Certanly we wll consder year 2000 R&D as the ndependent varable and year 200 profts as the dependent varable, as we expect that year 2000 R&D wll nfluence year 200 profts. The data, however, mght not support our expectatons. Ths s exactly why we wll thnk of year 2000 R&D as a potental nfluence. We have many reasons for dong the regresson. We wll certanly ask whether the potental nfluence turns out to be an actual nfluence. We wll want to quantfy the strength of the relatonshp. A very mportant use of regresson wll be n predcton. If we decde that the relatonshp between year 2000 R&D and year 200 profts s strong, we would use that relatonshp n predctng future profts based on pror R&D spendng. The word cause has been scrupulously avoded. We can descrbe somethng as a cause only f the data has been collected as part of a controlled randomzed experment. Economc and fnancal data are observatonal and not collected n an expermental framework. As much as we mght lke to say that spendng on R&D causes future proft, we smply do not have the logcal bass for such a clam. In the scatterplot that accompanes regresson work, t s customary to place the dependent varable on the vertcal axs and the ndependent varable on the horzontal axs. In the pctures above regardng home prce and floor area, the prce was the vertcal axs. We certanly want to thnk that area s a potental nfluence on prce. If the varable labels are x and y, t s customary to use y for the dependent varable and place t on the vertcal axs. 2

22 There s an nterestng excepton to ths, and you should be aware of t. Economsts frequently use P versus Q (prce versus quantty) graphs, and they place P on the vertcal axs. Ths layout would agree wth custom perhaps for thngs lke agrcultural commodtes, where Q precedes, and s a potental nfluence on, P. In other contexts, economsts consder supplers who set P, so that prce s a potental nfluence on Q, the quantty that sells; n such a case they have the dependent varable on the horzontal axs. In ts smplest form, regresson s executed as a straght-lne relatonshp, and we call the process lnear regresson. If the dependent varable s y and the ndependent varable s x, then the ftted regresson lne wll have the form y = b 0 + b x In ths form b 0 (the ntercept) and b (the slope) wll be numbers computed from the data. In our exploraton of the real estate prces, the ftted lne wll be Prce = b 0 + b Area The numerc verson, obtaned wth the help of a computer, s Prce = 88, Area You mght feel that the precson s a bt pretentous, and perhaps you d lke to report Prce = 88, Area 22

23 Here s the scatterplot, ths tme wth the ftted regresson lne supermposed: Prce = Area Prce Area 2500 We have a strong nterest n the estmated regresson slope, Ths represents the average movement n prce per unt movement n area. Specfcally, ths s to be nterpreted as $55.49 per square foot of area. Ths s the estmated margnal effect on prce of one addtonal square foot of area. The estmated ntercept, here 88,400, can be nterpreted n some problems, but probably not here. Some mght try to say that ths s the value of a house wth no area, meanng perhaps an empty buldng lot. However, the data base ncluded no buldng lots, and we should refran from such a speculaton. An mmedate use of ths regresson s for predcton. If a home wth 2,500 square feet of area comes onto ths market, we would predct that t would sell for 88, ,500 = 88, ,725 = 227,25 The home mght sell for more than $227,25 or t mght sell for less. We are, after all, just makng a predcton. 23

24 RESIDUALS Suppose that we use ths predcton method for a home that was part of our orgnal data set. The home lsted as the frst data pont had an area of,800 square feet. The ftted prce would be 88, ,800 = 88, ,882 = 88,282 We are gong to call ths a ftted prce rather than a predcted prce because ths home s n our data set and we know the prce at whch t sold. Indeed, ths prce was $82,400. The dfference between the actual observed prce (here $82,400) and the ftted prce ($88,282) s called the resdual. For ths home the resdual s 82,400-88,282 = -5,882 Relatve to ths regresson, ths partcular home sold for $5,882 less than t should have. Ths s not a mstake, t s not an error, t s not the result of neffectve negotaton. The data ponts do not le neatly on a perfect lne, so that some homes wll have postve resduals and some wll have negatve resduals. The table below lsts the resduals for all the homes n the data set. The values were obtaned by computer; the dfference between our -5,882 and the computer s -5,884.2 s related only to round-off ssues. Area Prce Resdual The resduals wll sum to zero; ths s a consequence of the formulas used to calculate the regresson. We can see that the ffth home sold for about $5,000 more than the ftted lne would suggest. The seventh home sold for about $25,000 less than the ftted lne would suggest. The resduals are non-zero smply because the ponts do not all fall on a straght lne. It s temptng to say that the ffth home was over-valued by the purchasers, but we should 24

25 resst ths nterpretaton. After all, the regresson work utlzed only floor area, and there are many other factors determnng prce. It wll not surprse you to learn that regresson models are used to relate stock prces to sets of ndependent varables. Companes wth negatve resduals mght be descrbed as undervalued, and therefore attractve purchases, but we have to be very careful about such judgments. After all, the regresson work mght be mssng varables that are relevant to the values of the companes. Of course, these companes mght really be undervalued, and certanly any company that produces a large negatve resdual should be examned closely! The work above has been descrbed as a ftted regresson lne, based on data. There s also a true regresson lne, nvolvng a model equaton wth random varables and a number of assumptons. We wll not here go nto the formalsm of the model equaton, but t s mportant to realze that our work wth data produces a ftted (or estmated) lne and not the guaranteed truth. Suppose, hypothetcally, that all the resduals turned out to be zero. Ths would ndcate that the dependent varable y could be predcted perfectly from the ndependent varable x. Ths means, of course, that there s a perfect accountng relatonshp between x and y and further that the correlaton between x and y would be + or - (dependng on the slope); see the ndented comments on page 20. Indeed, the resduals are related to the correlaton coeffcent. Ths s a complcated story, as ths s the equaton whch relates them: r 2 = 2 ( resdual) ( n ) Varance( y) The tells us that a set of large resduals produces a correlaton coeffcent close to zero. ADDITIONAL FORMULAS The calculatons to produce a ftted regresson lne are ntense, and we recommend that computer software be used, although, once agan, a sample hand calculaton mght mprove your understandng of what s gong on. There are nonetheless some nterestng quanttatve relatonshps that we can explore. Here s one: b = estmated regresson slope = Sample covarance of x and y Sample varance of x s xy = 2 sx 25

26 = n n ( x x)( y y) ( x x) 2 = s r s y x = Correlaton Standard devaton of Standard devaton of x y There s a lttle bt of algebra work behnd ths. Snce r = s xy sx s, t y sxy must happen that s xy = r s x s y. If we substtute ths nto b = 2 s, rsx s we get b = 2 s x y = s y r. s b 0 = estmated regresson ntercept = y - b x x x There s one further addtonal nterestng way to present a regresson lne. The ftted lne s y = b 0 + b x We can make the substtutons b 0 = y - b x and b = s r s y x to get ths: y s y y = r x s x x In ths expresson, x and y represent the varables (meanng the names of the horzontal and vertcal axes), whle the remanng tems ( x, s x, y, s y, r) are quanttes computed from the data. 26

27 THE REGRESSION EFFECT Let s consder the above form of the ftted regresson equaton wth regard to a predcton. We showed prevously that a home of 2,500 square feet would be predcted to sell for $227,25. We obtaned ths value as 88, ,500 = 88, ,725 = 227,25 but we could also obtan t by solvng for y n y 95,870 26,90 = 2,500, ,500,937 Observe that on the rght sde the fracton.3 shows that ths home s standard devatons above average n sze. One would thnk that the predcted prce would then be.3 standard devaton above average. However, the predcted prce s 227,25 95,870 only.6 standard devatons above average n prce. Indeed, we 26,90 see that wthn roundng The correlaton coeffcent n ths last result makes the predcted prce relatvely closer to average than was the floor area used to make the predcton! The floor area s somewhat hgh, and we predct the prce to be hgh, but not qute as hgh as the floor area. Ths partcular phenomenon s known as the regresson effect. It s a fact of statstcal lfe, and t tends to turn up over and over. It s not always recognzed. Suppose that a man s 6 9 tall, and suppose that he has a son. What heght would you predct that ths son would grow to? Most people would predct that he would be tall, but t would be qute unusual for hm to be as tall as hs father. Perhaps a reasonable predcton s 6 5. We are expectng the son to regress back to medocrty, meanng that we expect hm to revert to average values. Ths s of course a predcton on average. The son could well be even taller than hs father, but ths s unlkely. Ths regresson effect apples also at the other end of the heght scale. If the father s 5, then hs son s lkely to be short also, but the son wll probably be taller than hs father. The regresson effect s strongest when far away from the center. For fathers of average heght, the probablty s about 50% that the sons wll be taller and about 50% that the sons wll be shorter. 27

28 The regresson effect s everywhere. People found to have hgh blood pressure at a mass screenng (say at an employer s health far) wll have, on average, lower blood pressures the next tme a readng s taken. Mutual fund managers who have exceptonally good performances n year T wll be found to have not-so-great performances, on average, n year T +. Professonal athletes who have great performances n year T wll have less great performances, on average, n year T +. 28