newton’s first law of migration: the gravity model · having introduced these key migration...

Newton’s First Law of Migration:The Gravity Model

� INTRODUCTION

Places are connected with one another at the local, regional, and global scales throughsystems of spatial interactions. These interactions involve movements of ideas,information, money, products, and people. As the interconnectedness of the worldaccelerates, the number and intensity of these interactions increases. Millions ofmessages are sent daily across the Internet. Hugely popular American television showsand movies disseminate American culture overseas, and some foreign films and pro-gramming find an audience in the United States and Canada. Commodities and cap-ital freely move across national boundaries. And each year, more people becomeinternational migrants. Currently, 175 million people reside outside of the countryof their birth. The number of migrants in the world has more than doubled since1975, with most living in Europe (56 million), Asia (50 million), and North America(41 million).

Migration is defined as a permanent change in residence to outside of one’scommunity of origin. Conceptually, someone who moves to a new home within hisor her community but does not have to change his or her place of work, shop innew stores, find new doctors, and establish new friendships is considered a localmover but not a migrant. Offically, migration is defined as crossing an administra-tive boundary, such as between counties or states. Tourists, temporary residents,and seasonal workers may play important roles in some places, but they are not con-sidered migrants if they don’t intend to stay at least one year. There is, of course,some gray area regarding how far one has to move and how long one has to stay tobe considered a migrant, but that is just one of the factors that makes the study ofmigration so fascinating.

Migration can occur at many spatial scales, including rural-to-urban movementsfrom hinterlands to cities (Figure 4.1), urban-to-urban moves between regions, andglobal migration between countries. The size, composition, and spatial organizationof migration flows tell us a great deal about the places involved. In that people tendto move from less desirable places toward more desirable places, the system of migra-tion flows provides clues about how places stack up relative to one another. Placedesirability can result from economic factors such as job availability, high wages,and affordable housing and from noneconomic considerations such as a favorable

�CHAPTER4

© 2004 John Wiley & Sons, Inc. 85

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climate, clean air, low crime rates, nearness to friends and relatives, and the absenceof war and environmental disaster.

Migration patterns are the result of millions of individual and household deci-sions about where to live. For those who move, a combination of push and pull fac-tors triggers the decision to move. Push factors can include exorbitant housing costs,growing gridlock, rising crime rates, skyrocketing tax rates, a poor climate, and thelack of a satisfying, well-paying job. Pull factors can include the promise of a higherpaying job, a pleasant physical setting, the availability of affordable housing, a desir-able climate, or the lure of nearby family members. Sometimes what is a push forsome people is a pull for others. Take, for example, closeness to family. Many believethat living near family members provides a valuable and comforting social supportsystem; others see it as claustrophobic, stifling their independence. Similarly, a cli-mate that is too hot and a push for some can be just right and a pull for others. Highschool taxes can be perceived as desirable for a young family with children but asonerous for a childless single or an elderly couple. How we perceive various placecharacteristics and how much weight we attach to them is very much a personalmatter.

The effects of migration on places of origin and destination are influenced bya process called migration selectivity. Certain individuals are more likely to migratebased on their personal characteristics, including age, education, and other sociode-mographic characteristics. Age is the most important factor in influencing whethersomeone is a migrant or not (Figure 4.2). People are most prone to move duringtheir early adult years between the ages of 18 and 30. The average individual makesapproximately one-half of his or her 12 lifetime moves by age 25. During these youngadult ages, people leave their parents’ home to attend school, join the military ortake a job, leave college to find employment or change jobs, marry, and begin fam-ilies. All these life-course events are usually associated with changes in residence.Movement rates are also high among young children who typically have parents intheir 20s.

A second migration selectivity factor is education. People with higher levels ofeducation are more likely to make long-distance moves. Getting a college educa-tion often means moving to a new city and then returning or moving again upongraduation. In addition, education exposes us to new ideas and people from otherplaces. It also qualifies us for, and provides information about, a wide variety of jobsin many different geographical areas. The selectivity of migration alters the population

86 � Chapter 4. Newton‘s First Law of Migration: The Gravity Model

© 2004 John Wiley & Sons, Inc.

Figure 4.1 Rural-to-urban migrants at a train station in China.

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characteristics of origin and destination places. As a general rule, places experiencingnet out-migration lose a disproportionate share of their young, well-educated res-idents while areas of net in-migration gain such individuals. There is a doublewhammy for places experiencing out-migration. They lose not only population num-bers but also their youngest and best-educated residents. Especially troublesomeis that this process can snowball, making origins less attractive to future migrantsand less capable of retaining their current residents.

Migration also influences places through the tendency to form migrationstreams. People do not move randomly across the landscape. They move in well-defined channels from specific origins to particular destinations. Migration streamsoften occur between nearby places because it is cheaper, quicker, and easier for peo-ple to move short distances. The tendency for migration, or any other form of spa-tial interaction, to decrease with distance is called distance decay (Figure 4.3).

Migration streams result from information flows between origins and destina-tions. Letters, telephone calls, and return visits from earlier migrants communicateopportunity at the potential destination. These pioneer migrants assist newcomersin finding a place to live, getting a new job, and adjusting to a new community.Information about places also comes from newspapers, television, magazines, busi-ness contacts, and personal travel. However, most people know surprisingly littleabout the range of potential places to live. Their migration decisions are based ona narrow set of options dictated by first- or secondhand information about what itis like to live there.

Wherever a migration stream develops, a migration counterstream of peo-ple moving in the opposite direction occurs. Not everyone who migrates intends toremain permanently at the place of destination, for example, college students. Othersare unhappy with the circumstances of their move, their personal situation changes,or they are military or corporate personnel who are reassigned. Divorce might cre-ate a return migration. An elderly couple who moved from the North to a Sunbeltretirement community while in their 60s could return home when they are in their80s in fragile health and in need of family support. Also important to understand-

Introduction � 87


Age Selectivity of Migration:

Migration Rates 1999–2000

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4 9 14 19 24 29 34 39 44 49 54 59 64 69 74 79 84 89

Age

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Figure 4.2 Migration rates are highest for young adults in their early 20s.

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ing migration counterstreams is the presence of information linking the two places.Once, for whatever reason, the channels of communication are opened and inter-personal relationships are built, movement will occur in both directions, althoughnot necessarily at the same rate.

Migration streams involving faraway places get started in a variety of ways. Largestreams connecting New York and New Jersey with Florida arose after World WarII with the migration of retirees and snowballed as contact between the two areasgrew. Historically strong ties between California and Midwestern states began aslabor-force migration. The stream between Oklahoma and California originated withDepression-era Dustbowl migrants. The experience of Dustbowl migrants, eloquentlyportrayed in Steinbeck’s The Grapes of Wrath, changed conventional wisdom aboutmigrants from hardy pioneers in search of opportunity to disadvantaged families try-ing to survive (Figure 4.4).



Distance0

0Le

vel o

f Int

erac

tion

Figure 4.3 Distance decay curve showing decreasing interaction as distance increases.

Figure 4.4 A “Dustbowl” migrant family from Oklahoma recently arrived in California tojoin the harvest, November 1936.

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The most geographically focused migration streams in the United States todayare among newcomers to the country who are strongly attracted to immigrant com-munities or enclaves (see Chapter 12 for a fuller discussion of enclaves). Immigrantcommunities offer familiar language, food, music, and religious institutions. Theyalso help newcomers to locate all-important housing and jobs. Many immigrants findwork in businesses owned by their compatriots, or they establish their own smallbusinesses, providing goods and services to immigrant niche markets.

The internal migration streams of Cuban- and Mexican-born immigrants demon-strate the different conditions under which they migrated to the United States (Figure4.5). Mexicans are immigrants who, for the most part, moved voluntarily to the UnitedStates, largely for economic reasons. Cubans are refugees, meaning that they wereforced by political crisis to flee their homeland. Even today, 20 or 30 years aftertheir move to the United States, some Cubans continue to see themselves more asexiles hoping to return to Cuba than as immigrants seeking a permanent home andfuture in the United States. South Florida has emerged as the surrogate homelandfor Cuban émigrés. Much political activity is organized around the overthrow ofCastro’s regime, and high value is placed on preserving cubanidad or “Cubanness.”Cuban migration streams are strongly directed toward Florida; the counterstreamsare extremely weak. These uneven flows are redistributing the Cuban populationin favor of Florida. The Mexican migration system is quite different. Six of the 10largest interstate flows interconnect the three largest concentrations of Mexican-born population in California, Texas, and Illinois. Unlike the Cuban flows, the Mexicanflows are self-compensating. Streams and counterstreams are about equal in size,and, thus, very little population redistribution occurs as a result of Mexican inter-nal migration.

Having introduced these key migration concepts, we now shift our focus to themore applied task of predicting migration flows. Geographers use a mathematicalformula known as the gravity model because it resembles Isaac Newton’s formulafor the gravitational attraction between any two celestial masses, which you mighthave learned in physics class. Newton’s law has been adapted to social science researchfor the purpose of estimating the spatial interaction or movement between any twoplaces. Spatial interaction can take the form of trade, transportation, communica-tion, commuting, shopping, or, in the case of this chapter, migration.

The following example will help you to understand the idea behind the gravitymodel. Figure 4.6 shows the populations of several states and their distance fromCalifornia. Would you expect California to attract more migrants from North Carolinaor from South Carolina? Their distances are about equal, but North Carolina hastwice as many inhabitants. All other things being equal, you’d probably expect abouttwo times as many migrants from North Carolina because there are two times morepotential movers. Next, would you expect more migration to California from Arizonaor from Maryland? Their populations are both around 5.3 million, but Maryland is5 times farther away. Surely more people will move from Arizona, but probably not5 times more because each additional mile matters less and less. As shown in Figure4.3, distance decay tends to be nonlinear: steep at first but gradually flattening out.The first 100 miles reduces migration substantially, the second 100 miles less so,and the twentieth 100 miles (i.e., the difference between 1,900 miles and 2,000 miles)hardly matters to people at all.

In the gravity model formula, as in the California example in Figure 4.6, pop-ulation size and distance are used to explain the interaction flow, Iij, between origin

Introduction � 89


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10,000

4,0001,000

Number of Migrants,1985-1990

8,500

4,000

2,700

Number of Migrants,1985-1990

10 Largest Domestic Migration Streams of Persons Born in Mexico

10 Largest Domestic Migration Streams of Persons Born in Cuba

Figure 4.5 Examples of migration streams for two ethnic groups. Source: 1990 Census of Population, PublicUse Microdata Sample (PUMS), 5 Percent Sample. U.S. Census Bureau, Washington, DC.

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i and destination j. Unlike our example, however, the gravity model allows both sizeand distance to vary simultaneously:

where:

Iij = predicted interaction between origin i and destination jk = a scaling constantPi = a measure of size, usually population, for origin iPj = a measure of size, usually population, for destination jdij = distance between origin i and destination j� = an exponent that adjusts for the rate of distance decay unique to the

type of interaction being measuredLet’s look at the formula piece by piece. The mass or size variables in the numer-

ator of the fraction will have a positive relationship with spatial interaction. This meansthat as the population of the state increases, both for origins and destinations, theinteraction between them increases. Distance, being in the denominator, will be neg-atively or inversely relational to interaction, meaning interaction decreases as dis-tance increases. Dividing by distance creates a distance decay curve with the shapeshown earlier in Figure 4.3.

The other two factors in the formula are constants that are calculated statisti-cally to produce the most realistic estimates (we give them to you in this chapter).The k factor scales the relative levels of interaction between places, so its valuedepends on the type of interaction being measured: a large value of k could existfor phone calls per year, a medium value for air travelers per year, and a low value

Introduction � 91


Population

1,000,000

5,000,000

10,000,000 0 200 400100 Miles

Figure 4.6 Although North Carolina and South Carolina are the same distance from California, we expectmore migrants from North Carolina because it is larger. Although Arizona and Maryland have about the samepopulation, we expect more migrants from Arizona because it is closer to California.

I kP P

dij

i j

ij

=β

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for migrants per year. The � exponent affects how steeply interaction declines withdistance: the larger the �, the steeper the distance decay effect. For simplicity, wewon’t use � in this case study.

The basic gravity model can be modified to model all types of spatial interac-tion. For instance, if geographers suspect that high unemployment rates are a sig-nificant push factor, they can add them to the model and test them statistically. Amoving company such as Mayflower®or a trailer/truck rental company such as U-Haul® could use the gravity model to predict future migration patterns in orderto choose new office locations. Similarly, airlines use gravity-type models to predictpassenger flows, urban planners use them to predict commuting, and retailers usethem to predict shopping.

When we designed this activity, we hoped that you would learn not only howthe gravity model works but also how to think critically about models. Thinking crit-ically about models means neither blindly accepting their outputs nor completelyrejecting the model for not being perfectly true to reality. Thinking critically meansassessing the strengths and weaknesses of a model, judging where it fits well andwhere it doesn’t, and understanding what has been included in the model and whathas been omitted. Certainly, people are different than the atoms and planets stud-ied by physicists. Human actions are not mechanistically controlled by the size oftheir origins and destinations or the distances between them. In addition, the basicgravity model does not include migration selectivity factors such as age or educa-tion level, nor does it incorporate channelized migration streams and counterstreams.Nevertheless, human behavior is fairly predictable when the actions of millions ofpeople are aggregated, and certain general tendencies emerge that are well repre-sented by the gravity model.

After you have used the model to predict migration flows to your state or provincein Activity 1, you will learn how to assess the effectiveness of the gravity model usinggraphs and maps. This will give you an idea of where the model fits well and whereit doesn’t. Moreover, you can determine what factors in addition to population sizeand distance could be influencing migration patterns. The failures of the model willreveal to you as much about migration as its successes and possibly more. You willlearn that your state or province is more interconnected with some states than withothers, which in turn can tell you about its economy, history, and culture.



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GOALTo model spatial interaction, in this case migration, usingthe gravity model. You will use the gravity model to pre-dict the number of migrants to your state or province fromall other U.S. states or Canadian provinces. The accuracyof the model will be assessed, and residuals will bemapped to show where actual migration differs from whatthe gravity model predicts.

LEARNING OUTCOMESAfter completing the chapter, you will be able to:

• Apply principles of spatial interaction to patterns ofmovement.

• Identify the major source areas for migration to yourstate.

• Use functions of a spreadsheet.• Produce and interpret a scatter diagram.• Discriminate between positive and negative resid-

uals.• Identify outliers on a scatter diagram.• Think critically about models in human geography.

SPECIAL MATERIALS NEEDED• Computer with CD drive and Internet Explorer 5.0

and above. See Read Me.

BACKGROUNDSeeking a new life in a new place has always been a fun-damental part of the American dream. High levels ofmobility have been linked to settlement of the frontier, aninnate restlessness, the drive for change, and an inherentdynamism in American culture. Geographic mobility,defined as a move from one residence to another, is higherin the United States than in Western European countries,where many people have lived in the same area for manygenerations (Figure 4.7).

Higher-than-average levels of mobility found in theUnited States, Canada, Australia, and New Zealand sug-gest that something about the history and culture of thesecountries encourages movement. One explanation is thatthey are all high-immigration countries. Immigration fromabroad brings people with weak ties to the new place. Oneof the strongest predictors of whether people move in thefuture is whether they have moved in the past. Migrationbegets further migration in the sense that once ties to homeare broken, they are easier to break again. A second expla-nation is that the United States, Canada, Australia, andNew Zealand share cultures that value personal freedomabove loyalty to any particular group or place. A geographicmove is, at its essence, an exercise of such freedom. Finally,

in all four countries, land and housing costs are relativelycheap, and liberal government controls on housing codes,land use, and real estate markets make it easy for peopleto buy and sell homes, and thus to move.

Despite the popular conception that mobility is on therise and Americans are continually on the move, mobilityrates in the United States actually are at a post–World WarII low (Figure 4.8). During most of the 1950s and 1960s,20 percent of the population changed its residence everyyear. By the beginning of the 1980s, this figure was downto 16 percent. The most recent census figure was 16.1 per-cent in 1999–2000. The decline in mobility is attributed,in part, to the aging of the population. Older people arefar less likely to move than younger ones, and an older pop-ulation will have lower mobility rates than a younger one.Even among people in their 20s, however, mobility ratesare lower today than they were 50 years ago. One reasonis that we have become a nation of homeowners, and peo-ple who own their own home are far less likely to movethan renters. Also, rising labor force participation amongwomen and the growing number of dual-career householdsretard mobility because couples must consider the workand family responsibilities of both spouses in deciding tomove.

An exception to the overall decline in U.S. mobility wasduring the mid-1980s, when an upward spike in mobilityfollowed a sharp recession in which unemployment rateswere high, inflation skyrocketed, and the housing marketslumped. These conditions seriously curtailed the desireand ability to move. When the recession ended and inter-est rates fell, pent-up demand for movement brieflyreturned mobility rates to levels of the 1950s and 1960s.Since that unusual period, however, mobility rates havecontinued to decline.

Case Study � 93


� CASE STUDY

NEWTON’S FIRST LAW OF MIGRATION

Figure 4.7 This scene of a family packing theirbelongings in a moving van is a familiar one in thehighly mobile United States.

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The likelihood of moving varies markedly across majorregions of the United States. The Northeast has the low-est moving rate—11.7 percent—well below the nationalrate of 15.9 percent. It is followed by the Midwest at 15.1percent, the South at 17.1 percent, and the West at 18.5percent. The South and West have higher mobility in partbecause they have been destinations for recent migrantsfrom the Northeast and Midwest, and as noted earlier, oncesomeone moves, he or she is more likely to move again.The West, in addition, has a long tradition of transienceand impermanence dating from frontier days.

Major migration streams of the past 30 years movedpeople from the industrial core of the Northeast andMidwest to the South and West. Movements have ebbedand flowed with changes in the national and regionaleconomies, but population shifts generally have benefitedthe Sunbelt at the expense of the Frostbelt. The South hasbeen the major beneficiary of regional movements (morein-migrants than out-migrants), and the Northeast has beenthe most consistent loser of net migrants (more out-migrants than inmigrants) (Figure 4.9). In addition to theobvious attraction of warmer climates, migration tosouthern states was linked to the regional restructuring ofjobs. Cheap labor, energy, and land costs attracted low-wage industries, and the region attracted a disproportionateshare of cutting-edge industries such as electronics, com-puters, and communications technology. Also associatedwith the South’s growing attractiveness are changes in socialconditions and cultural attitudes in what’s commonly knownas the “New South.”

Broad regional shifts disguise smaller state-to-statemigration trends. The 15 largest state-to-state migrationflows reveal continued Frostbelt to Sunbelt movement;metropolitan-scale movements within the Northeast;connections between highly populated states such asCalifornia, Texas, New York, and Florida, and the emer-gence of California and Florida as redistributors of migrants(Figure 4.10). The largest stream in the national systemof migration connects New York with Florida. Movementbegan after World War II with the migration of retireesand increased thereafter as contact between the areas grew.Today’s streams consist of both retirees and labor forcemigrants. Movement within the highly populated urbanNortheast also results in streams from New York to NewJersey, New Jersey to New York, and New Jersey toPennsylvania.

California was a net loser of internal migrants in1998–1999. Despite the fact that many migrants movedfrom Texas, Arizona, and Washington to California, moremoved in opposite directions. California also sent manymigrants to the nearby states of Nevada and Oregon. A sim-ilar pattern has emerged in the relationship betweenGeorgia and Florida. In recent years, more migrants movedfrom Florida to Georgia than vice versa. Although Floridaattracts large numbers of migrants from New York andNew Jersey, it loses migrants to Georgia and other rapidlygrowing southern states. It redistributes migrants from theNorth to other states in the South in much the same waythat California traditionally has redistributed migrants fromthe Northeast and Midwest to other western states.



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1994/95

1999/2000

Annual Mobility Rates

Figure 4.8 Annual mobility rate equals the number of people who moved to a new residence between Marchof one year and March of the next, divided by the population age one or older. Note the lower rates in the lastfew decades since the peak years of the 1950s and 1960s. Source: U.S. Census Bureau.

� CASE STUDY (continued)

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Case Study � 95


Number of Migrants

54,000 - 51,000

42,000 - 37,000

33,000 - 30,000

27,000 - 23,000

Figure 4.10 Fifteen largest interstate migration streams, 1998–1999. Source: Raw data matrix from theInternal Revenue Service and U.S. Bureau of the Census.

Num

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1979/80 1984/85 1989/90 1994/95 1999/2000 1979/80 1984/85 1989/90 1994/95 1999/2000

1979/80 1984/85 1989/90 1994/95 1999/2000 1979/80 1984/85 1989/90 1994/95 1999/2000

Net in-migration Net out-migrationIn-migrants Out-migrants

Figure 4.9 In- and out-migration for four regions of the United States. Source: U.S. Census Bureau.


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The pattern of net migration rates (in-migrants minusout-migrants as a percentage of the population) shows largepopulation gains in Nevada, Arizona, and Colorado, andmore moderate gains in most southeastern, northern NewEngland, and northwestern states (Figure 4.11). Urbannortheastern states experienced net out-migration due tocontinued deindustrialization and economic restructuring(see Chapter 6 for a more complete discussion of theseprocesses). Also hard hit were states in the Great Plains,where the loss in agricultural jobs and failure to attract cutting-edge industries and services has led to depopula-tion of many rural areas. Some counties in the region arereverting to nineteenth-century population levels with twoor fewer persons per square mile. (Note: Net migrationrates for every country in the world can be found in theCountry Facts spreadsheet on the CD.)

Today’s migration patterns reflect the location of statesrelative to one another (nearby states tend to exchangemigrants), historical patterns of movement (i.e., longtimelinkages between Florida and New York and betweenCalifornia and Texas), the changing geography of economicopportunity in the nation, and the public’s perceptionsabout the attractiveness of places, including intangiblessuch as an agreeable climate, being near family and friends,and an ocean view. You are asked in this exercise to exam-ine recent migration flows between your state (orCanadian province) and all others in the nation in1998–1999 and to hypothesize about why your state orprovince is more connected to some than to others. Useyour basic knowledge of migration trends in the nation andyour knowledge of the circumstances of your particularstate or province.




Figure 4.11 Net migration rates for the United States (except Alaska and Hawaii), 1999.

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Name: _____________________________ Instructor: _________________________

Newton’s First Law of Migration: The Gravity Model

� ACTIVITY 1: PREDICTING MIGRATION WITH THE GRAVITY MODEL

Use the simplified format of the following gravity model to estimate migration flowsto your state in 1999 (or your province in 1998). The destination population term,Pj, has been left out because you will be looking at a single destination j—your stateor province—that is the same for all origins. The distance exponent also has beenleft out for simplicity.

where:

Mij = gravity model prediction of migration between origin i and destination jPi = population of origin state idij = distance from origin i to destination jk = a constant that adjusts the gravity model estimates so that the total num-

bers of actual and estimated migrants are approximately equal

You will obtain the data you need and perform calculations using a spreadsheet.If you’ve never used a spreadsheet, you will learn a valuable skill here. Just followthe step-by-step instructions.

A. Insert the CD into your computer. A window will automatically appear (ifthis doesn’t work, see the readme.txt file or the instruction sheet thatcame with the CD).

B. If Human Geography in Action has already been installed on your com-puter, click on Run from HD. If not, click either Install (for faster perfor-mance on your home computer) or Run from CD (on school labcomputers).

C. Click on the large Human Geography in Action logo to start.D. Click on Chapter Menu.E. Click on Chapter 4—Newton’s First Law of Migration.F. Click on Activity 1: Predicting Migration with the Gravity Model.G. Read the activity description and then click Continue.H. Choose your country, USA or Canada.I. Choose the destination state or province to which you wish to measure

migration. As you move the mouse over the names, the location on themap is highlighted. Click on your state or province.

You now will be looking at a spreadsheet with all the information youneed. You can scroll down the spreadsheet to look at all the values byusing the scroll bar. The columns are as follows:

Activity 1: Predicting Migration with the Gravity Model � 97


M kP

dij

i

ij

=

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Column A State State or province abbreviations.Column B Pi Population of each state in 1999 or each

province in 1998.Column C dij Distance in miles from the geographic cen-

ter of that state to the geographic center ofthe state you selected. Notice that your statehas a value of 0 because it is zero miles awayfrom itself.

Column D Pi/dij State or province population divided by thedistance to your state or province.

Column E Predicted The number of migrants, Mij, predicted byMigration the gravity model.

Column F Actual The actual number of migrants from that Migration state or province to your state or province

from 1998 to 1999. The U.S. data come fromthe Internal Revenue Service (IRS), whichtracks where claimants filed their returns in1998, opposed to where they filed in 1999.Any claimant crossing a state line betweenthose years is considered an interstatemigrant. Canadian data are estimated in thesame way and are obtained from StatisticsCanada for 1997 to 1998. Notice your ownstate or province has a value of 0, becausewe are not concerned here with movementswithin a state or province.

Column G Residual % Actual migration minus the predicted migra-tion, divided by the actual migration, andmultiplied by 100. These residuals show thepercentage error in your predictions.

Note: At the top of the screen, you are provided with the k coefficient that has alreadybeen calculated for your particular state or province.1

The spreadsheet software will compute values much like a calculator if you giveit a formula to use. In a spreadsheet, the letter of the column and the number ofthe row identify a cell. For instance, the population (Column B) of California (row6) is in cell B6. Multiplying or dividing a number by cell B6 is the same as multi-plying or dividing by the population of California.

J. Your first step is to divide the population of the first origin state (orprovince) by the distance between that origin and your state. Click oncell D2, the first empty cell where you will calculate Pi/dij. Because thepopulation of the first origin is in B2 and its distance from your state is inC2, you can calculate Pi/dij by entering the following formula:

= B2 / C2



1 For you math mavens, we estimated k by doing a least-squares linear regression of Mij on Pi /dijwith the constant term forced to zero (that is, using the functional form Y = kX, where Y = Mij and X = Pi /dij).

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The “=” is a code that tells the spreadsheet you are entering a formula,not a number. The formula simply says divide cell B2 by cell C2 andstore the answer in D2. You can type this formula directly in cell D2using the keyboard or create it using the formula buttons at the top right.Click the “=” button, then type B2, then click on the “÷” button, andthen type C2. Whichever way you enter the formula into the cell, if youdid so correctly, the Copy button in the upper left will highlight. Youhave now calculated your first value. The answer will appear in the cellitself. If you make a mistake, you will get an error message telling you theformula you have entered is incorrect and to try again. Click Try again,and edit the formula. If at any time you wish to return to the originalblank spreadsheet, go to the browser’s View menu and select Refresh.

K. Now comes one of the best features of spreadsheets—you can transferthis formula to the entire Column D. First, click on Copy in the upperleft. This copies the formula from cell D2 into a buffer, which the com-puter remembers. Second, click on the Column D header to highlightthe entire column. Third, click on the word Paste in the upper left. Thecomputer has now copied the formula from cell D2 into each of the cellsbelow it and has modified each formula to divide by the B and C columncells immediately to the left, rather than always dividing cell B2 by C2.You have just saved yourself much time compared with typing this for-mula 50 times.

Notice that the value for your state or province—infinity—is not avalid result. This is so because you tried to divide by zero (the distance),and zero can go into anything an infinite number of times. This is okay;you will not need a value from your own state, so ignore it.

L. Finish calculating the predicted migration to your state by multiplyingthe Pi/dij value in Column D by the coefficient k, which we have calcu-lated and provided to you for each state and province. Because theHuman Geography in Action software checks to make sure that you com-plete the spreadsheet correctly, k must be entered into the formulaexactly as it appears at the top of the spreadsheet. The easiest and safestway to do this is to use the k button in the upper right to enter the value.Whatever you do, don’t type the letter k. Click on cell E2 and type“=D2*” and then click on the k button (* means multiply in computerlanguage). Your formula should look something like:

= D2 * 3.0169 (for Alberta) = D2 * 0.2206 (for Alabama)

Follow the Copy and Paste commands from the previous step to copythis formula into all the cells for Column E. Column E is the predictedmigration to your state or province based on the gravity model. Thinkabout what these numbers mean. Based on the population of each stateor province and its distance from your state, you have predicted the num-ber of migrants in 1999 (or 1998 for Canada).

M. Your final step in completing the spreadsheet is to calculate residuals inColumn G. You will learn more about residuals and how to use them inActivity 3, but you must calculate them now while the spreadsheet is stillactive.

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A residual is the difference between the actual migration and the pre-dicted migration. Residuals indicate how well the gravity model predictedthe actual migrant flow. To calculate the residuals, simply subtract yourpredicted migration, Column E, from the actual migration, Column F.Then, to put the residuals in percentage terms, divide the result by theactual migration (Column F) and multiply by 100. In cell G2, enter thisformula exactly as shown here, and press Enter:

= 100 * (F2-E2) / F2

Again, use the Copy and Paste commands you learned in the previous steps tocopy the formula from cell G2 to all cells in column G. If done correctly, you willreceive a message on the screen that says you have completed the spreadsheet. ClickOK.

The numbers in Column G are interpreted as percentage errors. For instance,a –2.6 means that the actual migration was 2.6 percent less than the predicted migra-tion, as a percentage of the actual migration. Putting the residuals into percentageterms allows you to compare, on an equal footing, how well the gravity model pre-dicts migration from states of different sizes.

N. Click the Print button.O. Click on Activity 2: Scatter Diagram in the right margin. Do not close

the spreadsheet window; you will need to return to it later.



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Name: _____________________________ Instructor: _________________________


� ACTIVITY 2: SCATTER DIAGRAM

You now see a scatter diagram next to the U.S. or Canada map. A scatter diagramdepicts the relationship between two variables. One variable is measured on the x-axis (horizontal), and another is measured on the y-axis (vertical). Each dot repre-sents a different origin state or province. Each dot is placed at the intersection ofthat state’s x value and y value. The diagram therefore is a “scatter” of these dots,which show groupings or trends in the relationship between the two variables.

In this scatter diagram, the x-axis measures actual migration, and the y-axis mea-sures predicted migration (Figure 4.12). Perfectly predicted migration values fallexactly along the 45º line (i.e., the predicted value was exactly equal to the actualvalue). In the case of those dots that deviate from the 45º line, the gravity modelwas less successful in estimating migrant flows to your state. All points above theline had predicted values that are larger than the actual migration flows. Points abovethe line are therefore overpredicted and have negative residuals. All points belowthe line had predicted values that are smaller than the actual migration flows. Pointsbelow the line are therefore underpredicted and have positive residuals.

A. Beware of states that have x and y values far greater than any other state.To fit such extreme values in the upper-right corner of the graph, alarge number of other points usually end up getting “squished” into thebottom-left corner (see Figure 4.13). Eliminating the very large valuescan give you a more spread-out scatter where you can see each dot bet-ter. To eliminate an extreme value, click on that dot in the graph, whichwill highlight it. The same region will also be highlighted in red on themap (your state or province is shown in gray). Click Hide SelectedArea(s). Repeat as necessary. You can click Show All Areas to restore all

Activity 2: Scatter Diagram � 101


Actual Migration

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Figure 4.12 Over- and underpredicted areas of a scatter diagram.

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dots to the graph. Be sure to discuss in your write-up if you have elimi-nated an extreme value.

B. When you have finished customizing your graph, click the Print button. C. Return to the computer and identify up to five poorly predicted states

(outliers) on your graph. The outliers are dots that are farthest from theline of perfect prediction (Figure 4.13). Move your mouse over an outly-ing dot to highlight it in red on the map and display its name, its actualmigration to your state, and the migration predicted by the gravity model.By hand, label the outliers on your printed map (use two-letter abbrevia-tions if needed).

D. When you have finished, close the Activity 2: Scatter Diagram windowand return to the Activity 1: Spreadsheet window, where you should clickon Activity 3: Residual Map in the right margin.



Actual Migration

Label thesepoorly predictedstates (outliers)

Extreme value(candidate for deletionfrom graph)

Pre

dict

ed M

igra

tion

Figure 4.13 Extreme values to delete and outliers to label.

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Name: _____________________________ Instructor: _________________________


� ACTIVITY 3: RESIDUAL MAP

You now see a choropleth map showing the accuracy of the gravity model acrossspace. The difference between the actual value and the predicted value is called aresidual. We have defined our residuals in percentage terms so that the residualof a big state such as California can be compared with that of a small state such asVermont. The data displayed on the map are from the spreadsheet’s Residual col-umn, which was calculated as: 100 × (Actual Migration – Predicted Migration)/ActualMigration. Each state or province is shaded according to its residual. Move the mouseover the map to see each state’s name and residual.

A. The map’s default break point settings split the residual values into sixequal intervals. These default break points are not particularly good ones,because the map does not lend itself to easy interpretation. Interactivelydefine your own class break points using the graphic array to the left ofthe map. This graph shows the range of data on the x-axis, in this case thepercentage error between your predicted value and the actual migrationvalue for each state. The gray vertical line shows the zero value whereyour predicted migration values equal the actual migration values. The y-axis, which ranges from 0 to 50 states or 0 to 13 provinces, ranks the ori-gin states from highest to lowest residual. You can move your mouse overthe dots to see which states or provinces they represent.

The vertical red bars show the break points between classes. You canselect a bar by clicking on the top triangle with your mouse. Holdingdown the mouse button on the triangle, move it left or right to set newclass limits. The shading patterns between the bars match those of themap. When you move the bars, the break points in the boxes belowchange to reflect the new position. These boxes are also directly editable:click in a break point box, edit a value, and hit Enter. You will use thisinteractive graphic array and/or the editable boxes to make your final map.

B. You can follow a number of possible strategies in defining the breakpoints. First, you can define a break point at 0 (gray vertical line on thegraph) that divides the states or provinces into those with positive residu-als and those with negative residuals. You could then define two otherbreak points to separate the small positive residuals from the large, andthe small negative residuals from the large. You can choose to have onlyone or two extreme outliers in the large positive and large negativeclasses, or you can have many. A second strategy, related to the first,defines six classes with small, medium, and large residuals on both thepositive and negative side. A third strategy is to define a class that groupsresiduals that are close to zero (i.e., that are closely predicted by themodel) regardless of whether they are positive or negative. Fourth, you

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could use the graphic array to set class break points that divide the datainto “natural classes” or groupings. Vertical groupings in the array indi-cate a group of states with similar residuals. Setting your break points inthe empty horizontal gaps would avoid putting two states with very simi-lar percentage residuals into different categories (i.e., you could avoid“splitting hairs”). You can also experiment with the Equal Frequency andEqual Interval buttons.

Set the break points to make what you consider to be the most infor-mative choropleth map of the residuals. However you choose your limitsjustify your choices in the write-up in Activity 4.

C. You can change the number of classes using the menu in the lower left.By setting the number of classes to four and moving Break Point 1 to thelower limit and Break Point 3 to the upper limit, and positioning BreakPoint 2 at 0, it is possible to create a two-color map with one shade for allpositive residuals and another for all negative residuals.

D. At the bottom of the screen, pick a Color Scheme that best portrays theresidual classes. You should try to use shades that indicate residual sign(+ or –) and size.

E. Click on the Print button. Study your map and make any changes neces-sary. Sometimes colors look good on the screen, but print poorly. Makesure the categories are easily distinguished on the final map you print.Don’t use any two patterns that look very similar. Hand in the map withthis assignment.

F. When you have finished, close the Activity 1 and 3 windows, and clickthe Exit button at the top right of the Chapter 4 page. If you are on acampus network, log off your machine. Don’t forget your CD.



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Name: _____________________________ Instructor: _________________________


� ACTIVITY 4: EVALUATION

Type (double spaced) the answers to the following questions:

4.1. Report any extreme values eliminated from the scatter plot in Activity 2, andjustify your choice of color scheme and class break points for the choropleth mapof residuals in Activity 3.

4.2. Assess the gravity model’s ability to predict migrant flows during the study period.How close were the predicted values to the actual flows?

4.3. The gravity model assumes that distance is a barrier to migration. Specifically,how does distance act as a deterrent to migration?

4.4. Justify the use of population as the numerator in the gravity model function.Can you suggest a variable that might be preferable to population as a measure ofthe “sending” power of a state?

4.5. We have said that points that fall along the 45º line on the graph are predictedaccurately by the gravity model. What does it mean if a point is below the line? Abovethe line? Which one is overpredicted and which one is underpredicted? (Hint: Lookat the Residual column and compare those values with the location of the pointson the graph.)

4.6. Can you detect any spatial patterns (groups of states) on the map of residualsthat are overpredicted or that are underpredicted? What explanations can you sug-gest for these patterns?

4.7. Based on your answer to Question 4.6, how would you amend the gravity modelto more accurately predict migration to your state? Would you add any variables toaccount for factors other than population and distance?

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� FURTHER READINGSCastles, Stephen, and Mark J. Miller. 1998. The Age of Migration: International Population Movement in the Modern World.

New York: Guilford Press.Chant, Sylvia M. (ed.) 1992. Gender and Migration in Developing Countries. New York: Belhaven Press.Ellis, Mark, and Richard Wright. 1999. The Balkanization Metaphor in the Analysis of US Immigration. Annals of the Association

of American Geographers 88:686–698.Fielding, Tony. 1992. Migration and Culture, pp. 201–212 in Processes and Patterns Volume I: Research and Prospects, Tony

Champion and Tony Fielding (eds.). London: Bellhaven Press. Frey, William H. 1995. Immigration, Domestic Migration, and Demographic Balkanization in America: New Evidence for

the 1990s. Population and Development Review 22:741–763. Gober, Patricia. 1993. Americans on the Move. Population Bulletin 48:1–40. Long, Larry. 1988. Migration and Residential Mobility in the United States. New York: Russell Sage Foundation. McHugh, Kevin E. 2000. Inside, Outside, Upside Down, Backward, Forward, Round and Round; A Case for Ethnographic

Studies in Migration. Progress in Human Geography 24:71–89.McHugh, Kevin E., Ines M. Miyaris, and Emily H. Skop. 1997. The Magnetism of Miami: Segmented Paths in Cuban Migration.

Geographical Review 87:504–519.Pandit, Kavita, and Suzanne Davies Withers. 1999. Migration and Regional Restructuring in the United States. New York:

Rowman & Littlefield.Skeldon, R. 1995. The Challenge Facing Migration Research: A Case for Greater Awareness. Progress in Human Geography

19:91–96.Tocalis, Thomas R. 1978. Changing Theoretical Foundations of the Gravity Concept of Human Interaction, pp. 66–124 in

The Nature of Change in Geographical Ideas, Brian J. L. Berry (ed.). DeKalb, IL: Northern Illinois University Press. White, Paul, and Peter Jackson. 1995. (Re)Theorizing Population Geography. International Journal of Population Geography

1:111–123.White, Stephen E.1994. Ogallala Oases: Water Use, Population Redistribution, and Policy Implications in the High Plains

of Western Kansas, 1980–1990. Annals of the Association of American Geographers 84:29–45.



� DEFINITIONS OF KEY TERMSDistance Decay The declining intensity of an activity withincreasing distance from its point of origin.

Extreme Value A point on a scatter diagram that isroughly in line with the main trend but is separated fromthe main group of points because of its extremely high orlow value. Contrast with outlier.

Gravity Model A model to predict spatial interaction,where size (population) is directly related to interactionand distance is inversely related to interaction.

Migration A permanent change in residence to outsideone’s community of origin.

Migration Counterstream Migration that runs oppositeto a migration stream.

Migration Selectivity The tendency for certain types ofpeople to migrate. Age, education, and othersociodemographic characteristics are migration selectivityfactors.

Migration Stream A well-defined migration channel froma specific origin to a particular destination.

Net Migration Rate The percentage gain or loss ofpopulation due to migration. It is calculated as in-migrants

minus out-migrants divided by the total population, alltimes 100. Positive numbers indicate net gain; negativenumbers indicate net loss.

Outlier Point on a scatter diagram that lies far off thetrend line. Outliers on the graph correspond to cases thatare poorly predicted by the model. Outliers are not to beconfused with extreme values, which may lie far from anyother point but which are still close to the best-fitting line(see Figure 4.13).

Pull Factors Reasons to move to a particular place.

Push Factors Reasons to move from a particular place.

Residuals The difference between an actual observedvalue of some variable and its predicted value using thegravity model.

Scatter Diagram A scatter of dots showing therelationship between two variables. Each dot on the graphrepresents the x and y coordinates of a differentobservation or case.

Spatial Interaction Movements of ideas, information,money, products, and people between places.

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� WEB RESOURCESMartin, Philip. Migration News: migration.ucdavis.edu/mn. Migration Research Unit at the University College, London: www.geog.ucl.ac.uk/mru/.Miner & Silverstein, LLP. Predictive Gravity Modeling: www.msac.com/pptm/page2.html.United Nations Population Division: www.un.org/esa/population/.U.S. Census Bureau. Geographic Mobility/Migration: www.census.gov/population/www/socdemo/migrate.html.University of Pennsylvania. Mexican Migration Project: www.pop.upenn.edu/mexmig.

� ITEMS TO HAND INActivity 1: • The completed spreadsheet table Activity 2: • The scatter diagram, including the labels of poorly predicted states Activity 3: • The residual map Activity 4: • Typed answers to Questions 4.1–4.7

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newton’s first law of migration: the gravity model · having introduced these key migration...

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