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CHAPTER
4
Newton’s First Law of Migration:
The Gravity Model
INTRODUCTION
Places are connected with one another at the local, regional, and global scales through
systems of spatial interactions. These interactions involve movements of ideas,
information, money, products, and people. As the interconnectedness of the world
accelerates, the number and intensity of these interactions increases. Millions of
messages are sent daily across the Internet. Hugely popular American television shows
and movies disseminate American culture overseas, and some foreign films and programming find an audience in the United States and Canada. Commodities and capital freely move across national boundaries. And each year, more people become
international migrants. Currently, 175 million people reside outside of the country
of their birth. The number of migrants in the world has more than doubled since
1975, 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’s
community of origin. Conceptually, someone who moves to a new home within his
or her community but does not have to change his or her place of work, shop in
new stores, find new doctors, and establish new friendships is considered a local
mover but not a migrant. Offically, migration is defined as crossing an administrative boundary, such as between counties or states. Tourists, temporary residents,
and seasonal workers may play important roles in some places, but they are not considered 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 to
be considered a migrant, but that is just one of the factors that makes the study of
migration so fascinating.
Migration can occur at many spatial scales, including rural-to-urban movements
from hinterlands to cities (Figure 4.1), urban-to-urban moves between regions, and
global migration between countries. The size, composition, and spatial organization
of migration flows tell us a great deal about the places involved. In that people tend
to move from less desirable places toward more desirable places, the system of migration flows provides clues about how places stack up relative to one another. Place
desirability can result from economic factors such as job availability, high wages,
and affordable housing and from noneconomic considerations such as a favorable
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86 Chapter 4. Newton‘s First Law of Migration: The Gravity Model
Figure 4.1 Rural-to-urban migrants at a train station in China.
climate, clean air, low crime rates, nearness to friends and relatives, and the absence
of war and environmental disaster.
Migration patterns are the result of millions of individual and household decisions about where to live. For those who move, a combination of push and pull factors triggers the decision to move. Push factors can include exorbitant housing costs,
growing gridlock, rising crime rates, skyrocketing tax rates, a poor climate, and the
lack of a satisfying, well-paying job. Pull factors can include the promise of a higher
paying job, a pleasant physical setting, the availability of affordable housing, a desirable climate, or the lure of nearby family members. Sometimes what is a push for
some people is a pull for others. Take, for example, closeness to family. Many believe
that living near family members provides a valuable and comforting social support
system; others see it as claustrophobic, stifling their independence. Similarly, a climate that is too hot and a push for some can be just right and a pull for others. High
school taxes can be perceived as desirable for a young family with children but as
onerous for a childless single or an elderly couple. How we perceive various place
characteristics and how much weight we attach to them is very much a personal
matter.
The effects of migration on places of origin and destination are influenced by
a process called migration selectivity. Certain individuals are more likely to migrate
based on their personal characteristics, including age, education, and other sociodemographic characteristics. Age is the most important factor in influencing whether
someone is a migrant or not (Figure 4.2). People are most prone to move during
their early adult years between the ages of 18 and 30. The average individual makes
approximately one-half of his or her 12 lifetime moves by age 25. During these young
adult ages, people leave their parents’ home to attend school, join the military or
take a job, leave college to find employment or change jobs, marry, and begin families. All these life-course events are usually associated with changes in residence.
Movement rates are also high among young children who typically have parents in
their 20s.
A second migration selectivity factor is education. People with higher levels of
education are more likely to make long-distance moves. Getting a college education often means moving to a new city and then returning or moving again upon
graduation. In addition, education exposes us to new ideas and people from other
places. It also qualifies us for, and provides information about, a wide variety of jobs
in many different geographical areas. The selectivity of migration alters the population
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Introduction 87
Age Selectivity of Migration:
Migration Rates 1999–2000
40
35
Migration Rate (% of population who move)
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30
25
20
15
10
5
0
4
9
14
19
24
29
34
39
44
49
54
59
64
69
74
79
84
89
Age
Figure 4.2 Migration rates are highest for young adults in their early 20s.
characteristics of origin and destination places. As a general rule, places experiencing
net out-migration lose a disproportionate share of their young, well-educated residents while areas of net in-migration gain such individuals. There is a double
whammy for places experiencing out-migration. They lose not only population numbers but also their youngest and best-educated residents. Especially troublesome
is that this process can snowball, making origins less attractive to future migrants
and less capable of retaining their current residents.
Migration also influences places through the tendency to form migration
streams. People do not move randomly across the landscape. They move in welldefined channels from specific origins to particular destinations. Migration streams
often occur between nearby places because it is cheaper, quicker, and easier for people to move short distances. The tendency for migration, or any other form of spatial interaction, to decrease with distance is called distance decay (Figure 4.3).
Migration streams result from information flows between origins and destinations. Letters, telephone calls, and return visits from earlier migrants communicate
opportunity at the potential destination. These pioneer migrants assist newcomers
in finding a place to live, getting a new job, and adjusting to a new community.
Information about places also comes from newspapers, television, magazines, business contacts, and personal travel. However, most people know surprisingly little
about the range of potential places to live. Their migration decisions are based on
a narrow set of options dictated by first- or secondhand information about what it
is like to live there.
Wherever a migration stream develops, a migration counterstream of people moving in the opposite direction occurs. Not everyone who migrates intends to
remain permanently at the place of destination, for example, college students. Others
are unhappy with the circumstances of their move, their personal situation changes,
or they are military or corporate personnel who are reassigned. Divorce might create a return migration. An elderly couple who moved from the North to a Sunbelt
retirement community while in their 60s could return home when they are in their
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88 Chapter 4. Newton‘s First Law of Migration: The Gravity Model
Level of Interaction
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0
0
Distance
Figure 4.3 Distance decay curve showing decreasing interaction as distance increases.
ing migration counterstreams is the presence of information linking the two places.
Once, for whatever reason, the channels of communication are opened and interpersonal relationships are built, movement will occur in both directions, although
not necessarily at the same rate.
Migration streams involving faraway places get started in a variety of ways. Large
streams connecting New York and New Jersey with Florida arose after World War
II with the migration of retirees and snowballed as contact between the two areas
grew. Historically strong ties between California and Midwestern states began as
labor-force migration. The stream between Oklahoma and California originated with
Depression-era Dustbowl migrants. The experience of Dustbowl migrants, eloquently
portrayed in Steinbeck’s The Grapes of Wrath, changed conventional wisdom about
migrants from hardy pioneers in search of opportunity to disadvantaged families trying to survive (Figure 4.4).
Figure 4.4 A “Dustbowl” migrant family from Oklahoma recently arrived in California to
join the harvest, November 1936.
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Introduction 89
The most geographically focused migration streams in the United States today
are among newcomers to the country who are strongly attracted to immigrant communities or enclaves (see Chapter 12 for a fuller discussion of enclaves). Immigrant
communities offer familiar language, food, music, and religious institutions. They
also help newcomers to locate all-important housing and jobs. Many immigrants find
work in businesses owned by their compatriots, or they establish their own small
businesses, providing goods and services to immigrant niche markets.
The internal migration streams of Cuban- and Mexican-born immigrants demonstrate the different conditions under which they migrated to the United States (Figure
4.5). Mexicans are immigrants who, for the most part, moved voluntarily to the United
States, largely for economic reasons. Cubans are refugees, meaning that they were
forced by political crisis to flee their homeland. Even today, 20 or 30 years after
their move to the United States, some Cubans continue to see themselves more as
exiles hoping to return to Cuba than as immigrants seeking a permanent home and
future in the United States. South Florida has emerged as the surrogate homeland
for Cuban émigrés. Much political activity is organized around the overthrow of
Castro’s regime, and high value is placed on preserving cubanidad or “Cubanness.”
Cuban migration streams are strongly directed toward Florida; the counterstreams
are extremely weak. These uneven flows are redistributing the Cuban population
in favor of Florida. The Mexican migration system is quite different. Six of the 10
largest interstate flows interconnect the three largest concentrations of Mexicanborn population in California, Texas, and Illinois. Unlike the Cuban flows, the Mexican
flows are self-compensating. Streams and counterstreams are about equal in size,
and, thus, very little population redistribution occurs as a result of Mexican internal migration.
Having introduced these key migration concepts, we now shift our focus to the
more applied task of predicting migration flows. Geographers use a mathematical
formula known as the gravity model because it resembles Isaac Newton’s formula
for the gravitational attraction between any two celestial masses, which you might
have learned in physics class. Newton’s law has been adapted to social science research
for the purpose of estimating the spatial interaction or movement between any two
places. Spatial interaction can take the form of trade, transportation, communication, commuting, shopping, or, in the case of this chapter, migration.
The following example will help you to understand the idea behind the gravity
model. Figure 4.6 shows the populations of several states and their distance from
California. Would you expect California to attract more migrants from North Carolina
or from South Carolina? Their distances are about equal, but North Carolina has
twice as many inhabitants. All other things being equal, you’d probably expect about
two times as many migrants from North Carolina because there are two times more
potential movers. Next, would you expect more migration to California from Arizona
or from Maryland? Their populations are both around 5.3 million, but Maryland is
5 times farther away. Surely more people will move from Arizona, but probably not
5 times more because each additional mile matters less and less. As shown in Figure
4.3, 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, population size and distance are used to explain the interaction flow, Iij, between origin
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90 Chapter 4. Newton‘s First Law of Migration: The Gravity Model
10 Largest Domestic Migration Streams of Persons Born in Cuba
Number of Migrants,
1985-1990
10,000
4,000
1,000
10 Largest Domestic Migration Streams of Persons Born in Mexico
Number of Migrants,
1985-1990
8,500
4,000
2,700
Figure 4.5 Examples of migration streams for two ethnic groups. Source: 1990 Census of Population, Public
Use Microdata Sample (PUMS), 5 Percent Sample. U.S. Census Bureau, Washington, DC.
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Introduction 91
Population
1,000,000
5,000,000
10,000,000
0
100 200
400
Miles
Figure 4.6 Although North Carolina and South Carolina are the same distance from California, we expect
more migrants from North Carolina because it is larger. Although Arizona and Maryland have about the same
population, we expect more migrants from Arizona because it is closer to California.
i and destination j. Unlike our example, however, the gravity model allows both size
and distance to vary simultaneously:
Iij = k
where:
Pi Pj
dijβ
= predicted interaction between origin i and destination j
= a scaling constant
= a measure of size, usually population, for origin i
= a measure of size, usually population, for destination j
= distance between origin i and destination j
= an exponent that adjusts for the rate of distance decay unique to the
type of interaction being measured
Let’s look at the formula piece by piece. The mass or size variables in the numerator of the fraction will have a positive relationship with spatial interaction. This means
that as the population of the state increases, both for origins and destinations, the
interaction between them increases. Distance, being in the denominator, will be negatively or inversely relational to interaction, meaning interaction decreases as distance increases. Dividing by distance creates a distance decay curve with the shape
shown earlier in Figure 4.3.
The other two factors in the formula are constants that are calculated statistically 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 value
depends on the type of interaction being measured: a large value of k could exist
for phone calls per year, a medium value for air travelers per year, and a low value
Iij
k
Pi
Pj
dij
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92 Chapter 4. Newton‘s First Law of Migration: The Gravity Model
for migrants per year. The exponent affects how steeply interaction declines with
distance: the larger the , the steeper the distance decay effect. For simplicity, we
won’t use in this case study.
The basic gravity model can be modified to model all types of spatial interaction. For instance, if geographers suspect that high unemployment rates are a significant push factor, they can add them to the model and test them statistically. A
moving 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 order
to choose new office locations. Similarly, airlines use gravity-type models to predict
passenger flows, urban planners use them to predict commuting, and retailers use
them to predict shopping.
When we designed this activity, we hoped that you would learn not only how
the gravity model works but also how to think critically about models. Thinking critically about models means neither blindly accepting their outputs nor completely
rejecting the model for not being perfectly true to reality. Thinking critically means
assessing the strengths and weaknesses of a model, judging where it fits well and
where it doesn’t, and understanding what has been included in the model and what
has been omitted. Certainly, people are different than the atoms and planets studied by physicists. Human actions are not mechanistically controlled by the size of
their origins and destinations or the distances between them. In addition, the basic
gravity model does not include migration selectivity factors such as age or education level, nor does it incorporate channelized migration streams and counterstreams.
Nevertheless, human behavior is fairly predictable when the actions of millions of
people are aggregated, and certain general tendencies emerge that are well represented by the gravity model.
After you have used the model to predict migration flows to your state or province
in Activity 1, you will learn how to assess the effectiveness of the gravity model using
graphs and maps. This will give you an idea of where the model fits well and where
it doesn’t. Moreover, you can determine what factors in addition to population size
and distance could be influencing migration patterns. The failures of the model will
reveal to you as much about migration as its successes and possibly more. You will
learn that your state or province is more interconnected with some states than with
others, which in turn can tell you about its economy, history, and culture.
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Case Study 93
CASE STUDY
NEWTON’S FIRST LAW OF MIGRATION
GOAL
To model spatial interaction, in this case migration, using
the gravity model. You will use the gravity model to predict the number of migrants to your state or province from
all other U.S. states or Canadian provinces. The accuracy
of the model will be assessed, and residuals will be
mapped to show where actual migration differs from what
the gravity model predicts.
LEARNING OUTCOMES
After completing the chapter, you will be able to:
• Apply principles of spatial interaction to patterns of
movement.
• Identify the major source areas for migration to your
state.
• Use functions of a spreadsheet.
• Produce and interpret a scatter diagram.
• Discriminate between positive and negative residuals.
• 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.
BACKGROUND
Seeking a new life in a new place has always been a fundamental part of the American dream. High levels of
mobility have been linked to settlement of the frontier, an
innate restlessness, the drive for change, and an inherent
dynamism in American culture. Geographic mobility,
defined as a move from one residence to another, is higher
in the United States than in Western European countries,
where many people have lived in the same area for many
generations (Figure 4.7).
Higher-than-average levels of mobility found in the
United States, Canada, Australia, and New Zealand suggest that something about the history and culture of these
countries encourages movement. One explanation is that
they are all high-immigration countries. Immigration from
abroad brings people with weak ties to the new place. One
of the strongest predictors of whether people move in the
future is whether they have moved in the past. Migration
begets further migration in the sense that once ties to home
are broken, they are easier to break again. A second explanation is that the United States, Canada, Australia, and
New Zealand share cultures that value personal freedom
above loyalty to any particular group or place. A geographic
move is, at its essence, an exercise of such freedom. Finally,
in all four countries, land and housing costs are relatively
cheap, and liberal government controls on housing codes,
land use, and real estate markets make it easy for people
to buy and sell homes, and thus to move.
Despite the popular conception that mobility is on the
rise and Americans are continually on the move, mobility
rates in the United States actually are at a post–World War
II low (Figure 4.8). During most of the 1950s and 1960s,
20 percent of the population changed its residence every
year. By the beginning of the 1980s, this figure was down
to 16 percent. The most recent census figure was 16.1 percent in 1999–2000. The decline in mobility is attributed,
in part, to the aging of the population. Older people are
far less likely to move than younger ones, and an older population will have lower mobility rates than a younger one.
Even among people in their 20s, however, mobility rates
are lower today than they were 50 years ago. One reason
is that we have become a nation of homeowners, and people who own their own home are far less likely to move
than renters. Also, rising labor force participation among
women and the growing number of dual-career households
retard mobility because couples must consider the work
and family responsibilities of both spouses in deciding to
move.
An exception to the overall decline in U.S. mobility was
during the mid-1980s, when an upward spike in mobility
followed a sharp recession in which unemployment rates
were high, inflation skyrocketed, and the housing market
slumped. These conditions seriously curtailed the desire
and ability to move. When the recession ended and interest rates fell, pent-up demand for movement briefly
returned mobility rates to levels of the 1950s and 1960s.
Since that unusual period, however, mobility rates have
continued to decline.
Figure 4.7 This scene of a family packing their
belongings in a moving van is a familiar one in the
highly mobile United States.
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94 Chapter 4. Newton‘s First Law of Migration: The Gravity Model
CASE STUDY (continued)
Annual Mobility Rates
Percent moving to another residence per year
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25
24
23
22
21
20
19
18
17
16
15
19
19
44
/4
5
19
49
/5
0
19
54
/5
5
19
59
/6
0
19
19
64
/6
5
69
/7
0
/7
5
79
/8
0
84
/8
5
19
19
19
19
19
74
89
/9
0
94
/9
5
99
/2
00
0
Figure 4.8 Annual mobility rate equals the number of people who moved to a new residence between March
of one year and March of the next, divided by the population age one or older. Note the lower rates in the last
few decades since the peak years of the 1950s and 1960s. Source: U.S. Census Bureau.
The likelihood of moving varies markedly across major
regions of the United States. The Northeast has the lowest moving rate—11.7 percent—well below the national
rate of 15.9 percent. It is followed by the Midwest at 15.1
percent, the South at 17.1 percent, and the West at 18.5
percent. The South and West have higher mobility in part
because they have been destinations for recent migrants
from the Northeast and Midwest, and as noted earlier, once
someone moves, he or she is more likely to move again.
The West, in addition, has a long tradition of transience
and impermanence dating from frontier days.
Major migration streams of the past 30 years moved
people from the industrial core of the Northeast and
Midwest to the South and West. Movements have ebbed
and flowed with changes in the national and regional
economies, but population shifts generally have benefited
the Sunbelt at the expense of the Frostbelt. The South has
been the major beneficiary of regional movements (more
in-migrants than out-migrants), and the Northeast has been
the most consistent loser of net migrants (more outmigrants than inmigrants) (Figure 4.9). In addition to the
obvious attraction of warmer climates, migration to
southern states was linked to the regional restructuring of
jobs. Cheap labor, energy, and land costs attracted lowwage industries, and the region attracted a disproportionate
share of cutting-edge industries such as electronics, computers, and communications technology. Also associated
with the South’s growing attractiveness are changes in social
conditions and cultural attitudes in what’s commonly known
as the “New South.”
© 2004 John Wiley & Sons, Inc.
Broad regional shifts disguise smaller state-to-state
migration trends. The 15 largest state-to-state migration
flows reveal continued Frostbelt to Sunbelt movement;
metropolitan-scale movements within the Northeast;
connections between highly populated states such as
California, Texas, New York, and Florida, and the emergence of California and Florida as redistributors of migrants
(Figure 4.10). The largest stream in the national system
of migration connects New York with Florida. Movement
began after World War II with the migration of retirees
and increased thereafter as contact between the areas grew.
Today’s streams consist of both retirees and labor force
migrants. Movement within the highly populated urban
Northeast also results in streams from New York to New
Jersey, New Jersey to New York, and New Jersey to
Pennsylvania.
California was a net loser of internal migrants in
1998–1999. Despite the fact that many migrants moved
from Texas, Arizona, and Washington to California, more
moved in opposite directions. California also sent many
migrants to the nearby states of Nevada and Oregon. A similar pattern has emerged in the relationship between
Georgia and Florida. In recent years, more migrants moved
from Florida to Georgia than vice versa. Although Florida
attracts large numbers of migrants from New York and
New Jersey, it loses migrants to Georgia and other rapidly
growing southern states. It redistributes migrants from the
North to other states in the South in much the same way
that California traditionally has redistributed migrants from
the Northeast and Midwest to other western states.
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Case Study 95
1500
1000
500
Northeast
0
1979/80
1984/85
1989/90
1994/95
Number of Migrants (Thousands)
Number of Migrants (Thousands)
CASE STUDY (continued)
1999/2000
1500
1000
500
South
0
1979/80
1984/85
1989/90
1000
500
Midwest
1984/85
1989/90
1994/95
1999/2000
Number of Migrants (Thousands)
Number of Migrants (Thousands)
1500
0
1979/80
1999/2000
1500
1000
500
West
0
1979/80
1984/85
1989/90
Year
In-migrants
1994/95
Year
Year
1994/95
1999/2000
Year
Out-migrants
Net in-migration
Net out-migration
Figure 4.9 In- and out-migration for four regions of the United States. Source: U.S. Census Bureau.
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 the
Internal Revenue Service and U.S. Bureau of the Census.
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96 Chapter 4. Newton‘s First Law of Migration: The Gravity Model
CASE STUDY (continued)
The pattern of net migration rates (in-migrants minus
out-migrants as a percentage of the population) shows large
population gains in Nevada, Arizona, and Colorado, and
more moderate gains in most southeastern, northern New
England, and northwestern states (Figure 4.11). Urban
northeastern states experienced net out-migration due to
continued deindustrialization and economic restructuring
(see Chapter 6 for a more complete discussion of these
processes). 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 depopulation of many rural areas. Some counties in the region are
reverting to nineteenth-century population levels with two
or fewer persons per square mile. (Note: Net migration
rates for every country in the world can be found in the
Country Facts spreadsheet on the CD.)
Today’s migration patterns reflect the location of states
relative to one another (nearby states tend to exchange
migrants), historical patterns of movement (i.e., longtime
linkages between Florida and New York and between
California and Texas), the changing geography of economic
opportunity in the nation, and the public’s perceptions
about the attractiveness of places, including intangibles
such as an agreeable climate, being near family and friends,
and an ocean view. You are asked in this exercise to examine recent migration flows between your state (or
Canadian province) and all others in the nation in
1998–1999 and to hypothesize about why your state or
province is more connected to some than to others. Use
your basic knowledge of migration trends in the nation and
your knowledge of the circumstances of your particular
state or province.
Figure 4.11 Net migration rates for the United States (except Alaska and Hawaii), 1999.
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Page 97
Activity 1: Predicting Migration with the Gravity Model 97
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 flows
to 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 state
or province—that is the same for all origins. The distance exponent also has been
left out for simplicity.
Mij = k
where:
Pi
dij
Mij = gravity model prediction of migration between origin i and destination j
Pi = population of origin state i
dij = distance from origin i to destination j
k = a constant that adjusts the gravity model estimates so that the total numbers 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 follow
the step-by-step instructions.
A. Insert the CD into your computer. A window will automatically appear (if
this doesn’t work, see the readme.txt file or the instruction sheet that
came with the CD).
B. If Human Geography in Action has already been installed on your computer, click on Run from HD. If not, click either Install (for faster performance on your home computer) or Run from CD (on school lab
computers).
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 the
map is highlighted. Click on your state or province.
You now will be looking at a spreadsheet with all the information you
need. You can scroll down the spreadsheet to look at all the values by
using the scroll bar. The columns are as follows:
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98 Chapter 4. Newton‘s First Law of Migration: The Gravity Model
Column A
Column B
State
Pi
Column C
dij
Column D
Pi/dij
Column E
Predicted
Migration
Actual
Migration
Column F
Column G
Residual %
State or province abbreviations.
Population of each state in 1999 or each
province in 1998.
Distance in miles from the geographic center of that state to the geographic center of
the state you selected. Notice that your state
has a value of 0 because it is zero miles away
from itself.
State or province population divided by the
distance to your state or province.
The number of migrants, Mij, predicted by
the gravity model.
The actual number of migrants from that
state or province to your state or province
from 1998 to 1999. The U.S. data come from
the Internal Revenue Service (IRS), which
tracks where claimants filed their returns in
1998, opposed to where they filed in 1999.
Any claimant crossing a state line between
those years is considered an interstate
migrant. Canadian data are estimated in the
same way and are obtained from Statistics
Canada for 1997 to 1998. Notice your own
state or province has a value of 0, because
we are not concerned here with movements
within a state or province.
Actual migration minus the predicted migration, divided by the actual migration, and
multiplied by 100. These residuals show the
percentage error in your predictions.
Note: At the top of the screen, you are provided with the k coefficient that has already
been calculated for your particular state or province.1
The spreadsheet software will compute values much like a calculator if you give
it a formula to use. In a spreadsheet, the letter of the column and the number of
the row identify a cell. For instance, the population (Column B) of California (row
6) is in cell B6. Multiplying or dividing a number by cell B6 is the same as multiplying or dividing by the population of California.
J. Your first step is to divide the population of the first origin state (or
province) by the distance between that origin and your state. Click on
cell D2, the first empty cell where you will calculate Pi/dij. Because the
population of the first origin is in B2 and its distance from your state is in
C2, 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 /dij
with 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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Activity 1: Predicting Migration with the Gravity Model 99
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 and
store the answer in D2. You can type this formula directly in cell D2
using the keyboard or create it using the formula buttons at the top right.
Click the “=” button, then type B2, then click on the “÷” button, and
then type C2. Whichever way you enter the formula into the cell, if you
did so correctly, the Copy button in the upper left will highlight. You
have now calculated your first value. The answer will appear in the cell
itself. If you make a mistake, you will get an error message telling you the
formula 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 original
blank spreadsheet, go to the browser’s View menu and select Refresh.
K. Now comes one of the best features of spreadsheets—you can transfer
this formula to the entire Column D. First, click on Copy in the upper
left. This copies the formula from cell D2 into a buffer, which the computer remembers. Second, click on the Column D header to highlight
the entire column. Third, click on the word Paste in the upper left. The
computer has now copied the formula from cell D2 into each of the cells
below it and has modified each formula to divide by the B and C column
cells immediately to the left, rather than always dividing cell B2 by C2.
You have just saved yourself much time compared with typing this formula 50 times.
Notice that the value for your state or province—infinity—is not a
valid 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 multiplying
the Pi/dij value in Column D by the coefficient k, which we have calculated and provided to you for each state and province. Because the
Human Geography in Action software checks to make sure that you complete the spreadsheet correctly, k must be entered into the formula
exactly as it appears at the top of the spreadsheet. The easiest and safest
way 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 computer
language). 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 copy
this formula into all the cells for Column E. Column E is the predicted
migration to your state or province based on the gravity model. Think
about what these numbers mean. Based on the population of each state
or province and its distance from your state, you have predicted the number of migrants in 1999 (or 1998 for Canada).
M. Your final step in completing the spreadsheet is to calculate residuals in
Column G. You will learn more about residuals and how to use them in
Activity 3, but you must calculate them now while the spreadsheet is still
active.
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100 Chapter 4. Newton‘s First Law of Migration: The Gravity Model
A residual is the difference between the actual migration and the predicted migration. Residuals indicate how well the gravity model predicted
the actual migrant flow. To calculate the residuals, simply subtract your
predicted migration, Column E, from the actual migration, Column F.
Then, to put the residuals in percentage terms, divide the result by the
actual migration (Column F) and multiply by 100. In cell G2, enter this
formula exactly as shown here, and press Enter:
= 100 * (F2-E2) / F2
Again, use the Copy and Paste commands you learned in the previous steps to
copy the formula from cell G2 to all cells in column G. If done correctly, you will
receive a message on the screen that says you have completed the spreadsheet. Click
OK.
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 migration, as a percentage of the actual migration. Putting the residuals into percentage
terms allows you to compare, on an equal footing, how well the gravity model predicts 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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Activity 2: Scatter Diagram 101
Name: _____________________________
Instructor: _________________________
Newton’s First Law of Migration:
The Gravity Model
ACTIVITY 2: SCATTER DIAGRAM
You now see a scatter diagram next to the U.S. or Canada map. A scatter diagram
depicts the relationship between two variables. One variable is measured on the xaxis (horizontal), and another is measured on the y-axis (vertical). Each dot represents a different origin state or province. Each dot is placed at the intersection of
that 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 measures predicted migration (Figure 4.12). Perfectly predicted migration values fall
exactly along the 45º line (i.e., the predicted value was exactly equal to the actual
value). In the case of those dots that deviate from the 45º line, the gravity model
was less successful in estimating migrant flows to your state. All points above the
line had predicted values that are larger than the actual migration flows. Points above
the line are therefore overpredicted and have negative residuals. All points below
the line had predicted values that are smaller than the actual migration flows. Points
below the line are therefore underpredicted and have positive residuals.
Predicted Migration
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, a
large number of other points usually end up getting “squished” into the
bottom-left corner (see Figure 4.13). Eliminating the very large values
can give you a more spread-out scatter where you can see each dot better. To eliminate an extreme value, click on that dot in the graph, which
will highlight it. The same region will also be highlighted in red on the
map (your state or province is shown in gray). Click Hide Selected
Area(s). Repeat as necessary. You can click Show All Areas to restore all
es
lu n
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ic
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ct
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e
a ctu
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Actual Migration
Figure 4.12 Over- and underpredicted areas of a scatter diagram.
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102 Chapter 4. Newton‘s First Law of Migration: The Gravity Model
dots to the graph. Be sure to discuss in your write-up if you have eliminated 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 the
line of perfect prediction (Figure 4.13). Move your mouse over an outlying dot to highlight it in red on the map and display its name, its actual
migration to your state, and the migration predicted by the gravity model.
By hand, label the outliers on your printed map (use two-letter abbreviations if needed).
D. When you have finished, close the Activity 2: Scatter Diagram window
and return to the Activity 1: Spreadsheet window, where you should click
on Activity 3: Residual Map in the right margin.
Extreme value
(candidate for deletion
from graph)
Predicted Migration
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Label these
poorly predicted
states (outliers)
Actual Migration
Figure 4.13 Extreme values to delete and outliers to label.
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Activity 3: Residual Map 103
Name: _____________________________
Instructor: _________________________
Newton’s First Law of Migration:
The Gravity Model
ACTIVITY 3: RESIDUAL MAP
You now see a choropleth map showing the accuracy of the gravity model across
space. The difference between the actual value and the predicted value is called a
residual. We have defined our residuals in percentage terms so that the residual
of a big state such as California can be compared with that of a small state such as
Vermont. The data displayed on the map are from the spreadsheet’s Residual column, which was calculated as: 100 × (Actual Migration – Predicted Migration)/Actual
Migration. Each state or province is shaded according to its residual. Move the mouse
over the map to see each state’s name and residual.
A. The map’s default break point settings split the residual values into six
equal intervals. These default break points are not particularly good ones,
because the map does not lend itself to easy interpretation. Interactively
define your own class break points using the graphic array to the left of
the map. This graph shows the range of data on the x-axis, in this case the
percentage error between your predicted value and the actual migration
value for each state. The gray vertical line shows the zero value where
your predicted migration values equal the actual migration values. The yaxis, which ranges from 0 to 50 states or 0 to 13 provinces, ranks the origin states from highest to lowest residual. You can move your mouse over
the dots to see which states or provinces they represent.
The vertical red bars show the break points between classes. You can
select a bar by clicking on the top triangle with your mouse. Holding
down the mouse button on the triangle, move it left or right to set new
class limits. The shading patterns between the bars match those of the
map. When you move the bars, the break points in the boxes below
change 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 this
interactive graphic array and/or the editable boxes to make your final map.
B. You can follow a number of possible strategies in defining the break
points. First, you can define a break point at 0 (gray vertical line on the
graph) that divides the states or provinces into those with positive residuals and those with negative residuals. You could then define two other
break points to separate the small positive residuals from the large, and
the small negative residuals from the large. You can choose to have only
one or two extreme outliers in the large positive and large negative
classes, or you can have many. A second strategy, related to the first,
defines six classes with small, medium, and large residuals on both the
positive and negative side. A third strategy is to define a class that groups
residuals that are close to zero (i.e., that are closely predicted by the
model) regardless of whether they are positive or negative. Fourth, you
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104 Chapter 4. Newton‘s First Law of Migration: The Gravity Model
could use the graphic array to set class break points that divide the data
into “natural classes” or groupings. Vertical groupings in the array indicate a group of states with similar residuals. Setting your break points in
the empty horizontal gaps would avoid putting two states with very similar percentage residuals into different categories (i.e., you could avoid
“splitting hairs”). You can also experiment with the Equal Frequency and
Equal Interval buttons.
Set the break points to make what you consider to be the most informative choropleth map of the residuals. However you choose your limits
justify 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 the
lower limit and Break Point 3 to the upper limit, and positioning Break
Point 2 at 0, it is possible to create a two-color map with one shade for all
positive residuals and another for all negative residuals.
D. At the bottom of the screen, pick a Color Scheme that best portrays the
residual 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 necessary. Sometimes colors look good on the screen, but print poorly. Make
sure 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 with
this assignment.
F. When you have finished, close the Activity 1 and 3 windows, and click
the Exit button at the top right of the Chapter 4 page. If you are on a
campus network, log off your machine. Don’t forget your CD.
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Activity 4: Evaluation 105
Name: _____________________________
Instructor: _________________________
Newton’s First Law of Migration:
The Gravity Model
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, and
justify your choice of color scheme and class break points for the choropleth map
of 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 of
the “sending” power of a state?
4.5. We have said that points that fall along the 45º line on the graph are predicted
accurately by the gravity model. What does it mean if a point is below the line? Above
the line? Which one is overpredicted and which one is underpredicted? (Hint: Look
at the Residual column and compare those values with the location of the points
on the graph.)
4.6. Can you detect any spatial patterns (groups of states) on the map of residuals
that are overpredicted or that are underpredicted? What explanations can you suggest for these patterns?
4.7. Based on your answer to Question 4.6, how would you amend the gravity model
to more accurately predict migration to your state? Would you add any variables to
account for factors other than population and distance?
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DEFINITIONS OF KEY TERMS
Distance Decay The declining intensity of an activity with
increasing distance from its point of origin.
Extreme Value A point on a scatter diagram that is
roughly in line with the main trend but is separated from
the main group of points because of its extremely high or
low value. Contrast with outlier.
Gravity Model A model to predict spatial interaction,
where size (population) is directly related to interaction
and distance is inversely related to interaction.
minus out-migrants divided by the total population, all
times 100. Positive numbers indicate net gain; negative
numbers indicate net loss.
Outlier Point on a scatter diagram that lies far off the
trend line. Outliers on the graph correspond to cases that
are poorly predicted by the model. Outliers are not to be
confused with extreme values, which may lie far from any
other point but which are still close to the best-fitting line
(see Figure 4.13).
Migration A permanent change in residence to outside
one’s community of origin.
Pull Factors Reasons to move to a particular place.
Migration Counterstream Migration that runs opposite
to a migration stream.
Residuals The difference between an actual observed
value of some variable and its predicted value using the
gravity model.
Migration Selectivity The tendency for certain types of
people to migrate. Age, education, and other
sociodemographic characteristics are migration selectivity
factors.
Migration Stream A well-defined migration channel from
a specific origin to a particular destination.
Net Migration Rate The percentage gain or loss of
population due to migration. It is calculated as in-migrants
Push Factors Reasons to move from a particular place.
Scatter Diagram A scatter of dots showing the
relationship between two variables. Each dot on the graph
represents the x and y coordinates of a different
observation or case.
Spatial Interaction Movements of ideas, information,
money, products, and people between places.
FURTHER READINGS
Castles, 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.
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Items to Hand In 107
WEB RESOURCES
Martin, 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 IN
Activity 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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