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The Multigroup Entropy Index

(Also Known as Theil’s H or the Information Theory Index)

John Iceland1

University of Maryland

December 2004

1 These indexes were prepared under contract to the U.S. Census Bureau.

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Table of Contents

Summary......................................................................................................................................... 3

Data Source..................................................................................................................................... 3

Race and Ethnicity.......................................................................................................................... 4

Geographic Areas............................................................................................................................ 5

Residential Pattern Measures..........................................................................................................6

Dual-Group Entropy Indexes.......................................................................................................... 9

References....................................................................................................................................... 9

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Summary

This website contains three sets of residential-pattern indicators for 1980, 1990, and

2000.

1. The highlighted measure is the “multigroup entropy index,” which is also known as the

multigroup version of Theil’s H or the multigroup information theory index. This is a

measure of “evenness.”

2. “Diversity” scores are also available; these are used in the calculation of the multigroup

entropy index. A diversity score measures the extent to which several groups are present

in a metropolitan area, regardless of their distribution across census tracts.

3. Dual-group entropy indexes are included here, where the reference group consists of all

people not of the main group in question.2 Two-group entropy indexes are computed for

Non-Hispanic Whites, Non-Hispanic African Americans, Non-Hispanic Asians and

Pacific Islanders, Non-Hispanic American Indians and Alaska Natives, Non-Hispanics of

other races, and Hispanics.

Data Source

These indexes are based on data from the 1980, 1990, and 2000 decennial censuses (the

100 percent data). The main data issues involved in calculating racial and ethnic residential

patterns revolve around the definition of racial and ethnic categories, geographic boundaries, and

residential-pattern measures.

2 Dual-group entropy indexes were also included in the 2002 report by Iceland, Weinberg, and Steinmetz. In that

report, the entropy index indicated the segregation of each of several groups (Blacks, Hispanics, Asians and Pacific

Islanders, and American Indians and Alaska Natives) from non-Hispanic Whites.

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Race and Ethnicity

In 1977, the Office of Management and Budget (OMB) issued its Statistical Policy

Directive 15, which provided the framework for federal data collection on race and ethnicity to

federal agencies, including the Census Bureau for the 1980 decennial census. The OMB directed

agencies to focus on data collection for four racial groups – White, Negro or Black, American

Indian, Eskimo, or Aleut; and Asian or Pacific Islander – and one ethnicity – Hispanic, Latino, or

Spanish origin. The questions on the 1980 and 1990 censuses asked individuals to self-identify

with one of these four racial groups and whether they were Hispanic or not.3

After much research and public comment in the 1990s, the OMB revised the Nation’s

racial classification to include five categories – White, Black or African American, American

Indian or Alaska Native, Asian, and Native Hawaiian or other Pacific Islander. An additional

major change was to permit the self-identification of individuals as “one or more races.” While a

small fraction of the population had already been doing so on previous census forms, this new

directive made this practice permissible in data collection activities.

This change naturally challenges researchers to determine the best way to present

historically-compatible data. To facilitate comparisons across time, minority race/ethnicity

definitions that could be rather closely reproduced in the three different decades were used, and

which closely approximate 1990 census categories. Six mutually exclusive and exhaustive

categories were constructed: Non-Hispanic Whites, Non-Hispanic African Americans, Non-

Hispanic Asians and Pacific Islanders, Non-Hispanic American Indians and Alaska Natives,

Non-Hispanics of other races, and Hispanics. Having mutually exclusive and exhaustive

3The Population Censuses have a special dispensation from OMB to allow individuals to

designate “Some Other Race” rather than one of those specifically listed. Because of

Congressional directives, the decennial census questions also ask about specific Asian and

Pacific Islander races (e.g., Chinese).

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categories is essential for constructing a single multiracial index. For Census 2000, this involved

combining the Asian and Native Hawaiian or other Pacific Islander groups. In addition, non-

Hispanic people who identified themselves as being of two or more races in 2000 were also

categorized as “Other” since people could not mark more than one race in 1980 or 1990. Census

2000 figures indicate that 4.6 million, or 1.6 percent of the population, designated themselves as

multiracial (and non-Hispanic). Because of the relatively small number of multiracial people, the

impact of the creation of this category in Census 2000 on segregation is small.4 People who

reported being Hispanic were categorized as such, regardless of their response to the race

question.

Geographic Areas

Residential pattern indexes often measure the distribution of different groups across units

within larger areas. Thus, to measure residential patterns, one has to define both the appropriate

larger area and its component parts. The larger areas here are represented by metropolitan areas,

as these are reasonable approximations of housing markets. These are operationalized by using

independent and primary metropolitan statistical areas, referred to hereafter as metropolitan

areas, or MAs. To facilitate comparisons over time, the definition of MA boundaries in effect

during Census 2000 (issued by the Office of Management and Budget on June 30, 1999) were

4 As a way of testing the sensitivity of the information theory index calculated here to differences

in race categories, an alternative race classification scheme with the Census 2000 data was

tested: instead of the six categories described above, eight were constructed. The two extra were

created by splitting the Asian and Pacific Islander category into two (Asians, and Native

Hawaiians and Other Pacific Islanders), and splitting the non-Hispanic Other category into non-

Hispanic “Other,” and non-Hispanics who marked two or more races. The mean entropy index

for all 331 metropolitan in 2000 was 0.181 using six categories, and 0.180 using the eight

categories, indicating the very small effect of using these two alternatives. The correlation

between the two is over 0.99.

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used. Minor Civil Division-based MAs were used in New England. To address the second

geographic consideration, this analysis uses census tracts. These units are designed with the

intent of representing neighborhoods, are delineated with substantial local input, and thereby a

reasonable choice from a heuristic perspective.

In 2000, there were 331 MAs in the U.S. For this analysis, six MAs were omitted

(Barnstable-Yarmouth, MA, Flagstaff, AZ-UT, Greenville, NC, Jonesboro, AR, Myrtle Beach,

SC, and Punta Gorda, FL) because they had fewer than 9 census tracts and populations of less

than 41,000 in 1980. All other MAs used had populations of at least 50,000 in 1980, which is

typically one of the criteria for defining an area an MA.

Residential Pattern Measures

Residential pattern measures, usually referred to as “residential segregation” measures in

the social scientific literature, have been the subject of extensive research for many years, and a

number of different measures have been developed over time (e.g., see Massey and Denton,

1988; Iceland, Weinberg, and Steinmetz, 2002). Reardon and Firebaugh (2002) note that all

major reviews of such indexes limit their discussion to dichotomous measures (e.g. Duncan and

Duncan, 1955; James and Taeuber, 1985; Massey and Denton; 1988; White, 1986; Zoloth, 1976;

Massey, White, and Phua, 1996). The earliest of the multigroup indexes is the information theory

index (H) (sometimes referred to as the entropy index), which was defined by Theil (Theil, 1972;

Theil and Finezza, 1971).

The entropy index is a measure of “evenness”—the extent to which groups are evenly

distributed among organizational units (Massey and Denton 1988). More specifically, Theil

described entropy index as a measure of the average difference between a unit’s group

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proportions and that of the system as a whole (Theil 1972). H can also be interpreted as the

difference between the diversity (entropy) of the system and the weighted average diversity of

individual units, expressed as a fraction of the total diversity of the system (Reardon and

Firebaugh 2002).

The entropy score, which is a measure of diversity, and the entropy index, which

measures the distribution of groups across neighborhoods, are discussed below. A measure of the

first is used in the calculation of the latter. The entropy score is defined by the following

formulas, from Massey and Denton (1988). First, a metropolitan area’s entropy score is

calculated as:

where Πr refers to a particular racial/ethnic group’s proportion of the whole metropolitan area

population. All logarithmic calculations use the natural log.5

Unlike the entropy index defined below, this partial formula describes the diversity in a

metropolitan area. The higher the number, the more diverse an area. The maximum level of

entropy is given by the natural log of the number of groups used in the calculations. With six

racial/ethnic groups, the maximum entropy is log 6 or 1.792. The maximum score occurs when

all groups have equal representation in the geographic area, such that with six groups each would

comprise about 17 percent of the area’s population. This is typically not referred to as a measure

of “segregation” because it does not measure the distribution of these groups across a

metropolitan area. A metropolitan area, for example, can be very diverse if all minority groups

5 When the proportion of a particular group in a given census tract (Πr) is 0, then the log is set to

0. This is the preferred procedure here, as the absence of a group (or multiple groups) should

result in a 0 increase in the diversity score (where a higher score indicates more diversity).

E()ln[1/]

rr

r1

r

=∑

=

ΠΠ

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are present, but also very highly “segregated” if all groups live exclusively in their own

neighborhoods.

A unit within the metropolitan area, such as a census tract, would analogously have its

entropy score, or diversity, defined as:

where πri refers to a particular racial/ethnic group’s proportion of the population in tract i.

The entropy index is the weighted average deviation of each unit’s entropy from the

metropolitan-wide entropy, expressed as a fraction of the metropolitan area’s total entropy:

where ti refers to the total population of tract i, T is the is the metropolitan area population, n is

the number of tracts, and Ei and E represent tract i's diversity (entropy) and metropolitan area

diversity, respectively. The entropy index varies between 0, when all areas have the same

composition as the entire metropolitan area (i.e., maximum integration), to a high of 1, when all

areas contain one group only (maximum segregation). While the diversity score is influenced by

the relative size of the various groups in a metropolitan area, the entropy index, being a measure

of evenness, is not. Rather, it measures how evenly groups are distributed across metropolitan

area neighborhoods, regardless of the size of each of the groups.

Other multigroup segregation indexes exist, such as a generalized dissimilarity index and

an index of relative diversity. In a detailed review of 6 multigroup indexes (dissimilarity, gini,

entropy, squared CV (coefficient of variation), relative diversity, normalized exposure), Reardon

Ht(E E)

ET

ii

i1

n

=−

⎡

⎣

⎢⎤

⎦

⎥

∑

=

E()ln[1/]

iri ri

r1

r

=∑

=

ΠΠ

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and Firebaugh (2002) conclude that the entropy index is clearly the superior measure. They note,

for example, that entropy is the only index that obeys the “principle of transfers,” (the index

declines when an individual of group m moves from unit i to unit j, where the proportion of

persons of group m is higher in unit i than in unit j). The entropy index can also be decomposed

into its component parts. For these reasons, the entropy index was calculated here.

Dual-Group Entropy Indexes

In addition to the multigroup entropy index, indexes for particular groups are also

available here. These employ a two-group entropy index (H) calculation, which uses the same

formulas specified above, where the distribution of each of six groups in question (Non-Hispanic

Whites, Non-Hispanic African Americans, Non-Hispanic Asians and Pacific Islanders, Non-

Hispanic American Indians and Alaska Natives, Non-Hispanics of other races, and Hispanics) is

compared to the distribution of all other groups combined. In other words, the reference group

for these calculations consists of those who are not of the racial/ethnic group being considered.

Additional discussion and analyses of these indexes is contained in Iceland (2004).

References

Duncan, Otis Dudley and Beverly Duncan. 1955. “A methodological analysis of segregation

indexes.” American Sociological Review 20: 210-17.

Iceland, John. 2004. “Beyond Black and White: Residential Segregation in Multiethnic

America.” Social Science Research 33, 2 (June): 248-271.

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Iceland, John, Daniel H. Weinberg, and Erika Steinmetz. 2002. Racial and Ethnic Residential

Segregation in the United States: 1980-2000. U.S. Census Bureau, Census Special

Report, CENSR-3, Washington, DC: U.S. Government Printing Office.

James, David R. and Karl E. Taeuber. 1985. “Measures of segregation.” Sociological

Methodology 14: 1-32.

Massey, Douglas S. and Nancy A. Denton. 1988. "The Dimensions of Residential Segregation."

Social Forces 67:281-315.

Massey, Douglas S., White, Michael J., and Voon Chin Phua. 1996. "The Dimensions of

Segregation Revisited." Sociological Methods and Research 25, 2 (November): 172-206.

Reardon, Sean F., and Glenn Firebaugh. 2002. “Measures of MultiGroup Segregation.”

Sociological Methodology 32, 1 (January): 33-67.

Theil, Henri. 1972. Statistical decomposition analysis. Amsterdam: North-Holland Publishing

Company.

Thiel, Henri and Anthony J. Finezza. 1971. “A note on the measurement of racial integration of

schools by means of informational concepts.” Journal of Mathematical Sociology 1: 187-

94.

White, Michael J. 1986. “Segregation and diversity measures in population distribution.”

Population Index 52: 198-221.

Zoloth, Barbara S. 1976. “Alternative measures of school segregation.” Land Economics 52:

278-298.