International Journal of Forecasting 19 (2003) 595–601
www.elsevier.com / locate / ijforecast
Simple indicators of crime by time of day
Marcus Felson*, Erika Poulsen
School of Criminology, Rutgers University, 123 Washington Street, Newark, NJ 07102, USA
Abstract
Crime varies greatly by hour of day—more than by any other variable. Yet numbers of cases declines greatly when fragmented into hourly
counts. Summary indicators are needed to conserve degrees of freedom, while making hourly information available for description and
analysis. This paper describes some new indicators that summarize hour-of-day variations. A basic decision is to pick the first hour of the
day, after which summary indicators are easily defined. These include the median hour of crime, crime quartile minutes, crime’s daily
timespan, and the 5-to-5 share of criminal activity; namely, that occurring between 5:00 AM and 4:59 PM. Each summary indicator
conserves cases while offering something suitable to forecast.
 2003 International Institute of Forecasters. Published by Elsevier B.V. All rights reserved.
Keywords: Hour-of-day periodicity; Crime series data
1. Introduction
2. Background
Crime varies more by hour of day than by any
other predictor we know. Such variation is analyzed
all too seldom. Perhaps one reason for this neglect is
that hourly data produce too many categories, 168 h
per week. The result is too few cases per cell (this
loss of degrees of freedom impairs statistical analysis) and too many cells (this leads to very large
tables that are hard to understand).
This paper provides some simple indicators that
help gain a solution to these problems. However, a
larger problem needs additional work—how to think
about hourly variations in crime.
¨
Hagerstrand
(1973) showed how the individual
traverses a path through space–time in the course of
a day. The importance of these movements was
explained in social psychological terms by Bandura
(1985), who coined the term, ‘‘the psychology of
chance encounters.’’ Bandura described the intersection of individual paths in the course of a day and
how these chance intersections can change individual
lives and even history.
However, human ecology teaches us that many
encounters are not so random as one might think. In
his classic work, Hawley (1950) paid close attention
to hourly activity patterns and explained how they
are highly interdependent based on sustenance activities. More generally, Hawley distinguished three
features of time organization: tempos, rhythms, and
timing. A tempo is the number of events per unit of
time; that includes an annual crime rate or victimization rate. A rhythm is the periodicity of a time
*Corresponding author. Tel.: 1 1-973-353-5237.
E-mail addresses: felson@andromeda.rutgers.edu (M. Felson),
epoulsen@eden.rutgers.edu (E. Poulsen).
0169-2070 / 03 / $ – see front matter  2003 International Institute of Forecasters. Published by Elsevier B.V. All rights reserved.
doi:10.1016 / S0169-2070(03)00093-1
596
M. Felson, E. Poulsen / International Journal of Forecasting 19 (2003) 595–601
pattern. The monthly and seasonal cycles of crime
are examples of periodicities and are widely known
among criminologists (Harries, 1980). The hourly
periodicity of criminal behavior is generally known
but under-researched. Timing refers to the coordination or intersection of rhythms. The correspondence
between the rhythm of school activity and the
rhythm of delinquency in the course of a day is an
important example of timing (Felson, 2002).
Timing is more than description alone; it is best
understood in light of theories of how crime relates
to everyday life. For example, the environmental
criminology of Brantingham and Brantingham
(1993) helps us understand how the paths of offenders and victims might cross in space and time.
Geographers of crime also pay close attention to
hourly patterns and further assist us in putting this
information to use theoretically and empirically
(Harries, 1980; Rengert, 1996). The routine activity
approach pays close attention to hourly activities and
their link to crime opportunity (Felson, 2002).
These many theoretical ideas cover both space and
time, but the spatial dimension is far more frequently
researched. The reason for that might be that geographers have devised a variety of tools for mapping
activities in space and for summarizing spatial
processes statistically. Indeed, they have learned to
conserve degrees of freedom with measures of
central tendency and dispersion and with statistical
analyses linking variables to one another over space.
The study of temporal patterns, especially by hour of
day, has lagged behind. That lag is probably the
result of inadequate summary statistics for crime
patterns in time. The purpose of this paper is to
provide some summary statistics that are simple and
easy with which to work.
3. The first task
What is the first hour of the day? From the clock
viewpoint, one starts with 12:00 to 12:59 AM, but
that would ignore what we know about crime. At
that hour, many people are not yet straggling out of
urban bars, and parties are for some at a high swing.
In many places, alcoholic beverage consumption
accelerates after midnight in anticipation of closing
hours, and food consumption may well decline. A
majority of those driving cars or taking public transit
in the early hours of the morning might well have
high blood-alcohol levels. That makes them likely
offenders and targets of crime and ineffective guardians against it. Hence, it makes no sense at all for a
criminologist to treat midnight as the transition to a
new day.
Our first task is to figure out that transition time.
Crime statistics on hourly crime patterns suggest that
5:00 AM is probably the best time we have for the
beginning of a new day, at least for a criminologist’s
purposes. By that time, most of the substance abusers
and partygoers have either fallen asleep or have at
least gone home. The bars are closed. Working
people begin to wake, and a glimmer of light from
homes or sky sends cues that the window of crime
opportunity has largely closed.
Hence, a day lasts from 5:00 AM through 4:59
AM the next morning. We shall work with this
assumption for the duration of this paper, but we
recognize that not all nations, cities, or epochs will
fit this pattern precisely. Nor does crime trail away
equally every day of the week. However, such
variations should not lead one to abandon this useful
convention; it allows criminologists to compare
different places and decades. But first, one needs to
consider the span of data incorporated in formulating
hourly indicators.
4. The second task
In order to study hourly crime patterns, a
criminologist must decide what offenses to summarize and for what broader time period. For example,
one might wish to describe the hourly patterns for all
armed robberies in the city of Houston from 1990 to
1999, or one might wish to compare New York
City’s hourly aggravated assault patterns for September versus October of 2001.
The second task is more complicated than meets
the eye. Many offenses are not readily reported to
the police, so their hourly patterns are not known, or
are subject to so much error that the methods
presented in the current paper are probably unusable.
For example, burglaries are generally coded by the
hour the police are notified. Many people discover a
burglary when they come home after work or a trip.
M. Felson, E. Poulsen / International Journal of Forecasting 19 (2003) 595–601
Thus, hourly burglary data are often missing, unreliable or are coded as ‘‘sometime in the morning.’’ On
the other hand, alarm companies might have accurate
hourly data on alarms being set off and could even
be able to subtract false alarms to produce realistic
hourly burglary data files.
The problem with even the best hourly data,
though, is the voluminous number of categories, with
24 h per day and 168 h per week. That is why
summary indicators are essential for hourly analysis
and forecasting. The purpose of this paper is to
provide those summary indicators.
5. The median minute of crime
Having selected 5:00 AM as the first moment of
the day, we can now devise several simple indicators
for hourly patterns of crime. The first is the median
minute of crime, namely, that minute of the day by
which exactly half of the crimes have occurred. For
example, if the median minute of robbery is 6:13
PM, that means that exactly half the daily robberies
occur from 5:00 AM to 6:13 PM, and the rest from
6:13 PM to 4:59 AM the next morning. This simple
measure of central tendency tells us a good deal. For
example, an entire decade of Houston armed robberies could be summed up with this single indicator,
which in turn gives us an idea of how early or late
these offenses occur. In Scandinavia, one could test
the hypothesis that summer months have a much
later crime pattern than winter months with one
number for each. Using unpublished data some years
ago, the senior author noted that the median hour
was much earlier for crimes in Florida cities with
many retired persons than in other cities with more
normal age structures. Hence, the simplicity of the
measure does not mean it lacks the power to answer
questions. One can also calculate a mean minute as a
measure of central tendency by subtracting 5 h from
the time of crime, finding the mean of those times,
and then adding that time back.
597
could calculate a standard deviation about the mean
mentioned above. We think that quartiles offer a
simpler and more cogent way to study hourly
dispersion of crime and are more appropriate to the
problem at hand. We suggest that the most direct and
clearest way to study that dispersion is to find the
quartile minutes. After the median minute of crime is
known, the first half of the crime day is itself cut in
half by the same method to give the first quartile
minute. The second half of the crime day is then cut
in half to give the third quartile minute. With the
median minute of crime, these divide up the four
crime quartiles over the course of the day. Thus, if
the median minute of crime is 7:00 PM and the first
quartile minute is 4:30 PM, that means that 25% of
crimes occur from 5:00 AM to 4:30 PM, and another
25% from 4:30 PM to 7:00 PM. The third quartile
minute dissects the latter half of the day. Thus, the
first quartile minute, the median minute of crime, and
the third quartile minute give us a good idea of how
crime disperses over the day.
7. Crime’s daily timespan
Once we know the quartile minutes, it is elementary to calculate crime’s daily timespan. This is the
number of minutes between the first and third
quartile minute. Where crime is more dispersed over
the day, the daily timespan is higher. A narrow daily
timespan will be expected for smaller cities with less
extended nightlife. The median minute of crime and
the daily timespan together tell us a lot of information, even though they are but two numbers.
High school students appear to have an early median
minute of crime (around the time they get out of
school) and a narrow daily span of crime involvement (see Felson, 2002). Entirely different
patterns would be expected for older offenders
versus young, active offenders versus those who are
occasional, entertainment districts versus working
versus residential areas.
6. Crime quartiles
8. The 5-to-5 share of offenses
Measures of central tendency of course miss the
dispersion over the hours of the day. Of course, one
We have presented so far four summary indicators
of how crime distributes over the course of a day. To
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M. Felson, E. Poulsen / International Journal of Forecasting 19 (2003) 595–601
take a different tack, we now ask what share of
offenses have occurred by a particular time. We pick
5:00 PM as a cutoff time, since that vaguely tells us
when evening begins. What percent of offenses occur
by that time? We call this the 5 -to-5 share of
offenses. As evening and nighttime crime take over,
this indicator will decline. Technically speaking, this
number represents the percent of offenses that occur
from 5:00 AM to 4:59 PM. An early crime pattern
will push this indicator to higher levels.
9. Demonstration
The police departments of 13 middle-sized American cities have provided us with robbery data for the
years 1999–2001 or parts of those periods. These
cities include Akron, OH; Albany, NY; Cincinnati,
OH; Evansville, IN; Fort Wayne, IN; Hartford, CT;
Lincoln, NE; and Lowell, MA; Plano, TX; Rockford,
IL; South Bend, IN; Springfield, IL; and Tampa, FL.
The 2000 Census indicates that the largest of these
cities is Cincinnati, with a population of 331,285.
The smallest is Albany, with 95,658 inhabitants.
The exact definition of robbery in these data is as
follows: We have used all types of robberies for this
study which includes armed and unarmed robberies
as well as robberies of individuals and robberies of
commercial entities. We have classified these robbery
data by hour of day in order to calculate the
descriptive statistics discussed in this paper. Table 1
illustrates these calculations for the hourly pattern of
robberies in the year 2000 in Albany, NY.1
Although the table does not include every minute
of the day, its 24 h of data make it easy to see that
one-fourth of the robberies occur by the first quartile
minute, 3:00 PM. Another fourth occur by the
median minute of 8:30 PM. Three-fourths occur by
the third quartile minute 12:35 AM. The rest occur
between then and 4:49 AM. The timespan between
the first and third quartile minutes is a full 575 min,
1
This paper neglects standard errors. In future studies, we anticipate greater N’s per city and that this would be less an issue. The
formula for the standard error of a median can be found in
introductory textbooks. That same formula could be applied to
first and third quartile minutes.
Table 1
Illustration of how to calculate descriptive indicators for hourly
robbery patterns, Albany, NY, 2000
Hour of day Number Percent Cumulative Notes
of
of all
percent
incidents robberies
5:00–5:59
6:00–6:59
7:00–7:59
8:00–8:59
9:00–9:59
10:00–10:59
11:00–11:59
12:00–12:59
1:00–1:59
8
8
1
7
9
11
5
15
16
1.97
1.97
0.25
1.72
2.22
2.71
1.23
3.69
3.94
1.97
3.94
4.19
5.91
8.13
10.84
12.07
15.76
19.70
2:00–2:59
3:00–3:59
16
22
3.94
5.42
23.65
29.06
4:00–4:59
5:00–5:59
18
9
4.43
2.22
33.50
35.71
6:00–6:59
7:00–7:59
8:00–8:59
18
20
35
4.43
4.93
8.62
40.15
45.07
53.69
9:00–9:59
10:00–10:59
11:00–11:59
12:00–12:59
31
14
28
22
7.64
3.45
6.90
5.42
61.33
64.78
71.67
77.09
29
18
28
18
406
7.14
4.43
6.90
4.43
100
84.24
88.67
95.57
100.00
1:00–1:59
2:00–2:59
3:00–3:59
4:00–4:59
Total
afternoon hours
in boldface type
first quartile
minute 3:00 PM
5-to-5 share of
robberies 33.5%
median minute
8:30 PM
third quartile
minute 12:35 AM
daily timespan
575 minutes
or 9.5 h. About a third of the robberies occur by 5:00
PM, as indicated by the 5-to-5 share.
These indicators prove quite useful for comparing
the 13 cities. Although all of these cities have
something in common with regard to hourly robbery
patterns, they still differ in noticeable ways. Table 2
presents the descriptive indicators for the 13 cities,
all calculated in or around the year 2000. The table
orders the cities by the magnitude of their daily
timespans. For example, Albany’s timespan was 575
min, or 9.5 h. On the other hand, Springfield, IL, had
a daily timespan of only 402 min, or 6.5 h.
The first and third quartiles capture additional
M. Felson, E. Poulsen / International Journal of Forecasting 19 (2003) 595–601
599
Table 2
Descriptive indicators for hourly robbery patterns in 13 cities, 1999–2001
City*
Year
First
quartile
minute
Median
minute
Third
quartile
minute
Daily
timespan
(min)
The 5-to-5
share of
robberies (%)
Base
number of
robberies
Albany, NY
Evansville, IN
Tampa, FL
Cincinnati, OH
South Bend, IN
Akron, OH
Fort Wayne, IN
Rockford, IL
Hartford, CT
Plano, TX
Lowell, MA
Lincoln, NE
Springfield, IL
2000
2000
2000
2000
1999–2000
1999–2000
2000
2000
2000
1999–2000
2000–2001**
2000
2000
3:00 PM
3:30 PM
3:14 PM
2:45 PM
2:31 PM
3:18 PM
2:54 PM
3:44 PM
3:20 PM
3:00 PM
3:00 PM
5:08 PM
4:21 PM
8:30 PM
9:14 PM
9:00 PM
8:12 PM
8:00 PM
9:00 PM
8:32 PM
8:30 PM
8:16 PM
7:52 PM
7:00 PM
9:50 PM
8:30 PM
12:35 AM
12:58 AM
12:25 AM
11:45 PM
11:23 PM
12:00 AM
11:33 PM
11:59 PM
11:30 PM
10:55 PM
10:48 PM
12:22 AM
11:03 PM
575
568
550
540
532
521
519
494
489
475
468
433
402
33.5
29.3
30.4
34.2
33.6
31.9
32.2
27.5
30.2
32.6
30.1
24.7
27.9
406
133
2199
1533
801
1418
367
298
872
218
246
150
269
Note that the 5-to-5 share of robberies refers to those occurring from 5:00 AM to 4:59 PM.
* Cities are ordered by magnitude of their daily timespans.
** Lowell, MA, robberies include April 2000 through September 2001.
information not measured by the timespans. For
example, one-fourth of South Bend’s robberies
occurred by 2:31 PM, while in Lincoln, Nebraska,
the same share was not achieved until 5:08 PM. The
third quartiles varied rather less, but still were not
equal across cities. Lowell, MA, saw three-fourths of
its robberies occurring by 10:48 PM, while Evansville, IN, did not reach that mark until nearly 1:00 AM.
As Column 4 indicates, these same cities had the
earliest and latest median minutes of robbery: 7:00
PM for Lowell and 9:14 PM for Evansville. On the
other hand, the latest median minute was for Lincoln,
NE—9:50 PM. That was despite its being second
lowest in daily timespan.
The 5-to-5 share of robberies provides a somewhat
different summary of robbery time patterns. The
highest value on this indicator is calculated for
Cincinnati, with 34.2% of robberies occurring by
5:00 PM. Although Rockford, IL, is in the middle of
the distribution on the other indicators, it had one of
the lowest percentages on the 5-to-5 indicator.
crime and making comparisons. We only considered
one crime and a limited range of mid-sized cities, but
we believe that these indicators can in the future
assist researchers in describing and predicting how
crime distributes over time.
10.1. How crime timing distributes within cities
One might predict that entertainment districts of
cities will tend to have both later median minutes
and a wider timespan of crime on weekends, but a
narrower timespan Monday through Thursday. Business districts might have earlier median minutes, a
very early third quartile point, and narrower timespans. Residential areas would probably vary by
proximity to central and shopping areas and by
commuting patterns. Areas near high schools will
tend to have earlier median minutes and narrower
timespans. However, crimes carried out specifically
by more active young offenders might have an
earlier median minute but wider timespan.
10.2. How crime timing distributes among cities
10. General implications
We believe that these descriptive indicators serve
as useful tools for describing hourly patterns of
Cities with an older age structure will probably
have earlier median minutes and narrower timespans.
Cities with greater variance in age will probably
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M. Felson, E. Poulsen / International Journal of Forecasting 19 (2003) 595–601
have wider timespans for crime. Earlier bar closing
laws when enforced may produce earlier crime
medians. Cities with more liquor consumption in
cars and outdoors will tend to have later medians and
perhaps wider timespans, too. Cities with earlier
store closing hours will tend to constrain their crime
patterns in time as well.
10.3. How crime distributes among nations
Nations in the northern part of the Earth would
probably have, during the summer, later median
minutes and wider timespans for both property and
violent offenses. However, crimes of violence during
their winter months might well depend upon liquor
policies and enforcement. Nations with indoor garages will tend to constrain later car thefts, while
those lacking indoor parking will tend to have later
hours and wider timespans of auto theft. Gun availability will probably tend to widen the daily timespan
of robbery by making it easier for an offender to
accost someone and quickly succeed, even in light or
dusk.
10.4. Significance for policy
Knutsson (1994) has shown a major discrepancy
between the hourly patterns of crime and the hourly
levels of police assignment. That point offers us
strong evidence that ignoring these ‘‘details’’ leads to
serious waste of resources. Not only does demand
for service vary greatly over the day, but the types of
service demanded also vary. Earlier in the day, truant
officers and school crime officers might be needed.
Later, drug control expertise might be most relevant.
Still later, alcohol management personnel are central.
11. Implications for forecasting
Some years ago, the senior author discovered that
forecasting crime from 1963 to 1975 depended on
studying trends in crime settings and crime timing
(Cohen & Felson, 1979). The dispersion of activities
away from family and household settings produced a
major crime wave. That paper pointed towards a
forecasting strategy that emphasized time patterns of
activity in spirit, but lacked the data to carry out such
forecasting directly. In recent decades, substantially
more data on crime by hour of day has become
available. Such data can suitably be aggregated
according to the rules presented in the current paper.
As a result, we suggest that crime forecasting
strategy shift and make use of what we now know
about shifting activity patterns in the course of daily
life.
The central purpose of this paper is to argue that
forecasting strategies based on monthly, quarterly, or
annual crime totals miss the essential dynamic in
crime rate trends. Such breakdowns might be suitable for studying labor markets, housing starts, and
other economic shifts and cycles, but are less than
ideal for studying and forecasting changes in crime.
The econometric roots of crime forecasting, including that carried out by the senior author, can hinder
more than they help. The time is to move beyond
these roots and to recognize that crime has its own
dynamics, driven by the daily course of activities,
shifting by the hour.
That is not to say that monthly, quarterly, and
annual crime statistics should be dropped in forecasting. Indeed, summing hourly data to these larger
levels provides a way to have your small statistics
and aggregate them, too. Thus, an annual summary
of median hour of burglary can be calculated and
then put into a time series for a statistical analysis
over a longer timespan, taking into account the small
features of daily crime that drive crime rate trends
and cycles, even over the years.
A secondary purpose of this paper is to argue that
we no longer need to rely solely on spectral analysis
as a means for studying crime over detailed time.
This is not to deny its benefits when one lacks
hypotheses and a priori knowledge about the various
trends and cycles, nor is this to deprecate its beauty
in pulling out various cycles statistically, ones that fit
roughly what we know about daily activity patterns
over the calendar. However, we know much more
today about how the specifics of daily crime congeal
and disperse over time and, hence, how to build both
independent and dependent variables explicitly into
forecasting models.
The third purpose of this paper, by no means
small, is to help relieve the central frustration of
many forecasters who, perhaps, are tired of being
ignored by larger policy makers and substantive
M. Felson, E. Poulsen / International Journal of Forecasting 19 (2003) 595–601
scholars. These groups are not oriented towards the
more advanced methods and more difficult indicators. The simple ones presented here can help
sophisticated forecasters communicate beyond their
own ranks, while at the same time making good use
of forecasting methodologies.
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Biographies: Marcus FELSON is author of Crime and Everyday
Life, (Sage Publications), now in its third edition, and has
developed the ‘‘routine activity approach’’ to crime analysis. He is
also co-author (with Ronald V. Clarke) of Opportunity Makes The
Thief, published by the British Home Office. Professor Felson
graduated from University of Chicago and received his graduate
degrees from the University of Michigan. He is Professor of
Criminal Justice at Rutgers University.
Erika POULSEN is a PhD candidate in the Geography Department at Rutgers University, and is the research director for the
Crime Mapping Research Lab in the School of Criminal Justice,
Rutgers University. Her research involves applying geographic
techniques and methodologies for the spatial analysis of crime.