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Tables and graphs for monitoring temporal crime trends: Translating theory into practical crime analysis advice
Andrew P Wheeler John F. Finn Institute for Public Safety, USA
Abstract This article is a practical review on how to construct tables and graphs to monitor temporal crime trends. Such advice is mostly applicable to crime analysts to improve the readability of their products, but is also useful to general consumers of crime statistics in trying to identify crime trends in reported data. First, the use of percent change to identify significant changes in crime trends is critiqued, and an alternative metric based on the Poisson distribution is provided. Second, visualization principles for constructing tables are provided, and a practical example of remaking a poor table using these guidelines is shown. Finally, the utility of using time series charts to easily identify short- and long-term increases, as well as outliers in seasonal data using examples with actual crime data is illustrated.
Keywords Tables, time series, crime statistics, crime analysis, crime trends
Submitted 04 Oct 2015, accepted 14 Mar 2016
Crime statistics are invariably presented in tables and
graphs. Such statistics are regularly presented in police
departments, either internally at regular meetings or exter-
nally for crime reporting purposes and public information
awareness (Boba Santos, 2012; Manning, 2008; Wilson,
1957). Although several textbooks exist to help guide ana-
lysts in making and presenting crime maps, there is rela-
tively little advice about making tables and time series
graphs oriented specifically towards crime analysts and
police practitioners. This article fills that gap by providing
practical advice on presenting tables and graphs to monitor
temporal crime trends, and aims to translate statistical
advice and the science of how we visualize data into
improved crime analysis reports. This is important to
improve understanding and efficiency not only for crime
analysts, but also for those consuming the graphs, such as
police command staff and the general public.
The main motivation for monitoring crime over time is
typically to identify outlying trends in relation to historical
values, for example, the number of burglaries are high this
month. As an example, Figure 1 displays a table publicly
published by the New York City Police Department
(NYPD). Such tables are standard for CompStat meetings,
as well as disseminating crime statistics in public reports.
These tables are not idiosyncratic to CompStat, however,
and similar ones are used regularly to report crime
When a current crime trend is identified as being sub-
stantively high compared with historical numbers, presum-
ably the police department will take specific action to
attempt to reduce those types of crimes in the future. A
response can be as simple as notifying detectives that a
serial offender may be at large, or taking more extreme
measures such as allocating extra patrols to address the
problem. Showing falls in historical numbers can be used
as a performance tool to see if a particular action has effec-
tively decreased crime. Or the obverse when crime statis-
tics are increasing, as a way to pressure individuals within
Andrew P Wheeler, John F. Finn Institute for Public Safety, 421 New
Karner Rd. Suite 12, Albany, NY 12205, USA.
Email: [email protected]
International Journal of Police Science & Management
2016, Vol. 18(3) 159–172 ª The Author(s) 2016
Reprints and permission: sagepub.co.uk/journalsPermissions.nav
DOI: 10.1177/1461355716642781 psm.sagepub.com
the department to take some action (Eterno and Silverman,
2010). Finally, showing when crime has not substantively
increased compared with historical numbers is just as use-
ful. This would prevent extra resources being devoted to
chasing the noise, or provide a more accurate presentation
of crime statistics to the public. It is often the case that a
few crimes can be turned into a crime wave by the media
Effectively distinguishing what is high or low compared
with historical numbers is often demonstrated by showing a
Figure 1. Example CompStat report publicly released by the NYPD. Reports can be obtained from: www.nyc.gov/html/nypd/html/ crime_prevention/crime_statistics.shtml
160 International Journal of Police Science & Management 18(3)
percentage change. This article first presents reasoning to
explain why percentage change is a very poor statistic to
use when monitoring any time series, especially one of
crime counts. In place of percent change, an alternative
metric, a Poisson z-score, is provided. The article also dis-
cusses data visualization principles when constructing
tables, as well as the utility of graphics to monitor time
series. This will improve the utility of tables and graphs
for both crime analysts and command staff using such
tables internally, as well as presenting the information in
a more effective way to the public.
Use of tables and graphs in policing
Data-driven approaches to reducing crime and improving
safety, with examples such as CompStat (Silverman, 1999)
and hot-spot policing (Sherman and Weisburd, 1995), are
the norm in modern police agencies (Coldren et al., 2013;
Randol, 2014). The actuarial approach to allocating
resources strategically is not new—as far back as 1957,
O.W. Wilson suggested allocating resources to times with
more calls to the police (Wilson, 1957). The advancement
of data technology has changed the capabilities of statisti-
cal analysis, however, from long-term prospective planning
to more real-time analysis. Maps and graphs can be used as
managerial tools, either to take action in the face of an
emerging trend or as a measure of current performance
(Bonkiewicz, 2015; Davis et al., 2015).
The majority of the literature on police use of data has
focused on the adoption of crime mapping (Roth et al.,
2013), and hot-spot policing has regularly been shown to
be effective in reducing crime (Braga, 2001). Although
examples of pin maps can be found before modern mapping
technology (Harries, 1999), the introduction of the specific
crime analyst vocation is likely the foremost instrument to
adopting data-driven policing strategies (Boba Santos,
Less academic attention has been paid to the use of
tables and time series graphs in decision-making, although
they are just as integral and are regularly used in data
analysis. One counter example is Guilfoyle (2015), in
which a survey was conducted in presenting a table of
crime counts comparing one month to the prior month,
along with a percent change. Guilfoyle (2015) found that
the majority of officers used this simplistic information to
infer a more general trend that crime was increasing over-
all, although insufficient data was given to make such an
Whereas Guilfoyle (2015) focused on how statistics in
simple tables are interpreted by police actors, this article
focuses on simple improvements to tables and graphs to
improve the data supplied and the readability. This article
� a metric with various improvements over percent change in monitoring crime counts;
� data visualization principles applied to making tables; and
� constructing time series charts to monitor temporal crime trends.
Although each section has most practical utility to the
application of crime analysis, there are greater lessons in
interpreting temporal trends in crime counts for all police
agents, as well as the general public. In particular, know-
ing how percent change is a poor metric for evaluating
change in crime counts, and the possible swings in random
data, provide a warning for the consumers of crime sta-
tistics. Graphical advice applied to making tables is also
applicable to researchers, because most quantitative
research includes tables, and the advice given is not in
any way particular to crime analysts.
Percent change encourages chasing the noise
As shown in the example table from the NYPD (Figure 1),
percent changes based on prior crime counts are regularly
reported in tables of crime statistics. This section discusses
why using percent change to monitor temporal crime
trends, especially counts of crimes that occur infrequently,
is very problematic. The motivation to monitor temporal
crime trends is to identify when current numbers are out-
liers compared with historical numbers. The nature of per-
cent change makes it very difficult to know how large a
percent change is an outlier.
Any particular statistical estimate has variance. The
variance of an estimate will tell you how often that esti-
mate will have lower or higher measures. As a hypothe-
tical example, imagine a department writes an average of
100 traffic tickets per month. If the variance of that esti-
mate is 102, having a month with 80–90 tickets written
would be quite common. However, if the variance were
lower, say 42, months with 80–90 tickets would be very
rare. Knowing the variance of an estimate allows one to
identify changes that are likely by chance and those that
are not likely by chance.
The variance of any metric, including percent change,
will be some function of the variance of the items that make
up that metric. The formula for percent change is:
Percent Change ¼ Post� Pre Pre
A useful re-expression of this equation is:
Pre � Pre
Pre ¼ Post
Pre � 1
So the variance of percent change is based only on the
ratio of Post and Pre—which in our example are the counts
of crime per some time unit, such as months. The variance
of this ratio is not generally defined, but the more often the
value of Pre is close to zero, the larger the variance
becomes. (A related problem is that the measure is not
defined at all when Pre equals zero exactly.) So, the percent
change metric has a different variance depending on the
how often the crimes occur, and when the crimes occur
infrequently the variance is likely to be quite high (if it can
be calculated at all).
The implications of this point are simple to explain
using some examples. If the prior value is 5 crimes, and
this then increases by 3 to 8 crimes, the percent change will
be ð8� 5Þ=5 ¼ 3=5 ¼ 60%. However, if the prior value happened to be 4 crimes and this increased to 8, what
happens to be a minor difference in only one crime in the
baseline turns into a much larger percent change,
ð8� 4Þ=4 ¼ 4=4 ¼ 100%. Why does not having a defined variance matter? Percent
change makes it very easy to be fooled into thinking there is
a significant increase in crime compared with historical
numbers when really there is no change. This is called a
false positive, and any action taken by the police depart-
ment in response to a false positive will result in wasted
As an example, one might say that a 20% change is cause for concern, and if you knew the variance of the
percent change you could tell how often this 20% cut-off would produce a false positive. Typically, one wants to set
the rate of false positives very low, and standard scientific
conventions are either 5 times in 100 or fewer. With per-
cent changes, one cannot set a value to maintain a consis-
tent false-positive rate; for example, you cannot tell how
often a percent change of 20% or larger happens. It may happen 1 time in 100, 10 times in 100 or much more fre-
quently, even in random data. If one always assigns extra
patrols in response to a significant increase in crime in a
particular area following a monthly meeting, and the false-
positive rate was 1 in 100, you would only assign an extra
patrol by accident typically once every 8 years. However, if
the false-positive rate was 10 in 100, you would assign an
extra patrol by accident around once a year.
To illustrate this, I simulated a set of Poisson distributed
random variables with means of 5, 10, 20, 50 and 100
representing counts of crimes by months over 10 years.1
A Poisson distribution is often used to model counts of
events occurring in a particular interval. Crime data will
never be exactly Poisson distributed, but it is often a rea-
sonable approximation (Maltz, 1996).
By construction, these simulated variables have no
trends or outliers. Percent changes were then calculated
using the same month in the prior year as the Pre value,
so the simulation includes a total of 108 calculations of the
percent change. Figure 2 shows the histograms of these
statistics. Note by construction, this is what happens when
there are literally no changes in the distribution, any per-
cent change that might be flagged as noteworthy would
simply be chasing the noise. Subsequently, any response
by the police department to high percent changes would be
One can clearly see that the smaller mean of the time
series results in a much wider variance of percent changes.
The ranges for the samples here are listed in Table 1. In this
simulation, there is a clear bias for positive changes and,
even though the higher mean time series has a smaller var-
iance, the variance of the estimator still contains what would
likely be considered noteworthy swings of over 20%. That is, even for crimes with an average of over 100 occurrences
per month, percent changes of over 20% are not rare when the underlying distribution does not change.
The positive bias comes from the fact that percent
change is not symmetric. For example, an increase from
4 to 5 crimes is a 25% increase, whereas a decrease from 5 to 4 crimes is only a 20% decrease. This has the undesirable effect that one is more likely to think that a temporal crime
trend is significantly increasing than decreasing when using
percent change. Percent change is more likely to fool you
into thinking crime is increasing than it is decreasing.
If one were using these simulations to set the false-
positive rate, an absolute percent change of over 30% would be needed to have a false-positive rate under 10% for the time series with a mean of 100. The ranges from the
simulation in Table 1 provide an estimate for a false-
positive rate slightly under 1%. In my experience, swings of much smaller percent changes are typically considered
noteworthy. A 10% false-positive rate is not particularly good either. If one were doing a monthly report, a 10% false-positive rate would result in an average of over one
false positive per year. In practical situations, a much lower
false-positive rate would be needed for a metric to be an
effective signal for a police department to take action.
This is a problem because very large percent changes are
needed to be considered unusual compared with historical
numbers, but the statistic is not invariant to the distribution
of the prior time series. It is likely known that time series
with smaller counts have typically wider percentage
changes, but this makes it difficult to know when there is
a significant change in the prior time series. Not just sta-
tistically significant, but meaningfully significant. Consu-
mers of the statistics need to make mental guesses as to
what are reasonably large percent changes, and then vary
those guesses based on the baseline of historical numbers.
This presents an additional cognitive burden on the con-
sumer, and the percent change metric encourages chasing
the noise, especially for increases in crimes, because the
most variable percent changes will, on average, produce the
162 International Journal of Police Science & Management 18(3)
A simple alternative to percent change
Part of the motivation for using percent change in tables
is likely that it is both easily calculated and easily
understood, as well as historical inertia. Here, an alter-
native statistic that is easily calculated and has a better
behaved distribution with which to control false posi-
tives using typical crime data is presented, a Poisson
Presuming the crime series is a Poisson-distributed
random variable, a simple standardized metric can be cal-
2 � ð ffiffiffiffiffiffiffiffiffi
� p Þ
Where � is the long-term average of the distribution prior to Post. I refer to this metric as a Poisson z-score.
This normalizes the distribution of the statistic to have a
mean of 0 and a variance of *1, with higher means being closer to the normal approximation. This is because for
Poisson-distributed random variables, the mean is equal
to the variance. So, taking the square root makes the var-
iance of a Poisson-distributed variable with different means
the same. The multiplication by 2 is to transform the sta-
tistic to have a variance of 1, so that usual normal distri-
bution tables can be used to determine the approximate
false-positive rate. This statistic is also defined when the
prior value is zero. So, unlike percent change, one knows
how large a value of this statistic is needed to flag a change
in the time series as significant. In addition, the value does
not change even if the prior series averages 5 crimes a
month or 100 crimes a month.
Figure 3 presents a set of histograms using the standar-
dized metric instead of percent change using the same
simulated data as presented in Figure 2. Although the nor-
mal approximation is more accurate for the series with a
larger mean, one can see that the spread of the distributions
is more similar between the panels than the percent change,
are symmetric about zero, and are not biased. A typical
outlying value is then simply +2 for a false-positive rate of * 5%.
Considering that analysts are frequently monitoring
many series, a more stringent rule for a change in the prior
Figure 2. Histogram of percent change values for simulated random Poisson data with different means.
Table 1. Ranges of percent change based on simulation.
Mean Low % High %
5 –86 400 10 –73 200 20 –67 123 50 –31 61
100 –26 43
series may be considered. If one uses 3 as the rule to alert a
change as significant this only causes a false-positive rate
of slightly less than 3 in 1,000. (Considering only positive
changes it would be less than 2 in 1,000.)
The utility of the Poisson z-scores over percent change
are then as follows.
� The statistic is invariant to the baseline average in the crime series. So one does not need to make men-
tal adjustments to know what is a significantly large
percent increase based on how small or large the
prior series is.
� Increases and decreases are symmetric, so the statis- tic is not biased to flag increases as statistically sig-
nificant more often than decreases.
� A consistent false-positive rate for all crime series being monitored can be set. So, if one is monitoring
10 different series per week, and one only flags an
increase as statistically significant if it has a Poisson
z-score of over 3, this will produce a false-positive
rate around 2 times in 1,000. This will mean there is
only around 1 false positive in a year for the 10 dif-
ferent series (10 series times 52 weeks is 520 obser-
vations) using the Poisson z-scores. With percent
change, one will likely have a much larger number
of false positives, wasting many more resources on a
In reality, crime data are not exactly Poisson distributed.
But, the + 3 rule for the Poisson z-scores is later shown to have very close to the expected number of false positives
while monitoring a set of five different crimes series by
week over 2 years. This offers a substantial improvement
over percent change.
Although interpreting the statistic is more difficult than
interpreting percent change, it can be presented on a simple
scale showing when to flag an increase or decrease as being
significantly different from historical values. Instead of a
percent change being shown in a table, the Poisson z-score
can be inserted. This can be interpreted as an absolute value
of 2 or more being some evidence of change, an absolute
value of 3 or more being stronger evidence of change, and
an absolute value of 4 or more being quite strong evidence
of change. Positive values then signify an increase in crime,
and negative values signify a decrease in crime.
Making better tables
Just like a poor map, a poor table can hinder the reader from
interpreting the data contained therein correctly. In general,
Figure 3. Histograms of Poisson z-scores for simulated random Poisson data with different means.
164 International Journal of Police Science & Management 18(3)
one thinks of maps and graphs when discussing the princi-
ples of data visualization. But in terms of simply displaying
data, tables are a type of data visualization. They use rows
and columns to display numerical values, instead of the
points, bars or lines on a graph. Consequently, we can take
lessons on how to display graphs effectively and apply
those to how we make tables.
General data visualization principles applied to making
quality tables (or graphs) can be taken from gestalt theories
of visual perception (MacEachren, 2004). Although a
review of all of these principles is beyond the scope of this
article, the main principle of interest is the concept of prox-
imity, items closer together will be perceived as a group.
Using gestalt principles, one can construct tables to make
particular intended comparisons, or identify characteristics
that actively harm making particular comparisons.
In general, one wants to construct the table so that com-
parisons are made within perceived groups, because it
requires more effort to make comparisons between individ-
ual items in separate groups. A simple example is that
numbers are typically organized in columns. It is easier
to calculate the difference between two numbers that are
right aligned in one column than two numbers aligned in a
row. Although this section discusses ideas more specific
than proximity, such as the use of color, gridline placement
and ordering the columns, the general guiding principle in
designing effective tables is that the design should aid the
intended comparisons, not impede them.
As a practical example, this section takes a table and
reconstructs it using these general data visualization prin-
ciples specially oriented towards tables (Feinberg and Wai-
ner, 2011). Figure 4 shows a replica of a table (upper),
which is then reconstructed (lower). The motivating exam-
ple is left uncited, but it was taken from a recent award-
winning example in the statistical reports category of the
International Association of Crime Analysts. Although it is
a specific example, the reasoning around each of the
changes can be applied more generally.
When critiquing a particular data visualization, one
needs to be clear about what specific information the visua-
lization (or table here) is intended to convey (Kosara,
2007). The two main motivations for crime statistics tables
are to identify increases in crime trends and simple report-
ing of numbers (Behn, 2008; Dabney, 2010). The advice
given here is applicable to both.
The most obvious aspect of the table is the gratuitous use of
color. There are several reasons why the color scheme chosen
for the table is problematic. To describe the color scheme
itself, there are five colors in the table; dark blue for the
column and row headers, dark green for large decreases, yel-
low for small decreases, orange for small increases and red for
large increases. There are some inconsistencies in how the
colors are applied, for example, year-to-date homicides
increased from zero in 2012 to two in 2013, but those cells
are colored green and the percent change is colored orange, so
the exact data specifications used for the color scheme are
unclear. There is no legend in the report specifying what each
color represents, so these are best guesses.
For specific critiques of the color itself there are three
problems. First, people with red–green color blindness will
not be able to tell the difference between the green large
decrease and the red large increase anchors (a similar prob-
lem occurs when printing in black and white). Around 8% of the male population have some type of color deficiency
(MacEachren, 2004), with red–green deficiencies being the
Second, the color scheme has no perceptual basis for
rank ordering the values (Moreland, 2009), for example,
one is unlikely to know that dark green means a larger
decrease than yellow, and orange means a slight increase.
Yellow is a poor choice, because individuals frequently
rank the magnitude of colors based on saturation, at least
given certain slices of color ranges (MacEachren, 2004).
That is, darker colors are often taken to mean a larger
Ward 2012 2013 Z 2012 2013 Z
Hom 0 0 0 0 2 3 Rob-Bus 0 1 2 1 4 2 Rob-In 3 3 0 31 46 2 Rape 1 0 -2 5 4 0 Ass-Agg 12 14 1 93 94 0 Larceny 42 36 -1 262 262 0 Auto Th� 4 1 -2 26 25 0 Burg-Res 36 30 -1 191 189 0 Burg-Non 3 1 -1 11 9 -1 Burg-Bus 7 2 -2 23 16 -2 Total 108 88 -2 643 651 0
Year to DateMonth to Date
Figure 4. (Upper) Original replicated table based on an unnamed agency’s regular statistical reports. (Lower) Re-creation of the table based on data visualization principles. The original report did not contain any further descriptions of what the crime cate- gories specifically referred to.
change or be of greater interest, for example, bright red is
more important than light red or light blue. Yellow by its
nature has a high saturation, there is no such thing as dark
yellow, and so it commands disproportionate visual weight
compared with its ranking as only slightly decreasing. Yel-
low also tends to reproduce poorly when projected or in
print—so the yellow on the computer screen for the person
initially making the report is not likely to be the same
yellow as viewed in the report.
Third, color decreases the readability of the text (Few,
2008), which will again be exacerbated if the report is
printed. If a color scheme to visualize a range of values
is to be chosen, I would recommend one based on Color-
Brewer palettes (Harrower and Brewer, 2003), with the
blue to red diverging scheme appropriate for data that can
take on negative or positive values. However, the saying ‘if
you focus on everything you focus on nothing’, is particu-
larly apt here. Coloring the entire table visually emphasizes
everything, and so nothing immediately stands out to the
viewer. If the intent was for the red high increase cells to be
the most salient features of the table, use of the other colors
detracts from that goal.
In the re-creation of the table, the categories with the
biggest Poisson z-score in the year-to-date column are
sorted in descending order. The Poisson z-scores are
rounded to the nearest integer, because it is likely that the
error in using the z-scores does not justify accuracy to
tenths. Frequently, tables include an inappropriate number
of decimal places that decreases their readability, and
Feinberg and Wainer (2011) suggest no more than three
significant digits in the table if possible. Using a +2 rule, several changes would be flagged in the table, but using the
more conservative +3 the only change that is significant is homicides. It is likely that if an average of prior homicides
as opposed to zero were used it would not be flagged as a
significant increase. However, to illustrate how one can
highlight such a difference, a light orange shade is shown
in the year-to-date increase in homicides. This clearly high-
lights the year-to-date homicides against the rest of the
chart. This light orange looks similar to the light gray rows
when printing out in gray scale, so in addition, the values
are bold in that section. Even printed in gray scale, the bold
section stands out against the rest of the table (Feinberg and
Also, the table maintains the order of items based on
violent and non-violent crimes, and includes shared crimes
(e.g. Burg-Res adjacent to Burg-Non) next to each other.
Because the tables might be intended for looking up crime
statistics, it may be advisable to not sort the rows, but rather
to keep them at consistent locations. Otherwise data-based
sorting should be preferred over simple alphabetical order-
ing, because it can show patterns in the data and give pre-
cedence to particular categories. Here, sorting happens to
show that there were slight decreases for all of the month-
to-date property crimes. A clear alternative for crime sta-
tistics is to order the rows of the table by violent and non-
violent crimes. This likely facilitates look up to a greater
extent than does alphabetical ordering.
Zebra stripes are used to delineate headers, every
other row and the total row. It is easier to make com-
parisons down columns than across rows, so here zebra
stripes (Enders, 2007; Lee et al., 2014) help aid the
across the row comparisons, so one does not accidently
compare the month-to-date changes in auto-thefts with
year-to-date changes in burglaries. Although the evi-
dence for the utility of zebra stripes is slight, they have
been shown to be aesthetically preferred by consumers
The header row is simply given a larger font and
this, in addition to the slightly darker background
shade, provides a clear hierarchy for the columns. An
additional technique to create a hierarchy would be to
use a different font type, and bolder sans-serif fonts are
typically used for titles or headers (Lupton, 2010).
Using a larger font is sufficient here, however, and due
to the small space and printing, more exotic typefaces
are potentially problematic (Boba Santos, 2012). In
small type, the serifs in fonts also tend to be printed
poorly, so the sans-serif font Calibri (currently the
default in Excel) is a fine choice, although many others
would be sufficient. Not making the column or row
labels bold allows one to use bold type selectively to
highlight the increase in the year-to-date homicides.
More generally, it makes it easier to read the table and
know what columns and rows correspond to what spe-
Thin borders are selectively used to group month-to-
date and year-to-date comparisons. The use of grid lines
for every cell and row of the table tends to create Moiré or
scintillating patterns (Tufte, 2001). Again, the use of very
light borders groups the types of comparisons the table is
intended to make and prevents inappropriate comparisons
(like the z-score for the month-to-date against the year-to-
date numbers). Numbers are right aligned, but are padded
against the column so that they fall almost directly beneath
the column header. When using grid lines this is important,
because the right-aligned numbers often bump against the
edge of the table.
The use of color in the original table has the opposite
effect to the zebra stripes. It creates regions of similar color,
so it is more difficult to follow a single row (Lee et al.,
2014), and what comparisons are intended is confusing for
the viewer. For this reason, one should not use different
colors within a set of numbers intended to go together and
be compared, as is the case with the green homicide counts,
but the orange percent change in the original table.
166 International Journal of Police Science & Management 18(3)
Time series charts for monitoring crime series
Unfortunately, crime counts are never exactly Poisson dis-
tributed. Often the series have over-dispersion, which
occurs when the variance of the series is greater than the
mean (Berk and MacDonald, 2008; Osgood, 2000). Often
this occurs in crime data when there are many zeroes, and
then spurts of higher activity. Possible mechanisms that
cause this are crime sprees from the same offender(s) (Her-
ing and Bair, 2014), or reciprocating violence (Branting-
ham et al., 2012; Loftin, 1986). A Poisson series assumes
independence between the inter-arrival times of events, and
the above are realistic examples of dependence between
events. Less often a crime series has under-dispersion, the
variance is smaller than the mean, however, this does occur
when a time series shows strong persistence. An example of
a cause of under-dispersion may be a chronic, serial offen-
der operating on a regular time schedule (e.g. a chronic
burglar breaking into a few houses once every month).
Because of these deviations from a Poisson distribu-
tion, the normal approximation of the Poisson z-scores
suggested prior will not be perfectly accurate. In the case
of over-dispersion, the variance of the estimator will be
wider than usual, so there will be a larger number of false
positives. In the case of under-dispersion, the variance
will be smaller.
In light of these complexities, a simple, nonparametric
alternative is suggested: simply graphing the trends.
Figure 5 is the same time series chart of the random vari-
ables discussed above. It is clear from the chart that there
are no trends in the data, and typical month-to-month var-
iations are easily observed given the historical trends. Our
eyes do a better job than the statistics in identifying
abnormalities in the data (Buja et al., 2009; Maltz, 2010).
To illustrate the utility with actual crime data, Figure 6
is a small multiple chart (Cleveland, 1985; Tufte, 2001)
that was initially prepared for a weekly command staff and
intelligence sharing meeting. The actual weekly crime data
are shown as a light red line. The black line represents
the moving average of the prior 8 weeks (approximately
the prior 2 months), and the gray band is the +3 Poisson z-scores based on that moving average. The most recent
week is displayed as a red dot on top of all the other chart
elements, so the current week can be easily seen in refer-
ence to the gray bands.
The graph shows several trends in the high-density dis-
play. First, one can see that the red line is very noisy, but is
mostly covered by the Poisson z-scores, with the exception
of the lower bound for thefts of motor vehicles.2 This
Figure 5. Time series graph of simulated data.
shows with actual data that the Poisson z-scores are a rea-
sonable approximation to flag significant increases. Cover
here is a statistical property, in that the range of the Poisson
z-scores should contain the observed data a given propor-
tion of the time. The graph easily illustrates this property,
the observed data are covered if the red line is within the
gray bands. A metric with poor coverage would frequently
be outside the confidence bands, and so would flag patterns
too frequently as significant changes compared with the
The choice of a prior moving average of 8 weeks is ad
hoc, but has produced reasonable coverage of the historical
data. Effectively visualizing and conveying uncertainty in
the estimates are difficult (Spiegelhalter et al., 2011), but
with the moving average, one can see the typical values for
each crime, see any longer term trends and monitor whether
a particular week is outlying. The y-axis scale is trans-
formed to a square root scale, but labelled in the original
counts. This brings the different series closer to one another
in the small multiple charts, and also shows how the square
root stabilizes the variances of the different series.
Although the embellishments of the moving average and
the error bands introduce complexities into the chart, even a
chart of just the original time series shows a clearer picture
of the current data versus historical values than a single
number. However, the original data are very erratic, a com-
mon occurrence with low count crime data. Thus, the error
bands and the moving average actually simplify the data,
and provide a guide for the eye to identify both long-term
trends and whether the current series is outlying (Tukey,
Over a period of nearly 3 years, burglaries, motor vehi-
cle theft and violent interpersonal (the sum of aggravated
assaults and robberies) merely meander about a constant
mean. This suggests that any weekly variation in these
three series is simply noise, and there is effectively no
evidence to suggest that any of the series has shown sig-
nificant increases or decreases during the period. Any extra
Figure 6. Weekly crime statistics. The light red lines are the observed counts of crime. The black line is the moving average of the prior 8 weeks. The gray bands are plus and minus three Poisson z-scores based on the prior moving average. The red dot is the last observed week. Violent inter. includes aggravated assaults and robberies. Larcenies include only Part 1 larcenies. M.V., motor vehicle.
168 International Journal of Police Science & Management 18(3)
time devoted to analysis or problem-solving for these par-
ticular series in response to short-term fluctuations (up or
down) is equivalent to chasing dots on a time series graph
instead of dots on a map! Larcenies and thefts from motor vehicles show some
interesting crime waves that are partly explained by non-
traditional seasonal trends and the behavior of some serial
offenders. The town has a large enough college presence to
show marked increases when students come to and leave
town. A problem crime is frequently thefts of small elec-
tronics from motor vehicles (Brimicombe, 2012), with a
large proportion being from unlocked vehicles. Increases
in thefts from motor vehicles throughout 2012 brought
increased attention to community groups and the media,
and the thefts precipitously decreased to 10-year lows
beginning in 2013 after one prolific offender was appre-
hended, who later admitted to around 40 thefts in the prior
2 months. This precipitous fall in crime clearly showed that
the arrest and public awareness campaign were effective in
reducing thefts from motor vehicles.
The original simulated time series are flat, but more
realistic crime data will have seasonal trends, for example,
more crimes in the summer than in the winter. A conveni-
ent display for such data is a seasonal chart (Hyndman and
Athanasopoulos, 2014), where the months are on the x-axis
and each year gets a new line. Figure 7 displays a seasonal
chart showing burglaries per month for the years from 2004
through November 2014. The red line shows an outlier year
in 2011, and the blue line shows 2014 until November. In
retrospect, the 2011 outlying increases in burglaries were
due to thefts of copper from vacant buildings (Sidebottom
et al., 2014). They have since declined to an average of
around two per month. One can see that if such charts were
in use as of 2011, the numbers of burglaries in April
through August 2011 were substantially higher than in any
of prior 7 years.
Two years of interest are highlighted using brighter and
thicker lines, whereas past years are presented as lighter
gray lines that so they fade into the background (Kosslyn,
1994; Lofgren, 2012). The many lines are often compli-
cated, but drawing the lines as thinly as possible and mak-
ing them semi-transparent helps disentangle the spaghetti
and focus on the overall patterns. Instead of attempting to
draw confidence bands to represent the error in the prior
series, prior years are used as a guide stick to illustrate the
typical year-to-year variation by month.
This chart easily displays whether the new data is an
outlier compared with the prior years given the seasonality
in the series, as is the case with June 2014. Directly label-
ling the highlighted years within the charts allows the chart
to be printed in gray scale and still easily tell which line is
Both types of chart use the human eye to easily take into
account historical variation in the series. The time series
chart can identify long-term upward or downward trends,
especially with the aid of the moving average, whereas the
seasonal chart can identify if an observed count is an outlier
compared with prior counts around the same season in prior
years. Both give context to the current crime counts,
whereas a simple percent change (or any single metric) in
a table does not illustrate the nature of current rises or falls
in crime statistics. For example, whether the current count
is an outlier compared with historical numbers, or if there is
a long-term increase or decrease in the crime counts. Per-
cent change, in particular, is fragile to whether the baseline
value was high or low. If the prior year happened to be low,
one may spend an entire year having to explain large per-
cent increases. The example graphs as presented easily
bring such long-term trends to light.
Using analytics to monitor spatial and temporal trends is a
regular undertaking in modern police departments.
Although several textbooks are currently devoted to crime
mapping, much less attention has been paid to how to
appropriately monitor crime over time and present those
figures for general use. Rachel Boba Santos’s book, Crime
Analysis with Crime Mapping (2012) is one exception, but
even here much more effort is spent on crime mapping than
analyzing temporal trends. This is despite the fact that
monitoring crime over time is just as regular a task for
crime analysts and police planners.
This article first discusses the problematic aspects of
using percent change to monitor changes in crime statistics,
Figure 7. Seasonal charts of burglaries. The red line highlights 2011, which was a high year due to thefts of copper from vacant properties. The blue line is for 2014 as of November.
and provides an alternative metric to flag statistically sig-
nificant changes assuming the crime series is distributed as
a Poisson random variable. Using simulations demonstrates
how percent changes are problematic. Although crime data
are not likely to be exactly Poisson distributed, using an
example with actual crime data shows how the +3 Poisson z-scores have reasonable coverage rates. Although not
perfect, such an approach offers clear improvements over
percent change, while still being easy to calculate.
This article also discusses data visualization principles
designed to improve presentation in statistical tables, and
recreates a typical crime statistics table to make it more
readable. The article then presents the use of time series
charts to monitor complicated crime trends that cannot be
easily reduced to one simple metric. Two types of time
series graphs are illustrated; one with additional running
means and standard errors to visualize long-term upward
or downward trends, and a second seasonal chart that can
be used to spot outliers quickly even in highly seasonal
Although the article presents the utility of using time
series charts, it is not expected that they will replace the
standard reporting of crime statistics in tables. Tables are
important for reporting exact numbers, and it is the case
that some individuals may simply prefer and better under-
stand tables (Friendly and Kwan, 2011). This is why per-
centage change is shown to be a poor metric, and why the
alternative Poisson z-score is given in its stead. It is also
why data visualization principles in constructing tables are
given. Although such practices are not likely to be easily
changed, articulating the problems with current practices
and offering potential substitutes is the end goal of translat-
ing scientific research into practice.
Other procedures exist to monitor crime statistics, spe-
cifically control charting (Gorr and Lee, 2014; Rogerson
and Sun, 2001), however, visualizing the time series is a
simple and effective alternative. This is not a critique of
control charting, but the complexities of identifying appli-
cable limits with complicated seasonal data is a non-trivial.
Using the formula 2 � ð ffiffiffiffiffiffiffiffiffi
Þ is easily accom- plished in any spreadsheet program (Tollenaar and van der
Heijden, 2013). Plotting the data is easily accomplished in a
spreadsheet (Few, 2011), easily understood and provides a
powerful graphical tool to identify trends or outliers in the
current values of the crime series. Even if one does not
prefer the Poisson z-scores because crime data often do not
conform exactly to a Poisson distribution, graphing the
time-series data is a common-sense recommendation and
should be common place in statistical reports.
The practical advice in this article is most applicable to
crime analysts, but the utility of effectively presenting data
has uses for all consumers of the statistics. It prevents com-
mand staff or crime analysts from chasing noisy crime
trends. It prevents the general public from being misled
by false notions of increasing crime statistics—especially
when using percent change. Presenting statistics in an intui-
tive and simple to digest matter can also aid others in the
criminal justice organization who do not regularly use sta-
tistics to influence their behavior (Payne et al., 2013). One
should not simply generate such numbers rote (Manning,
2008), but mold such reporting to be an effective tool to aid
in making decisions in the organization.
I thank the Chief of Police for allowing me to use data for their
particular jurisdiction. I also thank Janet Stamatel for reviewing
an initial draft of the article. All views expressed in the article are
my own, and are not reflective of the Finn Institute or the police
department that supplied the data for the examples.
Conflict of interest
The author(s) declared no potential conflicts of interest with
respect to the research, authorship, and/or publication of this
The author(s) received no financial support for the research,
authorship, and/or publication of this article.
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172 International Journal of Police Science & Management 18(3)
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6 Ways to Look More Confident During a Presentation by Kasia Wezowski APRIL 06, 2017
Several years ago, colleagues and I were invited to predict the results of a start-up pitch contest in Vienna, where 2,500 tech entrepreneurs were competing to win thousands of euros in funds. We observed the presentations, but rather than paying attention to the ideas the entrepreneurs were pitching, we were watching the body language and microexpressions of the judges as they listened.
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We gave our prediction of who would win before the winners were announced and, as we and the audience soon learned, we were spot on. We had spoiled the surprise.
Two years later we were invited back to the same event, but this time, instead of watching the judges, we observed the contestants. Our task was not to guess the winners, but to determine how presenters’ non-verbal communication contributed to their success or failure.
We evaluated each would-be entrepreneur on a scale from 0-15. People scored points for each sign of positive, confident body language, such as smiling, maintaining eye contact, and persuasive gesturing. They lost points for each negative signal, such as fidgeting, stiff hand movements, and averted eyes. We found that contestants whose pitches were rated in the top eight by competition judges scored an average of 8.3 on our 15-point scale, while those who did not place in that top tier had an average score of 5.5. Positive body language was strongly correlated with more successful outcomes.
We’ve found similar correlations in the political realm. During the 2012 U.S. Presidential election, we conducted an online study in which 1,000 participants—both Democrats and Republicans—watched two-minute video clips featuring Barack Obama and Mitt Romney at campaign events delivering both neutral and emotional content. Webcams recorded the viewers’ facial expressions, and our team analyzed them for six key emotions identified in psychology research: happy, surprised, afraid, disgusted, angry, and sad. We coded for the tenor of the emotion (positive or negative) and how strongly it seem to be expressed. This analysis showed that Obama sparked stronger emotional responses and fewer negative ones. Even a significant number of Republicans—16%— reacted negatively to Romney. And when we analyzed the candidates’ body language, we found that the President’s resembled those of our pitch contest winners. He displayed primarily open, positive, confident positions congruent with his speech. Romney, by contrast, often gave out negative signals, diminishing his message with contradictory and distracting facial expressions and movement.
Of course, the election didn’t hinge on body language. Nor did the results of the start-up competition. But the right kinds of non-verbal communication did correlate with success.
How can you send out the same signals—and hopefully generate the same success? At the Center for Body Language, we’ve studied successful leaders across a range of fields and identified several positions which are indicators of effective, persuasive body language.
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Early in Bill Clinton’s political career he would punctuate his speeches with big, wide gestures that made him appear untrustworthy. To help him keep his body language under control, his advisors taught him to imagine a box in front of his chest and belly and contain his hand movements within it. Since then, “the Clinton box” has become a popular term in the field.
Holding the ball
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Gesturing as if you were holding a basketball between your hands is an indicator of confidence and control, as if you almost literally have the facts at your fingertips hands. Steve Jobs frequently used this position in his speeches.
When people are nervous, their hands often flit about and fidget. When they’re confident, they are still. One way to accomplish that is to clasp both hands together in a relaxed pyramid. Many business executives employ this gesture, though beware of overuse or pairing it with domineering or arrogant facial expressions. The idea is to show you’re relaxed, not smug.
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How people stand is a strong indicator of their mindset. When you stand in this strong and steady position, with your feet about a shoulder width apart, it signals that you feel in control.
This gesture indicates openness and honesty. Oprah makes strong use of this during her speeches. She is a powerful, influential figure, but also appears willing to connect sincerely with the people she is speaking to, be it one person or a crowd of thousands.
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The opposite movement can be viewed positively too—as a sign of strength, authority and assertiveness. Barack Obama has often used it to calm a crowd right after moments of rousing oration.
The next time you give a presentation, try to have it recorded, then review the video with the sound off, watching only your body language. How did you stand and gesture? Did you use any of these positions? If not, think about how you might do so the next time you’re in front of an audience, or even just speaking to your boss or a big client. Practice in front of a mirror, then with friends, until they feel natural.
Non-verbal communication won’t necessarily make or break you as a leader, but it might help you achieve more successful outcomes.
Kasia Wezowski is the founder of the Center for Body Language, the author of four books on the subject, and the producer and director of Leap, a documentary about the coaching profession.
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Copyright 2017 Harvard Business Publishing. All Rights Reserved. Additional restrictions may apply including the use of this content as assigned course material. Please consult your institution's librarian about any restrictions that might apply under the license with your institution. For more information and teaching resources from Harvard Business Publishing including Harvard Business School Cases, eLearning products, and business simulations please visit hbsp.harvard.edu.
How to Give a Data-Heavy Presentation by Alexandra Samuel OCTOBER 16, 2015
Data storytelling has become a powerful part of the communications toolkit, allowing both journalists and marketers to communicate key messages by using data and data visualization to drive articles, blog posts, and reports. But the power of data storytelling isn’t limited to written communication: you can also use data to deliver presentations that are both more credible and more visually compelling.
Knowing how to develop and deliver a data-driven presentation is now a crucial skill for many professionals, since we often have to tell our colleagues a story about the success of a new initiative,
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the promise of a new business opportunity, or the imperative of a change in strategy — stories that are much more compelling when they’re backed by numbers.
In the past four years, data has become a bigger and bigger part of my own presentations, since I frequently speak about data-driven projects like the new rules for the collaborative economy, and what social media analytics can’t tell you about your customers. I’ve enjoyed the luxury of working closely with data analysts, infographic designers, and my own in-house speechwriter, which has helped me pick up some tricks on what it takes to create a successful data-driven presentation.
As with any communication, start by thinking about your audience. Who are you presenting to, and how much do they know about the topic? If you’re presenting data on three different sales strategies to the sales team that’s been testing those approaches, you can plunge right in and show them what worked. If you’re reporting on that same experiment to another part of the organization, you need to provide a lot of context before you drop the bar charts in their laps; otherwise what looks like a clear story to you may simply confuse them. A good rule of thumb is to look at the legend on your charts: if you can’t count on the audience knowing what each item in the legend actually refers to, you need to spend some time on setup before you get to the numbers.
It’s easy to let the data overtake your presentation, so be sure you know the overall story you’re trying to tell, and use charts sparingly to support your story. You’re not trying to subdue your enemy through the sheer volume of data you can bring to bear on your argument; you’re using data strategically, when it provides clear and concrete evidence for the story you’re telling. I’ve found that audiences get overwhelmed by back-to-back data slides, so I try to intersperse charts with slides that convey my key point using images or a very few words of text. Show a photo of a shopping cart, and tell people that you now know which cash register displays are most likely to yield impulse purchases; then show the chart displaying the sales figures for different items. Follow that chart with an image or a few short bullets that emphasizes the actionable insights and implications of your data.
It’s rare that anyone will retain all the actual numbers in your presentation, so think about the words that capture the idea, insight, or conclusion you want them to hold onto. Instead of simply throwing up a bar chart that shows levels of employee engagement versus different working arrangements, build to that key chart with a story about the impact of working arrangements on employee satisfaction — illustrated by actual human examples, if possible.
And if there is a single number that really captures your key point — like “employees who work from home 1 day per week are 30% happier than the rest of our workforce” — then make not just that chart, but that specific data point very prominent in your deck. Highlight it in the relevant chart, and consider giving that single data point its own slide or bullet in your conclusion.
As you present, remember that it takes people some time to digest a chart or data table. Take the time to spell out the story you see in the data so that it’s clear to someone who hasn’t been poring over that dataset for the past six weeks. A simple statement like “in every region except the Southwest,
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email outperforms phone calls as a way of generating leads; in the Northeast, 5% of emails get a response, versus only 3% of phone calls” will help people understand what they are looking at and how they’re supposed to read your chart. Speak slower than you usually do, and consider pausing for a moment mid-chart, to allow people the time to absorb the data; even if you prefer to wait until the end of your presentation for questions, ask if anyone needs you to clarify the chart.
While clarifying statements are helpful, that doesn’t mean you can neglect the visuals. If all you do is produce your charts with a tool like Infogr.am or Tableau — both of which will produce charts that look a heck of a lot better than what Excel spits out — you’ll immediately improve your data-driven presentations. You may still need to restructure or reformat your charts to make them work on screen, however. Even if you’ve used shading to differentiate between categories in a printed document, it will be easier for people to distinguish between on-screen categories if they’re shown in different, contrasting colors.
Make sure your legend and data labels are printed in a large, visible font; if you’ve used an infographic design tool like Infogr.am, which generates beautiful charts but doesn’t let you adjust font sizes, you’ll have to add your own, larger labels when you’re producing your final deck. (My trick is to create those labels as individual text boxes in the color of my chart columns, so I can drop them on top of my columns and hide the original labels.)
If you don’t have the support of an infographic designer in creating your charts, get familiar with the very basic rules of good data visualization, like which types of charts to use for different purposes. And make sure that you don’t violate any data visualization principles when you squeeze your data onto a slide: if you simply can’t fit your entire chart onto a single slide in a way that is readable, it’s better to show highlights than to compromise the clarity of your data. A column chart that shows eight categories of clustered columns is going to be very crowded — but don’t you dare turn it into a set of pie charts instead: if there’s value in comparing categories, you want to keep that as a bar chart where all the categories are aligned on a single base line. Perhaps you don’t really need to show all eight categories — just the five most important ones. Or maybe you will have to break your chart into two successive slides; if so, organize your categories so that the most closely related categories are kept together on each slide.
Lastly, there is a lot of value in leaving people with a physical (or virtual) copy of your charts, so that they can look at the numbers more closely after your presentation. Since data-driven decks and reports tend to get circulated, make sure that any charts you include can stand on their own, without you speaking to them: note the source of your data, make your legend clear, and annotate your charts with callouts that show people how to make sense of a specific data point (“7 in 10 customers chose the blue package”).
If this is starting to sound daunting, don’t let a vision of the ideal data-driven presentation report keep you from using data to present your work or ideas to colleagues or peers. With data storytelling,
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excessively high standards can keep us from seizing the opportunity to make a good story better by backing it up with quantitative evidence.
You don’t have to have the perfect dataset or the world’s most beautiful infographics to make data storytelling a valuable part of your communications toolbox. All you need is to break down the wall that keeps math in one part of your brain, and storytelling in another.
Alexandra Samuel is a speaker, researcher and writer who works with the world’s leading companies to understand their online customers and craft data-driven reports like Sharing is the New Buying. The author of Work Smarter with Social Media (Harvard Business Review Press, 2015), Alex holds a Ph.D. in Political Science from Harvard University. Follow Alex on Twitter as @awsamuel.
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Copyright 2015 Harvard Business Publishing. All Rights Reserved. Additional restrictions may apply including the use of this content as assigned course material. Please consult your institution's librarian about any restrictions that might apply under the license with your institution. For more information and teaching resources from Harvard Business Publishing including Harvard Business School Cases, eLearning products, and business simulations please visit hbsp.harvard.edu.
The park behind James's house used to have a bike trail that was 12.347 kilometers long. Last year, the trail was reduced to 9.432 kilometers to make space for a meditation garden. What is the difference in the lengths of the old and the new bike trails?