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Technical ReportJune 1998, NCJ 167881 Displaying Violent Crime Trends Using Estimates from the National Crime Victimization Survey
By Michael D. Maltz, BJS Fellow Contents
The National Crime Victimization Survey Appendix I. Addtional information on basic statistical
principles HighlightsTrends in violent victimization, 1973-96
Note: Violent crimes included are rape and sexual assault,
robbery, aggravated assault, and simple assault. Data about murder are
not included because they do not come from a sample survey where precision
can be measured.
Intended for an audience that is not trained in statistics,
this report presents statistical information in a nontechnical format by
using graphical displays of violent crime trends which include the degree
of precision of these estimates. In addition, it discusses --
The National Crime Victimization Survey
Initiated in 1972, the National Crime Victimization Survey
(NCVS) is an annual sample survey of households in the United States conducted
by the Bureau of Justice Statistics (BJS). Currently, 45,000 households
are in the sample. All household members age 12 or over, or approximately
94,000 residents, are interviewed twice yearly about incidents in which
they were the victims of a crime during the previous 6 months.
The NCVS was created to obtain information on both crimes that are reported
to the police and those that are not. It complements the police-reported
crime data found in the FBI's Uniform Crime Reports (UCR). Because the
UCR and NCVS programs are conducted for different purposes, use different
methods, and focus on somewhat different aspects of crime, the information
they produce together provides a more comprehensive panorama of the Nation's
crime problem than either could produce alone. (For additional information
about the NCVS and the UCR, see The
Nation's Two Crime Measures.)
The NCVS measures property crimes and the violent crimes of rape and
sexual assault, robbery, aggravated assault, and simple assault. Because
the NCVS data are the result of interviews with victims, the survey does
not provide data about the violent crime of murder. The FBI's UCR serves
as the source of murder data.
Frequently it is not possible or practical to survey everyone
in a population, particularly if the population is the Nation as a whole.
Sampling, a commonly used technique, gathers information from a portion
of a population and develops estimates that can be generalized to the whole
population. Most reputable survey organizations select people to be interviewed
at random to ensure that their sample is representative of the entire population.
The entire population need not be examined to get a good reading of its
characteristics. A sample from the population would do. For example, to
determine if there is enough salt in a well-stirred bowl of soup, a teaspoonful
will suffice. Moreover, the sample size is independent of the population
size. A single teaspoonful would also suffice to check the salt in a large
vat of (well-stirred) soup.
Sampling has a number of advantages:
The second is frequency of occurrence of the behavior being measured.
For example, election-year polls are sample surveys in which a small sample
of voters (usually about 2,000) is asked about their candidate preferences.
A sample of this size is adequate for this purpose because most people
voice a preference. But a much larger sample is needed to estimate violent
crime rates such as that for robbery because, despite their frequency,
the vast majority of people do not experience a violent victimization
in any 6-month period. Since the sample used to estimate victimizations
in the NCVS must be large enough to capture a sufficient number of incidents,
the NCVS interviews 94,000 individuals.
The survey is administered to a random sample of the population, which
results in two benefits: first, the collected data will be more likely
to reflect the characteristics of the entire population; and second, the
precision of the estimates generated by the sample can be calculated,
to determine how close they come to the true numbers. This precision is
important in comparing two estimates from sampled data. Without knowledge
of this precision, conclusions cannot be drawn from sampled data.
Statistical significance is a standard applied to a comparison of two estimates. For example, surveys of voter preferences are usually reported along with their margin of error, such as "plus or minus 3 percentage points." If the survey results in one candidate receiving 46% of the expected vote and another candidate receiving 44%, it might be reported that "the election is too close to call." Given the 3 percentage point margin of error, a statistician would say that "the difference between the two candidates is not statistically significant." Trends in violent victimization rates, 1973-96
Note: The violent crimes included are rape and sexual
assault, robbery, aggravated assault, and simple assault. The light gray
area indicates that because of changes made to the victimization survey,
data prior to 1992 are adjusted to make them comparable to data collected
under the redesigned methodology. Data for 1995 and beyond are based on
collection year (see Criminal Victimization
1996: Changes 1995-96 with Trends 1993-96 ).
For the NCVS, the number of people who report violent victimizations
to interviewers can be quite small, which limits our ability to declare
year-to-year changes as statistically significant. The figure above shows
the estimated violent crime victimization rates from 1973 to 1996. In the
figure, violent crime appears to be increasing from 1992-93 and 1993-94
then decreasing from 1994-95 and 1995-96. By assessing the precision of
these estimates -- as shown below -- conclusions can be drawn that there
were statistically significant decreases from 1994 to 1995 and 1995
to 1996. However, the individual year-to-year increases from 1992 to 1993
and 1993 to 1994 were not statistically significant. Therefore,
a simple chart of the best estimate like the one above is inadequate for
drawing conclusions about year-to-year changes.
Not all statistically significant findings may be substantively
significant -- important in the context of the subject matter studied.
For example, in 1996 the daytime theft rate was 39.7 per 1,000 households
and the nighttime theft rate was 37.5. The difference between these rates
is statistically significant. Substantively, however, there is really
little difference since households are about as likely to experience a
theft at either time of day.
Sometimes people assume that large differences are statistically significant
because they would be substantively significant if they were true. For
example, the NCVS estimates that the rate for rape and sexual assault
dropped considerably, about 17%, from 1995-96. A drop of this magnitude
would be substantively significant if it were statistically
significant. However, because it is based on a relatively small number
of incidents, it is not statistically significant. The data do
not provide sufficient evidence that this decrease occurred. (See chart
below.)
Statistical significance only has meaning when referring to a random
sample, although it is often misused with samples that are not random.
Trends in violent victimization, 1973-96
To view the data in this chart, go to table
1.
These figures depict both the NCVS estimates of violent
crime and their precision. They display the same trend data as in
the chart above but the scale is changed to highlight the trend line. The
first figure depicts only the best estimates based on the NCVS sample and
does not reflect the range of possible values where the actual number could
fall.
Each bar in the middle figure shows the range within which the true victimization
rate is likely to fall for that year. Because the estimates are based
on samples, their precision depends on the sample size: the larger the
sample, the better the estimate and the smaller the range bars. The samples
were larger before 1992, which is why the earlier range bars are shorter
than those after 1992. (The 1992 range bar is very large due to the size
of the sample that year, as explained below.)
The bars reflect the range within which the true rate is likely to fall.
When the bars are shorter, there is a greater likelihood that the true
rate will fall close to the best estimate. There is a considerable
likelihood (68% probability) that the true victimization rate lies within
the range represented by the darkest segment of the bar. There is a greater
likelihood (90%) that the true victimization rate lies within the expanded
range represented by the two darkest segments of the bar. The full bar
includes the range within which the true value is highly likely
(95%) to lie. For example, while the best estimate in 1996 is 42 violent
crime victimizations per 1,000 persons age 12 or over, there is a 95%
likelihood that the true value of the victimization rate lies between
40 and 44.
Some year-to-year changes are so large that contiguous bars do not touch
(1980-81, 1982-83, 1990-91, 1994-95, and 1995-96), suggesting statistically
significant increases and decreases. Where there is a lot of overlap (1973-76
and 1986-90), the year-to-year changes may be too small to be statistically
significant.
The bottom figure, which overlays the range bars with the trend line,
puts the trend line in context. Even though the victimization rates have
a range of possible values, general trends are readily apparent. Violent
crime rates increased from the early 1970's to the early 1980's, then
fell until around 1986. For several years in the late 1980's, violent
crime rates were stable, but increased in the early 1990's and fell after
1994 through 1996.
To view the data for this chart, go to table
2. To determine whether a change from year to year is statistically significant, the precision of the year-to-year changes must be assessed. The figure above depicts the estimated annual percent change in violent crime rates from 1973-74 (the top bar) through 1995-96 (the bottom bar). For example, from 1990 to 1991 violent crime increased by an estimated 9% and from 1995 to 1996 it decreased by an estimated 10%. The range bars represent the range within which the true annual percent change from year to year is likely to fall. If a bar does not cross the No change line, we are reasonably certain a change occurred. If a bar crosses the No change line, there is a possibility that there was no change. The degree of certainty depends on how much of the bar crosses the line. BJS standards, based on general social science practice, specify the degree of certainty that is acceptable with sampled data. For the most part, the BJS standard of confidence is 95%, meaning there is a possibility that the difference may be due to chance but that this possibility is less than 5%. BJS terms findings in the 90% range as marginal or as indicating some evidence of a change. The 68% range is presented because it is a standard used by statisticians. (See Appendix I for further explanation.) Year-to-year changes in the NCVS violent crime estimates often are not statistically significant. Based on the 95% standard, there was a statistically significant change in the violent crime rates for 8 of the 23 periods (1976-77, 1979-80, 1980-81, 1982-83, 1985-86, 1990-91, 1994-95, and 1995-96). To highlight the significant increases and decreases, the range bars are outlined and the estimate is represented by a large dot. There is some evidence that the violent crime rate increased from 1986 to 1987, but the likelihood that this change occurred is not as great as that for the other years because the last segment of the range bar (between 90% and 95%) crosses the No change line rather than clearing the line. If the range bar only extended to the 90% level (the two darkest segments of the bar), the bar would not cross the No change line. For this year, the estimate is marked with a black square. The range bars for all of the other periods intersect the No change line at a segment representing less than 90% probability (within the two darkest segments). Although a change may have occurred, the probability of such a change does not meet the BJS minimum standard of certainty. Therefore, these changes are not statistically significant and these estimates are marked with a small dot.
*The change in murder rates is presented as a point because
the rates are not derived from sampled data and their precision cannot
be calculated. Murder rates are for all ages.
The figure above shows the 1995-96 percent change in the
victimization rates for the category of total violent crime and the types
of crime that comprise it: rape and sexual assault (hereafter referred to
as rape), aggravated assault, simple assault, robbery, and murder.
The crime categories are displayed vertically according to their 1996 rates.
The highest rate is total violent crime (sum of all types). Among the crime
types, the rate for simple assault is the highest and the rate for murder
is the lowest.
As in the prior figure, the range bars represent the range within which
the true percent change from year to year is likely to fall. Murder is
included for comparison. The value for the change in murder rates is given
as a point (represented by a diamond) and not a range of estimates, because
murder rates are derived from nonsampled data and consequently their precision
cannot be calculated.1
The range bar for simple assault is clear of the No change line,
so there is a 95% likelihood that it declined from 1995 to 1996. The best
estimate of the decrease was 11%. Since the No change line intersects
the darker segments of the range bars for rape, robbery, and aggravated
assault, the BJS standards of certainty are not met, so it cannot be concluded
that a change in these rates occurred from 1995 to 1996.
The length of the range bars varies considerably from crime to crime,
depending on the rarity of the event. There are many fewer rapes than
simple assaults, and therefore the range bar for rape is much longer than
that for simple assault, indicating less certainty in the estimate.
Note that even though the rape rate appears to have decreased the most
(over 17%), its range bar crosses the No change line inside the
90% region. Therefore, the evidence for a decrease is too weak to accept,
because there is sufficient likelihood that there was actually no change
-- or even a slim possibility of an increase.
With the current rape rate and NCVS sample size, it could not be concluded
that there was a year-to-year change in the rape rate unless the change
in the rape rate was greater than 23%. Conversely, for a 10% change in
the rape rate to be statistically significant, a sample of over 500,000
U.S. residents, more than 5 times the current sample size, would be needed.
This appendix describes some of the basic principles used in this report. It is intended to be introductory rather than exhaustive.
The ranges represented by the range bars were calculated
using methods based on statistical inference. The examples given below are
based on a simple random sample, in which every member of the population
being studied has the same probability of being in the sample.2
Suppose a city had an equal number of males and females. If 64 people
are randomly selected, approximately 32 would be expected to be female.
It would not be surprising if the sample actually contained between 27
and 37 females, but it would be extremely unlikely if all 64 turned out
to be female: that would be equivalent to tossing a coin 64 times and
getting tails each time.
A measure of this likelihood is the standard error (SE), which
reflects how close to the true value (in this case, 32) the estimate
is expected to be. It is not expected that a sample will always produce
that number, but it should be close.
The probability that the estimate will fall within a range can be calculated
using standard errors. According to the methods of statistical inference,
if several samples were drawn from the population, 68% of them would fall
within one standard error of the true value. In this example, with a sample
size of 64, the standard error is 4, so 68% of the time the expected number
of females in the sample would be 32 + or - 4 or between 28 and 36.3
Another way of putting it is that if 1,000 different
samples of size 64 were drawn from the city's population, in about 680
of them the number of females would be between 28 and 36.
In addition, about 90% of the samples would fall within 1.645 standard
errors of the true value and that 95% of the samples would fall within
1.96 standard errors of the true value. Social scientists generally report
findings based on the probability that the estimate occurred within a
range defined as the confidence interval. BJS usually reports two confidence
intervals, the 90% and 95% ranges, to reflect its standards of confidence;
in this report, we also include the 68% range to show one standard error.
The table below gives the ranges in terms of the number of females expected
in samples of size 64, with different levels of confidence.
Number of females expected in a sample, for different confidence levels
More often the value of interest is not the expected number
of occurrences but the percent of occurrences. In this
example the most likely percent of occurrences is 50% of the sample (32
of 64). Sixty-eight percent of the samples are expected to be within 1 standard
error (4/64, or 6.25%) of 50%.4 The
following table shows how the expected range in frequency of occurrence
varies by confidence level.
Percentage of females expected in the sample, for different confidence
levels
This table shows two percentages simultaneously, but they
are entirely different: the first column refers to confidence levels expressed
as percentages and the third column refers to the percentages of females
in the sample. The first line in the table indicates that that 68% of the
time a random sample is drawn, it is expected to contain between 43.75%
and 56.25% females. The table also shows that 95% of such samples will include
between 37.75% and 62.25% females; or conversely, that fewer than 5% of
the samples will be outside this range.
Usually the sample percentage is close to the actual percentage in the
population; the larger the sample the more accurate the estimate. This
is as true for measuring criminal victimization as it is for population
characteristics. Even though the actual percentage of people who were
victims is unknown, a sufficiently large sample will produce a sample
percentage quite close to the actual percentage.
The example above shows that, given the frequency of occurrence of a
characteristic in the population, the frequency of occurrence of
a characteristic in the sample can be estimated. The more important
problem is the inverse problem of statistical inference: given
the frequency of occurrence of a characteristic in the sample,
what is the frequency of occurrence of that characteristic in the population?
The question answered by the NCVS is, given the number of incidents captured
by the survey, what is the crime victimization rate for the Nation as
a whole?
Again, the standard error is used to gauge how close the estimate is
to the actual frequency of occurrence. For example, if the people in the
sample experienced a rape rate of 1.4 incidents per 1,000 people, then
the actual value in the population is close to this, or is at least within
the relevant confidence interval. In the 1996 NCVS, about 132 of the 94,000
people sampled experienced a rape, which translates into a rate of 1.4
per 1,000 and a standard error of .14 per 1,000 people.5
Therefore, the rate of occurrence of rape in the
U.S. population is between about 1.26 and 1.54 per 1,000 at the 68% confidence
level, as shown below.
Expected rate of rape victimization, for different confidence levels
The same approach can be used in analyzing year-to-year
changes. The estimated 1995-96 change in the incidence of rape was a 17.65%
reduction (or -17.65%), with a standard error of 11.94%. This would produce
the following confidence intervals:
Sample percent change in the rate of rape victimization, for different
confidence levels
Note that at both 90% and 95% confidence levels, the ranges include zero.6 Therefore, the possibility that there was no true change cannot be excluded so these results are not statistically significant at those confidence levels. The apparent reduction of 17.65% in rape incidence may just reflect normal variation between the sample and the population. When the confidence interval does not include zero (as with simple assault in the earlier figure), the year-to-year change is (statistically) distinguishable from zero and therefore is statistically significant.
In the above examples no mention was made of the size of
the population, because the sample size is not based on the size of the
population from which it was drawn. Estimating the victimization rate in
the United States requires the same sample size as estimating the victimization
rate (at the same precision) in, say, Illinois.
To understand why this is so, suppose a jar contained black and white
beans that were thoroughly mixed.7
If a cupful of beans (the sample) was scooped from
the jar, close to the same color proportions would be found in the cup
as in the jar. Now suppose there was a carload full of thoroughly mixed
black and white beans in the same proportions as in the jar. A cupful
scooped from the carload would have close to the same color proportions
as the carload, and the proportions of the two cups would be expected
to be equally close to the proportions of their "parent" populations.
In other words, the size of the population does not affect the proportions
in the cup, as long as the beans are thoroughly mixed. Randomly sampling
people throughout the country is the logical equivalent of thoroughly
mixing the beans; since people around the country cannot be "mixed"
to distribute all different types of people according to the needs of
the survey, the survey "moves around" to accomplish the same
objective.
If the size of the sample is close to the size of the population, the
precision improves. For example, suppose a sample of 94,000 (as for the
NCVS) is used, but the city being sampled has a population of 94,000 as
compared to the 280 million population of the United States. In this case,
the sample's victimization rate would be precisely the city's true victimization
rate, and the standard error would be 0 (perfect precision). Most samples
are only a small fraction of the population. The standard error starts
to shrink when that fraction approaches 20% of the population.8
The NCVS sample size is sufficiently large to permit a limited amount
of disaggregation, depending on the frequency of the event; the larger
the sample size, the larger the number of incidents and the greater the
ability to estimate the variation in rate by subcategories. The approximately
132 rape victimizations in the 1996 sample may be adequate to furnish
statistically significant differences by race, but is too small to determine
whether variation by race and age is statistically significant. For simple
assault, however, its rate of 26.6 per 1,000 is based on approximately
2,500 events, which is sufficiently large to estimate variation by sex,
race, and age categories.
The effect of the sample size on standard error
can be noted in the size of the range bars in the earlier
figures and in the SEs given in table 1. The
SE increased gradually from 1973 to 1991 as a result of decreases in the
sample size for budgetary reasons. In 1992, when the NCVS was redesigned,
the SE was dramatically larger than in prior years because of the use
of a half-sample. For calibration purposes the old and new surveys were
each administered to half the sample; in this way the effect of the redesign
on victimization rates could be gauged. The two samples could not be combined,
so the estimates were based on that half of the sample given the new survey
and are less precise.
Sampling is not the only source of error in NCVS data; however,
it is the only one whose magnitude can be estimated. Other sources of error
in the NCVS include:
Other types of data collection also have error. For example, the FBI's
Uniform Crime Reports (UCR) is a voluntary system and not all jurisdictions
report crime data. To provide national estimates of reported crime, the
FBI must account for the jurisdictions that did not provide data. They
estimate the crime for the nonreporting jurisdictions based on similar
jurisdictions that have reported. Since the reporting jurisdictions are
likely to have experienced more crime than those that did not report,
the estimated crime rate may be biased upward, but the magnitude of this
bias is unknown.
Appendix II. Data used in the graphics
Table 1 contains the data used in preparing the first five
figures . The standard errors (SEs) are calculated using the formulas and
procedures described in Appendix II of Criminal Victimization in the
United States 1994. The data for B-Value and r (the Greek letter
rho) are included for archival purposes since they were used in calculating
both the SE for the violent crime victimization rate and for its year-to-year
change. Note that the procedure for calculating the SE changed in 1993,
the year after the NCVS underwent a major change in its collection procedures.
Table 2 contains the data (calculated from the data in table 1) used to
plot the figure on year-to-year change from 1973-96.
Table 3 contains the data used to plot the figure on annual
percent change by crime type.
Some of the numbers in these tables may vary from other published numbers
because of differing collection periods and the use of summed rather than
aggregate data. For the most part, the differences do not affect the conclusions
made about the data.
1To account for the relatively
few agencies that do not provide complete data, homicide rates for the Nation
as a whole must be estimated. (See Appendix I, Nonsampling
error.)
2The NCVS employs
a very complex, random stratified multistage cluster sample, but the general
principles discussed in this report apply to it as well. See Appendix
II of the BJS report, Criminal Victimization in the United States,
1994, for a description of the sample construction.
3For a simple
random sample, the standard error is equal to 4For a simple
random sample, the standard error for a percentage equals 5The SE is larger
than would be expected if the NCVS was a simple random sample; see Appendix
II of the BJS report, Criminal Victimization in the United States,
1994.
6This is also
noted in the figure on annual percent change by crime
type, where two of the three segments of the range bar for rape (90%
and 95%) cross the No change line.
7 Deming, W.
E. "Sample Surveys: The Field." In The International Encyclopedia
of Statistics, Vol. 2. The Free Press, New York, 1978, pp. 867-885.
Cited in Wright, T., "Sampling and Census 2000: The Concepts,"
American Scientist, 86, 3 (May-June 1998), pp. 245-253.
8Blalock, H.
M., Jr., Social Statistics, 2nd Edition, McGraw-Hill, New York,
1972, p. 514.
Criminal Victimization,
1973-95, NCJ 163069, 4/97.
Criminal Victimization 1996: Changes
1995-96 with Trends 1993-96, NCJ 165812, 11/97.
Criminal Victimization in the United
States 1994, NCJ 162126, 5/97.
The Nation's Two Crime Measures,
NCJ 122795, 11/95, also presented on pages 399-400 of the FBI's Crime
in the United States, 1996.
About the NCVS redesign
The Effects of the Redesign on Victimization
Estimates, NCJ 164381, 4/97.
National Crime Victimization Survey
(NCVS) Redesign: Press Release, NCJ 151169, 10/94.
National Crime Victimization Survey
(NCVS) Redesign: Fact Sheet , National Crime Victimization Survey
Redesign: Technical Background, National Crime Victimization Survey
(NCVS) Redesign: Questions & Answers, NCJ 151171, 10/94.
The Bureau of Justice Statistics is the statistical agency of the U.S.
Department of Justice. Jan M. Chaiken, Ph.D., is director.
Marianne W. Zawitz, BJS statistician, and Michael Maltz, BJS Visiting
Fellow and Professor of Criminal Justice, University of Illinois at Chicago,
wrote this report. It was edited by Tom Hester. BJS is grateful for the
thorough reviews and thoughtful comments provided by Alfred Blumstein,
Heinz School of Public Policy and Management, Carnegie Mellon University;
Michael Maxfield, School of Criminal Justice, Rutgers University; Stephen
Fienberg, Departments of Statistics and Social Sciences, and Laura J.
Dugan, Heinz School of Public Policy and Management, Carnegie Mellon University;
John Richters, Developmental Psychopathology Research Branch, National
Institute of Mental Health; James Nolan and Yoshio Akiyama, Federal Bureau
of Investigation; Jay Hoover, Office of Policy Development, and Stephen
Shandy, Criminal Division, U.S. Department of Justice. Within BJS, assistance
was provided by Lawrence Greenfeld, Michael Rand, and Cheryl Ringel.
June 1998, NCJ 167881
This report belongs to the BJS Technical Report series. The previous
publication in the series was Effects of the Redesign on Victimization
Estimates, April 1997, NCJ164381. Readers who want additional data,
analyses, and graphs about criminal victimization in the United States
should access the BJS Internet Web site: http://www.ojp.usdoj.gov/bjs/.
Data presented in this report may be obtained from the National Archive
of Criminal Justice Data at the University of Michigan, 1-800-999-0960.
The archive may also be accessed through the BJS Web site. When at the
archive site, search for data sets ICPSR 6406, ICPSR 7635, ICPSR 8608,
and ICPSR 8864.
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where N is the sample size and p is the probability of the
occurrence. In this case SE = 
=
4.
.