5.1 Percentages, Crime Rate Changes & Clearance Rates

5.1 Percentages, Crime Rate Changes & Clearance Rates

Modern law enforcement relies heavily on quantitative data to direct patrol operations, measure investigative efficacy, and allocate public safety resources. Under contemporary administrative models such as CompStat (Computer Statistics) and intelligence-led policing, command staff and patrol supervisors continuously monitor fluctuations in reported offenses, clearance percentages, and localized crime trends. Percentages are also a basic arithmetic skill: FPSI's own PACT sample includes a salary that rises by a fixed percentage each year.

Percentage word problems do not test abstract university mathematics. They present everyday departmental scenarios: comparing year-over-year burglaries across patrol sectors, calculating the clearance rate of a detective division, or standardizing crime frequencies across jurisdictions with differing populations. Excelling in this domain requires mastery of percentage mechanics, strict adherence to baseline rules, and vigilance against deceptive distractor choices designed to exploit common arithmetic shortcuts.

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The Mathematical Foundation: Percentages as Base-100 Ratios

A percentage is a standardized mathematical expression representing a fraction of 100. The word originates from the Latin per centum, meaning "by the hundred." In law enforcement administration, percentages transform disparate raw numbers into directly comparable proportions.

Converting between fractions, decimals, and percentages is the foundational arithmetic skill required for rapid scratch-paper calculations:

1. Fraction to Decimal: Divide the numerator (top number) by the denominator (bottom number). $\frac{3}{8} = 3 \div 8 = 0.375$
2. Decimal to Percentage: Multiply the decimal by 100 (shift the decimal point two places to the right) and append the percent symbol (%). $0.375 \times 100\% = 37.5\%$
3. Percentage to Decimal: Divide the percentage by 100 (shift the decimal point two places to the left) and remove the percent symbol. $62.5\% = \frac{62.5}{100} = 0.625$

During timed civil service examinations where electronic calculators are prohibited, recognizing standard fractional equivalencies saves critical testing seconds:

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The Critical Baseline Rule for Percentage Change

The most important percentage skill for police math problems is percentage change—measuring how much a specific crime category has increased or decreased between two reporting periods (such as comparing 2025 to 2026, or Month 1 to Month 2).

The Fundamental Formula

Every percentage change calculation is governed by one immutable rule: the change must always be divided by the original (baseline) value, never by the new value.

$\text{Percentage Change} = \frac{\text{New Value} - \text{Original Value}}{\text{Original Value}} \times 100\% = \frac{\Delta V}{V_{\text{original}}} \times 100\%$

Where:

• $\text{New Value}$ is the incident count in the later period.
• $\text{Original Value}$ is the historical baseline count from the earlier period.
• $\Delta V$ represents the absolute difference (change in raw incidents).

Directionality: Increase vs. Decrease

• If $\text{New Value} > \text{Original Value}$, the numerator is positive, representing a percentage increase.
• If $\text{New Value} < \text{Original Value}$, the numerator is negative, representing a percentage decrease.

Comparative Derivation Table

To see why the baseline value matters mathematically, examine how dividing by the wrong number distorts both increases and decreases:

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Successive Percentage Changes

When a value changes by a percentage more than once, each change applies to the new value, not the original. Salary raises are the classic example, and FPSI's sample math items include one.

$\text{New Value} = \text{Original} \times (1 + r_1) \times (1 + r_2)$

Example: A recruit starts at $42,000 and receives a 4% raise at the end of each of the first two years.

1. After year one: $\text{\textdollar}42{,}000 \times 1.04 = \text{\textdollar}43{,}680$.
2. After year two: $\text{\textdollar}43{,}680 \times 1.04 = \text{\textdollar}45{,}427.20$.

The trap answer adds the two raises as one 8% raise on the starting salary: $\text{\textdollar}42{,}000 \times 1.08 = \text{\textdollar}45{,}360$. That is $67.20 too low, because the second raise is also earned on the first raise.

Hand-calculation shortcut: multiplying by 1.04 is the same as adding 4%. Find 1% by moving the decimal two places ($420), multiply by 4 ($1,680), and add it to $42,000 to get $43,680. Repeat with the new value: 1% of $43,680 is $436.80, times 4 is $1,747.20, and $43,680 + $1,747.20 = $45,427.20.

Decreases work the same way. A 10% cut followed by a 10% increase does not return to the starting value: 100 becomes 90, and 90 increased by 10% is 99.

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Investigative Clearance Rates

In public safety administration, the clearance rate measures an agency's success in resolving reported offenses. Under the Federal Bureau of Investigation (FBI) Uniform Crime Reporting (UCR) Program and the National Incident-Based Reporting System (NIBRS), reported offenses are cleared in one of two standardized ways:

1. Cleared by Arrest: At least one person has been arrested, formally charged with the offense, and turned over to the court for prosecution.
2. Cleared by Exceptional Means: Law enforcement has encountered an element beyond its control that precludes arresting and prosecuting the offender. To clear an offense exceptionally, the agency must satisfy all four of the following strict criteria:
   • Positively identified the offender;
   • Gathered sufficient evidence to support an arrest and prosecution;
   • Located the offender's exact whereabouts so they could be taken into custody; and
   • Encountered an insurmountable legal or physical barrier outside police control (such as the death or suicide of the offender, the victim's formal refusal to cooperate after the suspect has been identified, or a foreign government's refusal to extradite).

The Clearance Rate Formula

$\text{Clearance Rate} = \frac{\text{Total Crimes Cleared}}{\text{Total Crimes Reported}} \times 100\% = \frac{\text{Cleared by Arrest} + \text{Cleared by Exceptional Means}}{\text{Total Reported Offenses}} \times 100\%$

Step-by-Step Clearance Problem

Scenario: During the second quarter, the Major Crimes Division logged the following investigative figures for Sector Charlie:

• Reported Homicides: 20
• Reported Robberies: 80
• Reported Aggravated Assaults: 150
• Total Cleared Homicides: 15 (12 by arrest, 3 exceptional)
• Total Cleared Robberies: 32 (28 by arrest, 4 exceptional)
• Total Cleared Assaults: 90 (75 by arrest, 15 exceptional)

Calculation:

1. Homicide Clearance Rate: $\frac{15}{20} \times 100\% = 0.75 \times 100\% = 75.0\%$
2. Robbery Clearance Rate: $\frac{32}{80} \times 100\% = \frac{4}{10} \times 100\% = 40.0\%$
3. Aggravated Assault Clearance Rate: $\frac{90}{150} \times 100\% = \frac{3}{5} \times 100\% = 60.0\%$
4. Overall Sector Violent Crime Clearance Rate:
   • Total Reported = $20 + 80 + 150 = 250$
   • Total Cleared = $15 + 32 + 90 = 137$ $\text{Overall Rate} = \frac{137}{250} \times 100\% = \frac{137 \times 4}{250 \times 4} = \frac{548}{1{,}000} = 54.8\%$

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Per Capita Crime Rates: Standardizing Across Populations

Comparing raw crime numbers between different cities, precincts, or reporting districts is fundamentally flawed because larger populations naturally generate higher absolute volumes of incidents. A metropolitan city of 500,000 residents reporting 1,000 burglaries is experiencing a far lower burglary risk than a small town of 20,000 residents reporting 200 burglaries.

To make meaningful comparisons, police statisticians calculate per capita crime rates. In the United States, crime rates are conventionally expressed as the number of offenses per 100,000 residents.

The Per Capita Formula

$\text{Crime Rate per } 100{,}000 = \frac{\text{Total Reported Offenses}}{\text{Jurisdiction Population}} \times 100{,}000$

If an examination question asks for the rate per 1,000 or per 10,000 residents, substitute that multiplier accordingly:

$\text{Crime Rate per } 1{,}000 = \frac{\text{Total Reported Offenses}}{\text{Jurisdiction Population}} \times 1{,}000$

Comparative Population Example

Consider two municipal police jurisdictions evaluating their annual violent crime data:

• City of Oakridge:
  • Population: $75{,}000$
  • Reported Violent Crimes: $420$
• City of Pineville:
  • Population: $120{,}000$
  • Reported Violent Crimes: $624$

Step 1: Calculate Oakridge Per Capita Rate: $\text{Rate}_{\text{Oakridge}} = \frac{420}{75{,}000} \times 100{,}000$ Simplify the fraction by dividing numerator and denominator by 1,000: $\text{Rate}_{\text{Oakridge}} = \frac{420 \times 100}{75} = \frac{42{,}000}{75}$ Executing division on scratch paper ($42{,}000 \div 75 = 560$): $\text{Rate}_{\text{Oakridge}} = 560\text{ violent crimes per } 100{,}000\text{ residents}$

Step 2: Calculate Pineville Per Capita Rate: $\text{Rate}_{\text{Pineville}} = \frac{624}{120{,}000} \times 100{,}000 = \frac{624 \times 100}{120} = \frac{624 \times 10}{12} = 52 \times 10 = 520$ $\text{Rate}_{\text{Pineville}} = 520\text{ violent crimes per } 100{,}000\text{ residents}$

Conclusion: Even though Pineville recorded 204 more raw crimes than Oakridge ($624$ vs. $420$), Oakridge actually has a higher violent crime rate ($560$ vs. $520$ per $100{,}000$). This demonstrates why per capita standardization is indispensable for objective civil service problem solving.

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Multi-Crime Trend Analysis: Worked Examination Walkthrough

Examine the following departmental reporting table, a typical data-interpretation practice problem:

Question Walkthroughs

Problem A: Largest Percentage Increase

Which offense category experienced the largest percentage increase from Year 1 to Year 2?

1. Identify categories with increases (Year 2 > Year 1):
   • Aggravated Assault: from 200 to 250 ($+50$)
   • Motor Vehicle Theft: from 160 to 208 ($+48$)
2. Compute percentage increase for Aggravated Assault: $\frac{250 - 200}{200} \times 100\% = \frac{50}{200} \times 100\% = 25.0\%$
3. Compute percentage increase for Motor Vehicle Theft: $\frac{208 - 160}{160} \times 100\% = \frac{48}{160} \times 100\% = \frac{3}{10} \times 100\% = 30.0\%$
4. Result: Although Aggravated Assault had a larger raw numerical increase ($50$ vs. $48$), Motor Vehicle Theft experienced the larger percentage increase ($30.0\%$ vs. $25.0\%$).

Problem B: Year 2 Clearance Rate for Motor Vehicle Theft

What was the total clearance rate for Motor Vehicle Theft in Year 2?

1. Sum total cleared offenses: $\text{Total Cleared} = 42\text{ (arrest)} + 10\text{ (exceptional)} = 52$
2. Divide by Year 2 reported offenses ($208$): $\text{Clearance Rate} = \frac{52}{208} \times 100\%$
3. Simplify the fraction: $\frac{52}{208} = \frac{1}{4} = 0.25 = 25.0\%$
4. Result: The clearance rate was exactly $25.0\%$.

By systematically applying these principles, verifying historical baselines, and maintaining scratch-paper accuracy, candidates can solve any percentage or rate problem presented on the examination.