Ch2. WorkKeys Numerical Reasoning — Basic Statistics, Data Analysis, and Fast Calculation

Overview of Numerical Reasoning

Numerical Reasoning is one of the core competencies measured by WorkKeys and similar US workforce readiness assessments (O*NET, ACT WorkKeys, SHRM competency frameworks). It measures the ability to think mathematically and calculate accurately in a workplace context.

On these assessments, numerical reasoning is not a test for math prodigies. Basic calculation and data interpretation are the heart of the section.

4 Sub-Competencies:

Sub-Competency	Description
Basic Arithmetic	Four operations, ratios, unit conversion
Basic Statistics	Mean, median, mode, variance, standard deviation
Data Interpretation	Reading and analyzing numbers in tables and graphs
Chart Construction	Converting raw data into charts

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Basic Arithmetic: High-Frequency Patterns

Percentages and Ratios

Ratio = Part / Whole × 100

Percent Change = (Comparison Period − Base Period) / Base Period × 100

Example: Revenue $1,000,000 in 2023 →$1,250,000 in 2024 Percent change = (1,250,000 − 1,000,000) / 1,000,000 × 100 = 25%

Absolute change vs. percent change — a classic trap:

• Absolute change: $250,000 (difference in dollar amount)
• Percent change: 25% (relative shift)

Rate, Distance, and Time

Distance = Rate × Time

Time = Distance / Rate

Workplace application: Output = Processing Rate × Time

Concentration Calculations (Weighted Mix)

Concentration (%) = Solute / Solution × 100

Mixing problems: use weighted average

Example: 100 g of 20% saltwater + 200 g of 10% saltwater — what is the combined concentration? → (20×100 + 10×200) / (100+200) = (2,000+2,000)/300 = 13.3%

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Basic Statistics

Measures of Central Tendency

Concept	Calculation	Key Trait
Mean	Sum / Count	Sensitive to outliers
Median	Middle value when sorted	Robust to outliers
Mode	Most frequent value	Useful for categorical data

Why mean alone is not enough:

Example: [1, 2, 3, 4, 100] → Mean = 22, Median = 3

A mean of 22 is far from most values. When extreme values exist (like income distribution), the median gives a more realistic picture.

Measures of Dispersion

Variance: The average squared distance of each value from the mean.

Standard Deviation = √Variance

A larger standard deviation means values are more spread out around the mean.

Workplace applications: quality control, performance distribution, risk assessment

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Data Analysis Strategy

Common Chart Types

Bar chart: Compare magnitudes across categories

Line graph: Trend over time (when does it rise or fall?)

Pie chart: Composition ratios

Table: Multi-dimensional data (year × category)

5-Step Data Analysis Strategy

Step 1. Read the title: What does this chart show? (Subject, time frame, units)

Step 2. Check the axes: What do the horizontal and vertical axes mean, and what are the units? (thousands, millions, etc.)

Step 3. Identify the overall trend: Is the general direction up, down, or mixed?

Step 4. Spot anomalies: Sudden spikes or drops, reversals, outliers

Step 5. Compare against answer choices: Does the claim in each answer choice actually hold up in the chart?

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Common Chart Traps

Trap 1: Absolute Value vs. Ratio Confusion

The chart shows absolute values, but an answer choice claims “has the highest ratio.”

Example: A may have a larger absolute number than B, but B’s share of the total could still be higher.

Trap 2: Absolute Change vs. Percent Change

Year-over-year dollar differences (absolute change) vs. year-over-year growth rate (percent change) are not the same.

Trap 3: Base Year Confusion

“Compared to 2020” and “compared to the prior year” are different. Always confirm which year serves as the baseline.

Trap 4: Unit Confusion

$10,000 units vs.$1,000,000 units — always verify units before calculating!

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Habits That Reduce Calculation Errors

1. Estimate first: Get a rough answer before calculating precisely. If the answer choices are far apart, a quick estimate can eliminate options.

Example: 1,234 × 198 ≈ 1,200 × 200 = 240,000 (exact: 244,332)

2. Unify units: Keep units consistent throughout and verify the final unit.

3. Verify with back-calculation: Confirm using an inverse operation or a different method.

4. Process of elimination: Remove obviously impossible answer choices first.

5. Question extreme answers: If your answer seems unreasonably large or small, recalculate.

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High-Frequency Calculation Types

Work Rate Problems

A alone takes x days; B alone takes y days. How long together?

Daily work rate: A = 1/x, B = 1/y, Together = 1/x + 1/y

→ Days to completion = 1 / (1/x + 1/y) = xy / (x+y)

Average Speed

A trip from A→B at 60 mph, and B→A at 40 mph. What is the average speed for the round trip?

Average speed = 2 / (1/60 + 1/40) = 2 / (5/120) = 48 mph

(Not the arithmetic average of 50 mph — this is a time-weighted harmonic mean.)

Proportional Division

Divide 120 in the ratio A:B = 3:5:

• A = 120 × 3/(3+5) = 120 × 3/8 = 45
• B = 120 × 5/8 = 75

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Study Checklist

• Can explain and calculate the difference between absolute change and percent change
• Can explain the difference between mean, median, and mode
• Can apply the 5-step data analysis strategy
• Can recognize and avoid all 4 common trap types
• Can accurately solve work rate and average speed problems