A Comprehensive Guide to Measurement, Finance, and Geometry
MEASUREMENT AND STATISTICS
Accuracy and Precision
Accuracy: How close a measured value is to the actual value.
Precision: The smallest measurement possible on a measuring tool.
Measures of Central Tendency
Mean: Average (sum of all values divided by the total number of values).
Median: Middle value when data is arranged from least to greatest.
Mode: Most frequent value.
Trimmed Mean: Mean calculated after discarding a certain percentage of the highest and lowest values.
Weighted Mean: Mean calculated by assigning weights to different values based on their importance.
Other Statistical Concepts
Outliers: Values that are significantly different from the rest of the data.
Tolerance: The acceptable range of variation in a measurement.
Uncertainty: The margin of error associated with a measurement.
Example: Precision, Uncertainty, and Tolerance
a. Ruler marked in cm
Precision = 1 cm
Uncertainty = +/- 0.5 cm
Tolerance = 1.5 – 0.5 = 1 cm
b. Tape measure in 1/2 inch increments
Precision = 0.5 in
Uncertainty = +/- 0.25 in
Tolerance = 0.75 – 0.25 = 0.5 in
Calculating Tolerance and Nominal Value
Tolerance = Maximum Value – Minimum Value
Nominal Value (NV) = Minimum Value + (1/2) * Tolerance
Finding Percentiles and Percent Scores
Percentile Formula: PR = (B/n) * 100 or PR = (B + 0.5E/n) * 100
Where:
- B = Number of scores below the reference score
- n = Total number of scores
- E = Number of scores equal to the reference score
Percent Score: (Score / Total Score) * 100
PROBABILITY AND BUSINESS FINANCE
Key Terms
Breakeven Point: The point at which total revenue equals total costs.
Expected Value: A prediction of the average outcome of a situation with uncertainty.
Expenses: Costs incurred by a business (e.g., salaries, supplies, advertising).
Fixed Costs: Costs that remain constant each month (e.g., rent, utilities).
Variable Costs: Costs that fluctuate (e.g., raw materials).
Goods: Products sold by a business.
Odds: A comparison of favorable outcomes to unfavorable outcomes.
Probability: The likelihood of an event occurring.
Revenue: Income generated from sales.
Services: Tasks performed by a business.
Types of Probability
Experimental Probability: Based on past observations (P = Desired Events / Total Observed Events).
Theoretical Probability: Based on possible outcomes (P = Desired Events / Total Possible Events).
Example: Probability
John rolls a six-sided die.
a. Theoretical probability of rolling a 6:
1/6, 0.167, 16.7%, 1 out of 6
b. Experimental probability (results: 6, 4, 6, 6, 1, 6):
4/6 = 2/3, 0.667, 66.7%, 2 out of 3
Expected Value Calculation
Expected Value = (Probability as Decimal * Winning Amount) – Cost
Odds
Odds in Favor: Favorable Outcomes : Unfavorable Outcomes
Odds Against: Unfavorable Outcomes : Favorable Outcomes
Breakeven Point Calculation
Months to Breakeven = Loan Amount / Monthly Profit
GEOMETRY AND TRIGONOMETRY
Types of Triangles
Acute Triangle: All angles are less than 90 degrees.
Obtuse Triangle: One angle is greater than 90 degrees.
Right Triangle: One angle is exactly 90 degrees.
Equilateral Triangle: All sides are equal in length.
Isosceles Triangle: Two sides are equal in length.
Scalene Triangle: All sides have different lengths.
Types of Quadrilaterals
Quadrilateral: Four-sided polygon.
Parallelogram: Two pairs of parallel sides.
Rectangle: Four right angles.
Square: Four right angles and four equal sides.
Rhombus: All sides are equal in length.
Trapezoid: One pair of parallel sides.
Isosceles Trapezoid: Two non-parallel sides are equal in length.
Kite: Two pairs of adjacent sides are equal in length.
Polygons
Triangle: 3 sides
Quadrilateral: 4 sides
Pentagon: 5 sides
Hexagon: 6 sides
Heptagon: 7 sides
Octagon: 8 sides
Nonagon: 9 sides
Decagon: 10 sides
Dodecagon: 12 sides
Formulas
Sine Law
sin(A) / a = sin(B) / b = sin(C) / c
Cosine Law
Finding an Angle
cos(A) = (b² + c² – a²) / (2bc)
cos(B) = (a² + c² – b²) / (2ac)
cos(C) = (a² + b² – c²) / (2ab)
Finding a Side
a² = b² + c² – 2bc * cos(A)
b² = a² + c² – 2ac * cos(B)
c² = a² + b² – 2ab * cos(C)
Internal Angles of a Polygon
Sum of Internal Angles = 180(n – 2) degrees, where n is the number of sides.
Central Angles of a Regular Polygon
Central Angle = 360 / n degrees, where n is the number of sides.
Number of Diagonals in a Polygon
Number of Diagonals = n(n – 3) / 2, where n is the number of sides.
Applications
Triangles: Construction, design (strong and rigid).
Quadrilaterals: Construction, easily divided into triangles.
Sine Law: Calculating distances and angles in triangles without a right angle (e.g., roof trusses).
Cosine Law: Similar applications to Sine Law, particularly useful for finding the third side of a triangle when two sides and the included angle are known (e.g., flight paths).
Polygons: Building walls, tessellations.
HOME FINANCE
Key Terms
Assessed Value: Estimated value of a property for tax purposes.
Closed Mortgage: A mortgage that doesn’t allow prepayment without penalty.
Deductible: Amount paid out of pocket before insurance coverage kicks in.
Down Payment: Initial payment made when purchasing a property.
Emergency Maintenance: Urgent repairs.
Fixed-Rate Mortgage: Interest rate remains constant.
Frontage: Length of property bordering a street.
Home Insurance: Protection against damage to a property.
Interest Rate: Cost of borrowing money.
Mortgage Insurance: Required when the down payment is less than 20%.
Preventative Maintenance: Regular upkeep to prevent major issues.
Principle: Amount of money borrowed.
Property Appraisal: Professional assessment of a property’s value.
Survey: Determines property boundaries.
Variable-Rate Mortgage: Interest rate fluctuates.
Mortgage Calculations
Monthly Payment = (Principle * Table Value) / 1000
Total Payments = Monthly Payment * 12 * Number of Years
Total Interest = Total Paid – Total Borrowed
Gross Debt Service Ratio (GDSR)
GDSR = (Monthly Mortgage Payment + Monthly Heating Costs + Monthly Property Tax) / Gross Monthly Income * 100
Affordability
Start with Gross Monthly Income and use the GDSR to determine the maximum affordable mortgage payment and home price.
Reducing Mortgage Costs
- Increase monthly payments.
- Increase down payment.
- Make payments more frequently (bi-weekly or weekly).
- Make lump sum payments.
Property Taxes
Monthly Property Tax = Annual Tax Bill / 12
Portioned Assessment = Assessed Value * Portion Percentage
Property Tax = (Portioned Assessment * Mill Rate) / 1000
Local Improvement Costs
Costs for improvements like sidewalks, streetlights, and sewer repairs, often calculated based on frontage.
Home Insurance Factors
- Location (metro, protected, semi-protected, unprotected)
- Coverage type (standard, comprehensive)
- Deductible
Monthly Home Ownership Costs
- Mortgage/Rent
- Insurance
- Property Tax
- Utilities
Emergency vs. Preventative Maintenance
Emergency Maintenance: Unexpected repairs (e.g., furnace breakdown).
Preventative Maintenance: Regular upkeep (e.g., furnace cleaning, hot water tank inspection) to prevent emergencies.
Buying vs. Renting
Buying
Advantages: Ownership, potential investment, customization.
Costs: Mortgage, property tax, insurance, utilities, maintenance.
Renting
Advantages: No property tax or homeowner’s insurance, fewer responsibilities.
Costs: Rent, utilities.
Loan Payment Breakdown
Payment – Interest = Principle
Payment – Principle = Interest
Principle + Interest = Payment
Old Unpaid Balance – Principle = New Unpaid Balance
VEHICLE FINANCE
Types of Insurance
All-Purpose Insurance: Comprehensive coverage for daily use.
Leisure Insurance: Limited coverage for occasional use (e.g., 2 times a month).