Ratios, Percents, and Operations with Real Numbers: Foundations for Statistical Analysis

Ratios, Percents, and Operations with Real Numbers (1.6)

Ratios

Ratios are fundamental tools in statistics for comparing quantities and understanding relationships between variables.

• Definition: A ratio is a comparison of two quantities, expressed as a to b, a:b, or as a fraction $\frac{a}{b}$.

• Example: If a person has 6 cats and 2 dogs, the ratio of cats to dogs is $\frac{6}{2} = \frac{3}{1}$, or "3 cats to 1 dog."

• Unit Ratio: A unit ratio is a ratio with a denominator of 1, written as $\frac{a}{b}$ with $b=1$.

Example: Finding a Unit Ratio

• Given: Average annual charge for tuition and fees was $9970 at public four-year colleges and $3570 at public two-year colleges.

• Unit ratio: $\frac{9970}{3570} \approx 2.79$

• Interpretation: The average annual charge at public four-year colleges is about 2.79 times that at public two-year colleges.

Comparing Ratios

Ratios are often used to compare affordability, efficiency, or other relationships between groups or regions.

• Median: The median is the middle value in an ordered list of numbers, or the average of the two middle values if the list has an even number of entries.

Example: Comparing Home Affordability by Region

Region	Median Sales Price $	Median Household Income $	Unit Ratio
Northeast	402,800	59,210	6.80:1
Midwest	269,700	54,267	4.97:1
South	257,700	49,655	5.19:1
West	333,900	57,688	5.79:1

• Interpretation: The lower the unit ratio, the more affordable the homes are in that region.

• Order of Affordability: Midwest > South > West > Northeast.

• Key Point: Affordability depends on both price and income, not just the price alone.

Percents

Percents are used to express ratios as parts per hundred, which is essential for interpreting statistical data.

• Definition: Percent means "for each hundred": $a\% = \frac{a}{100}$

Converting Between Percents and Decimals

• To write a percentage as a decimal, divide by 100 (move the decimal two places left).

• To write a decimal as a percentage, multiply by 100 (move the decimal two places right) and add a percent symbol.

• Examples:

  • 86% = 0.86

  • 7% = 0.07

  • 0.125 = 12.5%

Finding the Percentage of a Quantity

To find the percentage of a quantity, multiply the decimal form of the percentage by the quantity.

• Formula: $\text{Percentage of Quantity} = \text{Decimal Form} \times \text{Quantity}$

• Examples:

  • 7% sales tax on $1500: $0.07 \times 1500 = 105$

  • 20% of 2300 students: $0.20 \times 2300 = 460$

Percent Change

Percent change measures the relative increase or decrease from an initial value to a final value, a common calculation in statistical analysis.

• Formula: $\text{Percent change} = \frac{b - a}{a} \times 100$

  • $a$ = initial value, $b$ = final value

• Examples:

  • Disney stock: $\frac{112.21 - 104.33}{104.33} \approx 0.076 = 7.6\%$

  • Coca-Cola stock: $\frac{49.22 - 44.41}{44.41} \approx 0.108 = 10.8\%$

• Interpretation: Percent change is more meaningful than absolute change when comparing quantities with different starting values.

Multiplying and Dividing Real Numbers

Understanding the rules for multiplying and dividing real numbers is essential for accurate statistical calculations.

• Product/Quotient of Different Signs: The result is negative.

• Product/Quotient of Same Signs: The result is positive.

• Examples:

  • $7 \times (-4) = -28$

  • $(-0.2) \times 0.3 = -0.06$

  • $-5 \times (-6) = 30$

  • $-10 \div 2 = -5$

  • $-\frac{1}{6} \div -\frac{3}{5} = \frac{5}{18}$

Equal Fractions with Negative Signs

Negative signs in fractions can be written in different positions without changing the value of the fraction.

• If $b \neq 0$, then: $- \frac{a}{b} = \frac{-a}{b} = \frac{a}{-b}$

Additional info: These foundational concepts are essential for understanding more advanced statistical topics such as descriptive statistics, probability, and inferential statistics. Mastery of ratios, percents, and operations with real numbers is critical for interpreting data and performing statistical calculations accurately.