Statistics: Mean and Median Concepts

Statistics

• Mean: the average value of a set of numbers
  • Calculated by summing all values and dividing by the number of values
  • Example: mean of {2, 4, 6, 8, 10} = (2 + 4 + 6 + 8 + 10) / 5 = 30 / 5 = 6
• Median: the middle value of a set of numbers when arranged in order
  • If the number of values is odd, the median is the middle value
  • If the number of values is even, the median is the average of the two middle values
  • Example: median of {2, 4, 6, 8, 10} = 6 (middle value)
• Mode: the value that appears most frequently in a set of numbers
  • Example: mode of {2, 4, 6, 4, 10} = 4 (appears twice)
• Range: the difference between the largest and smallest values in a set of numbers
  • Example: range of {2, 4, 6, 8, 10} = 10 - 2 = 8

Indices

• Index notation: a shorthand way of writing repeated multiplication
  • Example: 2³ = 2 × 2 × 2 = 8
• Index laws:
  • Product of powers: aⁿ × aⁿ = aⁿ⁺ⁿ
  • Power of a product: (ab)ⁿ = aⁿbⁿ
  • Power of a power: (aⁿ)ⁿ = aⁿ⁺ⁿ
  • Zero index: a⁰ = 1
• Simplifying indices:
  • Example: 2³ × 2⁴ = 2³⁺⁴ = 2⁷

Cube Root

• Cube root: the value that, when multiplied by itself three times, gives the original number
  • Notation: ³√x or x¹/³
  • Example: ³√8 = 2 (because 2 × 2 × 2 = 8)
• Properties of cube roots:
  • Product rule: ³√(xy) = ³√x × ³√y
  • Quotient rule: ³√(x/y) = ³√x / ³√y

Statistics

• Mean: Represents the average of a set of numbers obtained by dividing the sum of all values by their count.
• Median: The central number in a sorted list; if odd count, it's the middle number; if even, it averages the two middle numbers.
• Mode: Identifies the most frequently occurring number in a dataset, useful for understanding common trends.
• Range: Shows the dispersion of values, calculated by subtracting the smallest number from the largest.

Indices

• Index Notation: A method to express repeated multiplication, simplifying expressions through exponents, e.g., 2³ for 2 multiplied by itself three times.
• Index Laws:
  • Product of Powers: When multiplying same bases, add their exponents.
  • Power of a Product: When raising a product to a power, distribute the exponent to each factor.
  • Power of a Power: To raise a power to another power, multiply the exponents.
  • Zero Index: Any non-zero number raised to zero equals one.
• Simplifying Indices: Combine exponents by adding them in multiplication scenarios, e.g., 2³ × 2⁴ simplifies to 2⁷.

Cube Root

• Cube Root: Represents a number that, when tripled multiplied by itself, equals the original number. Denoted as ³√x or x¹/³.
• Properties of Cube Roots:
  • Product Rule: The cube root of a product equals the product of the cube roots of the individual numbers.
  • Quotient Rule: The cube root of a quotient equals the quotient of the cube roots.