Statistics Basics

Statistics

• Types of Data:
  • Qualitative (categorical): describes characteristics (e.g., colors, shapes)
  • Quantitative (numerical): describes amounts (e.g., length, time)
• Measures of Central Tendency:
  • Mean: average value
  • Median: middle value (when data is in order)
  • Mode: most frequent value
• Measures of Variability:
  • Range: difference between largest and smallest values
  • Interquartile Range (IQR): difference between Q3 and Q1 (75th and 25th percentiles)

Algebra

• Variables and Constants:
  • Variables: letters or symbols representing unknown values
  • Constants: numbers
• Equations and Inequalities:
  • Equations: statements with an equal sign (=)
  • Inequalities: statements with an inequality sign (>, <, ≥, ≤)
• Linear Equations:
  • Standard form: Ax + By = C (A, B, and C are constants)
  • Slope-Intercept form: y = mx + b (m is slope, b is y-intercept)

Multiplication and Division

• Properties of Multiplication:
  • Commutative: a × b = b × a
  • Associative: (a × b) × c = a × (b × c)
  • Distributive: a × (b + c) = a × b + a × c
• Division:
  • Definition: a ÷ b = c if and only if a = c × b
  • Rules:
    • Division by zero is undefined
    • Division is the inverse operation of multiplication

Statistics

• Qualitative Data: describes characteristics, such as colors, shapes, and categories
• Quantitative Data: describes amounts, including length, time, and numerical values
• Mean: the average value of a dataset, calculated by summing values and dividing by the number of values
• Median: the middle value in a dataset when arranged in order, providing a middle point
• Mode: the most frequent value in a dataset, which may not exist in every dataset
• Range: the difference between the largest and smallest values in a dataset, providing a sense of spread
• Interquartile Range (IQR): the difference between the 75th percentile (Q3) and 25th percentile (Q1) in a dataset, providing a more robust measure of spread

Algebra

• Variables: letters or symbols used to represent unknown values or quantities, often used to solve equations
• Constants: numbers that do not change, often used in equations and formulas
• Equations: statements that express the equality of two expressions, often containing variables, constants, and various operations
• Inequalities: statements that express a relationship between two expressions, such as "greater than" or "less than", often containing variables, constants, and various operations

Multiplication

• Multiplication is a mathematical operation that represents repeated addition.
• Multiplication is denoted by the multiplication symbol (×) or an asterisk (*).
• The commutative property of multiplication states that the order of numbers being multiplied does not change the result.
• The associative property of multiplication states that the order in which numbers are multiplied does not change the result.
• The distributive property of multiplication states that multiplication distributes over addition.
• There are two types of multiplication: scalar multiplication and matrix multiplication.
• Scalar multiplication involves multiplying a number by a scalar value.
• Matrix multiplication involves multiplying matrices.

Division

• Division is a mathematical operation that represents sharing or grouping.
• Division is denoted by the division symbol (÷) or a forward slash (/).
• The order of numbers being divided changes the result, making division non-commutative.
• The order in which numbers are divided changes the result, making division non-associative.
• There are three types of division: whole number division, fractional division, and long division.
• Whole number division involves dividing whole numbers.
• Fractional division involves dividing fractions or decimals.
• Long division is a step-by-step procedure for dividing one number by another.