Lecture Notes: Statistics, Geometry, and Trigonometry

Statistics

Basics

• Population vs. Sample:
  • Population: Entire group.
  • Sample: Subset of the population.
• Group Data: Uses intervals (e.g., 11-15).
• Ungrouped Data: Lists specific numbers.

Frequency Table

• Columns: Interval, Frequency (f), Midpoint (x), f(x).
• Calculating Mean:
  • Add a column for midpoints (x).
  • Multiply frequency by midpoint (f(x)).
  • Mean = Sum of f(x) / Sum of f.

Measures of Central Tendency

• Mode: Most frequently occurring number.
• Median: Middle value when data is ordered.
• Mean: Average (sum of values / number of values).

Measures of Spread

• Range: Highest value - Lowest value.
• Interquartile Range: Q3 - Q1.
• Standard Deviation: Spread of data from mean.

Probability

• Basic Probability: Desired outcomes / Total outcomes.
• Probability Space: Used when multiple events occur (e.g., dice rolls).

Diagrams

• Pie Charts: Use protractor to draw sectors according to angles.
• Bar Graphs vs. Histograms:
  • Bar Graphs: Discrete data, bars don't touch.
  • Histograms: Continuous data, bars touch.

Geometry

Basic Concepts

• Lines and Planes: Infinite in length, planes are flat surfaces.
• Angles: Measured in degrees.
• Polygons: Shapes with straight sides (e.g., triangles, quadrilaterals).

Circle Theorems

• Angle in a Semicircle: Always 90°.
• Angles from the Same Chord: Are equal.
• Cyclic Quadrilateral: Opposite angles sum to 180°.
• Tangent-Chord Theorem: Angle between tangent and chord equals angle in opposite segment.

Transformations

• Translation: Slide using a vector.
• Reflection: Flip over a line.
• Rotation: Turn around a point.
• Enlargement: Scale by a factor.

Trigonometry

Pythagorean Theorem

• Formula: c² = a² + b².

SOA CAH TOA

• SOA: sin(θ) = opposite/hypotenuse.
• CAH: cos(θ) = adjacent/hypotenuse.
• TOA: tan(θ) = opposite/adjacent.

Sine and Cosine Rules

• Sine Rule: For non-right triangles with opposite angles and sides.
• Cosine Rule: For non-right triangles with three sides and one angle.

Bearings

• Drawings: Use North as reference, measured clockwise.
• Common Angles: Use trigonometry to solve for unknowns.

These notes cover essential concepts in statistics, geometry, and trigonometry, focusing on practical application and solving problems using these mathematical principles. Remember to practice drawing diagrams and solving past paper questions to reinforce understanding.