Mechanics 1 Lecture Notes

Statistical Modelling

• Definition: A simplification of a real-world situation represented by equations to make predictions.
• Examples:
  • Rolling a die with equal probability for each face.
  • Heights of adults modeled by a curve.
• Advantages:
  • Quick and easy to produce.
  • Enhances understanding and prediction.
  • Useful for controlling situations like timetables.
• Disadvantages:
  • Simplifies situations, may only apply to specific values.

Representation of Sample Data

• Variables:
  • Qualitative: Non-numerical (e.g., colors).
  • Quantitative: Numerical (e.g., length, time).
  • Continuous: Any value within a range.
  • Discrete: Only specific values (e.g., shoe size).
• Frequency Distributions: Best represented in tables.
• Cumulative Frequency: Adding frequencies progressively.

Graphical Representation

• Stem and Leaf Diagrams: Visual representation of data.
• Grouped Frequency Distributions: No gaps between classes in discrete distributions.
• Cumulative Frequency Curves and Histograms: Visual distribution without gaps for continuous data.

Measures of Central Tendency

• Mode: Most frequent value or class.
• Mean: Sum of values divided by number of values.
• Median: Middle value in ordered list.
• When to Use:
  • Mode for qualitative data.
  • Median for skewed quantitative data.
  • Mean for symmetric data.

Measures of Spread

• Range: Difference between largest and smallest values.
• Interquartile Range: Difference between upper and lower quartiles.
• Variance and Standard Deviation:
  • Variance is the square of standard deviation.
  • Measures spread of distribution.

Probability

• Relative Frequency: Experimental probability.
• Sample Spaces and Events: Set of all possible outcomes.
• Probability Rules: Using Venn diagrams and tree diagrams.

Correlation

• Scatter Diagrams: Shows correlation visually.
• PMCC: Measures strength and direction of linear relationship.
• Coding: Does not affect PMCC.

Regression

• Explanatory and Response Variables: Variables in experiments.
• Regression Line: Line minimizing the sum of squares of deviations.
• Interpretation: Contextual understanding of slope and intercept.

Discrete Random Variables

• Definition: Must take numerical values.
• Probability Distributions: Similar to frequency distributions.
• Expectation: Expected mean of a distribution.

Normal Distribution

• Standard Normal Distribution: Mean 0, standard deviation 1.
• General Normal Distribution: Defined by mean and standard deviation.

Context Questions and Answers

• Accuracy: Guidelines for numerical answers.
• Statistical Models: Uses and process.
• Histograms: When to use based on data type.
• Regression: Interpretation and use of slope/intercept.

Appendix

• PMCC and Regression Line Proofs: Mathematical proofs and coding effects.

This document provides a comprehensive overview of statistical concepts and techniques, focusing on data representation, measures of central tendency and spread, probability, correlation, regression, and both discrete and normal distributions. The material is intended to aid in understanding core statistical principles and applications.