Statistics and Probability Concepts: A Comprehensive Guide

Posted on May 7, 2024 in Mathematics

T.1 Descriptive Statistics

Variables

Absolute frequency (fi): The number of times a value (xi) appears in a dataset.

Relative frequency (hi): The proportion of times a value (xi) appears, calculated as fi/n, where n is the total number of observations.

Cumulative absolute frequency (Fi): The number of times values are less than or equal to xi.

Cumulative relative frequency (Hi): The proportion of values less than or equal to xi, calculated as Fi/n.

Frequency Tables with Grouped Data

Class intervals (Ii, Si): The lower and upper limits of a range of values.

Class mark (xi): The representative value of an interval, calculated as (Ii + Si) / 2.

Class size (ci): The length of the interval, calculated as Si – Ii.

Number of classes: The number of intervals used to group the data.

Graphic Representation of Data

  • Qualitative data: Pie charts and bar charts.
  • Quantitative data without grouping: Frequency polygons and cumulative frequency polygons.
  • Quantitative data with grouping: Histograms and cumulative frequency histograms.

Summary Data – Measures of Position

These measures describe how a variable is distributed across its possible values.

Measures of Central Tendency

These measures identify a central point in the data.

  • Mean: The average value, calculated as the sum of all values divided by the number of values.
  • Median: The middle value when data is ordered, dividing the data into two equal halves.
  • Mode: The most frequently occurring value.

Other Measures of Position

  • Quartiles (Q1, Q2, Q3): Values that divide the ordered data into four equal parts.
  • Percentiles (P1, P2, …, P99): Values that divide the ordered data into one hundred equal parts.

T.2 Probability

Probability theory describes and explores the behavior of random phenomena. A phenomenon is random if its outcome cannot be predicted with certainty.

Key Concepts

  • Sample space: The set of all possible outcomes of a random phenomenon.
  • Event: A subset of the sample space, representing a specific outcome or group of outcomes.

Operations on Events

  • Union (A U B): The event containing all outcomes in A, B, or both.
  • Intersection (A ∩ B): The event containing outcomes that belong to both A and B.
  • Disjoint events: Events with no common outcomes.
  • Difference (A – B): The event containing outcomes in A but not in B.

Probability

The likelihood of an event occurring, expressed as a number between 0 and 1.

Axioms of Probability

  1. The probability of any event is between 0 and 1.
  2. The probability of the entire sample space is 1.
  3. If two events have no common outcomes, the probability of their union is the sum of their individual probabilities.

Properties of Probability

  1. P(A’) = 1 – P(A), where A’ is the complement of event A.
  2. P(A U B) = P(A) + P(B) – P(A ∩ B).
  3. P(A – B) = P(A) – P(A ∩ B).

Laplace’s Rule

If a sample space contains N equally likely outcomes, the probability of each outcome is 1/N. If event A contains k outcomes, then P(A) = k/N.