Important Definitions: All Exams

Note on Statistical Concepts

Page 1: Definitions

  • Symbols

    • Proportion: A number between -1 and 1.

    • Percentage: A number between 0 and 100.

    • Population: The entire set of units or objects we want to study.

    • Variable: An attribute of a unit or object.

    • Parameter: Summary measure calculated from a population.

    • Statistic: Summary measure calculated from a sample.

    • Sample Mean (x̄): Average of sample data.

    • Population Mean (μ): Average of population data.

    • Mean: Average value.

    • Median: Middle number in a sorted list.

    • Mode: Measurement that occurs most frequently.

Page 2: Variability

  • Variability: Measures the extent to which data points diverge.

    • Range: Difference between the largest and smallest values.

    • Standard Deviation: Positive square root of the variance.

      • Population Standard Deviation (σ): Standard deviation for the entire population.

      • Sample Standard Deviation (s): Standard deviation for a sample.

    • p-th Percentile: The value below which p% of the data falls.

    • Z-Score: Standardized score indicating how many standard deviations an element is from the mean.

Page 3: Probability Concepts

  • Sample Point: Basic outcome of an experiment.

  • Sample Space: Collection of all sample points of an experiment.

  • Event: Collection of sample points.

  • Mutually Exclusive Events: Events that cannot occur simultaneously.

  • Independent Events: The occurrence of one event does not affect the probability of another.

Page 4: Random Variables

  • Binomial Random Variable (X): Number of successes in n trials.

    • Trials are identical and independent.

    • Only two possible outcomes per trial (success or failure).

    • Probability of success remains constant across trials (p), with failure probability (q) = 1 - p.

  • Poisson Random Variable (X): Number of events occurring in a given time, area, or volume.

    • Probability of an event occurring is constant across units.

    • The occurrence of events is independent of each other.

    • Mean number of events in each unit is a constant value denoted by λ (lambda).

Page 5: Summary Statistics

  • Parameter: Numerical descriptive measure of a population.

  • Sample Statistic: Numerical descriptive measure calculated from observations in a sample.

  • Chart Parameters:

    • Sample Mean (M)

    • Standard Deviation (σ)

    • Proportion (P)

  • Central Limit Theorem (CLT): States that the distribution of sample means approaches a normal distribution as the sample size increases, typically with a sample size of 30