Summary of Statistics Concepts

What is Statistics?

  • Statistics: The science of collecting, analyzing, presenting, and interpreting data.
  • Types of data:
    • Quantitative: Measures how much or how many.
    • Qualitative: Labels for categories of like items.

Types of Statistics

  • Descriptive Statistics: Organizes and summarizes data.

    • Uses tables/graphs (e.g., averages) for summaries.
    • Descriptive values:
    • Parameter: Descriptive value for a population.
    • Statistic: Descriptive value for a sample.

  • Inferential Statistics: Uses sample data to make inferences about populations.

    • Limited information due to sampling.
    • Examples: Hypothesis testing, Regression Analysis.

Population and Sample

  • Population: Entire group of individuals (e.g., class size and academic performance of all freshmen).
  • Sample: A selected representation of the population for study.

Variables in Statistics

  • Variable: Any characteristic that can be measured or counted (e.g., age, income).
  • Types of Quantitative Variables:
    • Discrete: Indivisible categories (e.g., class size).
    • Continuous: Infinitely divisible (e.g., time, weight).

Levels of Measurement

  1. Nominal: Categorizes and labels variables without order.
  2. Ordinal: Ranks categories in order (e.g., satisfaction levels).
  3. Interval: Equal intervals between measurements but no true zero (e.g., temperature).
  4. Ratio: All properties of interval plus true zero (e.g., weight, height).

Sample Size

  • Sample size is the proportion of the population studied.
  • Use Slovin's formula:
    n = \frac{N}{1 + Ne^2}
  • Where:
    • n = Sample Size
    • N = Total Population
    • e = Margin of Error

Probability Sampling Techniques

  1. Simple Random Sampling: Each member has a chance to be included.
  2. Systematic Sampling: Members selected at regular intervals (k = N/n).
  3. Stratified Sampling: Population divided into strata, samples taken from each stratum.

Non-Probability Sampling Techniques

  • Includes quota, purposive, and convenience sampling.

Frequency Distribution

  • Describes occurrences of distinct values.
  • Grouped and Ungrouped types exist.

Graphical Presentations

  • Present data in bar graphs, pie charts, etc.

Descriptive Statistics Measures

  • Mean: Average value.
  • Median: Middle score when ordered.
  • Mode: Most frequent score.
  • Range: Difference between largest and smallest values.
  • Standard Deviation: Indicates variability around the mean.

Normal Distribution

  • Bell-shaped curve where most data cluster around the mean.
  • Characteristics: Symmetric, unimodal, mean=median=mode.

Z-Scores

  • Measures how many standard deviations a score is from the mean.
  • Formula: z = \frac{x - \bar{x}}{s}

Correlation

  • Measures statistical relationship between two variables.
  • Pearson's r: Measures linear correlation.
    r = \frac{n(\Sigma xy) - (\Sigma x)(\Sigma y)}{\sqrt{[n(\Sigma x^2) - (\Sigma x)^2][n(\Sigma y^2) - (\Sigma y)^2]}}
  • Spearman's rank: Non-parametric measure of correlation. \rho = 1 - \frac{6 \Sigma d^2}{n(n^2 - 1)}
    • Where d is the difference in ranks.