Statistics: Descriptive and Inferential

Descriptive Statistics

• Definition: Summarizes and describes the features of a data set.
• Key Measures:
  • Central Tendency: Indicates the center of a data set.
    • Mean: Average of all data points.
    • Median: Middle value when data points are ordered.
    • Mode: Most frequently occurring value.
  • Dispersion: Describes the spread of data.
    • Range: Difference between the maximum and minimum values.
    • Variance: Average squared deviation from the mean.
    • Standard Deviation: Square root of variance, indicating average distance from the mean.
  • Shape of Distribution:
    • Skewness: Measure of asymmetry in data distribution.
    • Kurtosis: Measure of the "tailedness" of the distribution.
• Visual Representation:
  • Histograms: Bar graphs representing frequency distribution.
  • Box Plots: Visual representation of the median, quartiles, and outliers.

Inferential Statistics

• Definition: Makes predictions or inferences about a population based on a sample.
• Key Concepts:
  • Population vs. Sample:
    • Population: Entire group of interest.
    • Sample: Subset of the population used for analysis.
  • Estimation:
    • Point Estimation: Single value estimate of a population parameter.
    • Interval Estimation: Range of values, usually expressed as a confidence interval.
  • Hypothesis Testing:
    • Null Hypothesis (H0): Statement of no effect or no difference.
    • Alternative Hypothesis (H1): Statement indicating the presence of an effect or difference.
    • Type I Error: Rejecting a true null hypothesis.
    • Type II Error: Failing to reject a false null hypothesis.
  • P-Value: Probability of obtaining results at least as extreme as the observed results, under the assumption that the null hypothesis is true.
  • Confidence Levels: Commonly used levels include 90%, 95%, and 99% which indicate the likelihood that the true parameter lies within the interval.
• Techniques:
  • T-tests: Compare means between two groups.
  • ANOVA: Compare means among three or more groups.
  • Regression Analysis: Examines relationships between variables.

Descriptive Statistics

• Summarizes and describes features of a data set, allowing for an understanding of its main characteristics.

• Central Tendency measures:

  • Mean: Average calculated by summing all data points and dividing by the number of points.
  • Median: The middle value when data points are arranged in order, dividing the data set into two equal halves.
  • Mode: The value that appears most frequently in the data set.

• Dispersion measures the spread of the data:

  • Range: The difference between the maximum and minimum values, indicating how far apart values are.
  • Variance: Average of the squared deviations from the mean, representing data spread.
  • Standard Deviation: The square root of variance, illustrating the average distance of data points from the mean.

• Shape of Distribution includes:

  • Skewness: Quantifies the asymmetry of the data distribution, indicating whether data points are concentrated on one side of the mean.
  • Kurtosis: Measures the "tailedness" or sharpness of the distribution, indicating the presence of outliers.

• Visual Representations of data distributions:

  • Histograms: Bar graphs that display frequency distribution of data, showing how data points are distributed across different intervals.
  • Box Plots: Illustrate median, quartiles, and potential outliers, providing a visual summary of the data's distribution.

Inferential Statistics

• Enables predictions or inferences about a broader population based on analysis of a sample.

• Population vs. Sample distinction:

  • Population: The complete group of individuals or items being studied.
  • Sample: A smaller subset selected from the population, used for analysis to draw conclusions about the population.

• Estimation techniques:

  • Point Estimation: A single value estimate of a population parameter, providing a best guess.
  • Interval Estimation: A range of values (confidence interval) suggesting where the true population parameter lies with a certain level of confidence.

• Hypothesis Testing framework:

  • Null Hypothesis (H0): Assumes no effect or difference exists; tested against alternative hypotheses.
  • Alternative Hypothesis (H1): Proposes that there is an effect or difference.
  • Type I Error: Incorrectly rejecting a true null hypothesis (false positive).
  • Type II Error: Failing to reject a false null hypothesis (false negative).

• P-Value: The probability of observing results as extreme as the ones obtained, assuming the null hypothesis is true; used to determine statistical significance.

• Confidence Levels: Common levels include:

  • 90%, 95%, and 99% indicate the likelihood that the population parameter lies within the specified confidence interval.

• Statistical Techniques for analysis include:

  • T-tests: Used to compare means between two groups, examining their statistical significance.
  • ANOVA (Analysis of Variance): Used for comparing means among three or more groups.
  • Regression Analysis: Investigates relationships between variables, helping in prediction and modeling.