Descriptive Statistics Overview

Descriptive Statistics

• Continuous Data Distributions: Data can be either odd-shaped or bell-shaped.
• Odd-Shaped Data: Use the median to describe centrality and quartiles to describe spread because mean is skewed by the outliers.
• Bell-Shaped Data: Use the mean for central tendency and the standard deviation for spread.
• Skewness impact: Right skew (positive): mean > median; Left skew (negative): mean < median.

Categorical vs. Continuous Data

• Categorical Data: More than two responses, non-numeric, ordered (ordinal) or unordered (nominal). Examples include country of birth or Likert scales.
• Continuous Data: Data can take any value in a given range, numerical, with infinite resolution. Examples include weight, age, and height.

Data Visualization

• Categorical Data (1 variable): Bar Chart
• Categorical Data (2 variables): Clustered Bar Chart
• Continuous Data (1 variable): Box Plot or Histogram
• Continuous Data (2 variables): Scatterplot
• Continuous & Categorical Data: Two Boxplots or two Histograms

Hypothesis Testing Assumptions

• Independent Samples T-test:
  • Normal distribution of continuous variables (check histograms).
  • Independent samples (individuals within samples and between samples).
  • Equal variances.
• Chi-Square Test:
  • Categorical data.
  • Independent observations (individuals within/between groups).
  • Expected frequencies (in each category) > 5.
• Sign Test:
  • Ordinal or continuous data.
  • Independent observations.
  • Paired samples.
  • Symmetrical differences (differences between paired observations).
• Mann-Whitney U test:
  • Independent groups.
  • Rankable data.
  • Compares two groups, not normally distributed categorical data (ordinal).
• Wilcoxon Test:
  • Paired samples.
  • Ordinal categorical data (rankable data).

Interpretation of p-values

• p < 0.05: Reject the null hypothesis; statistically significant.
• p ≥ 0.05: Accept the null hypothesis; not statistically significant.
• Lack of significance: Does not confirm equivalence, may indicate insufficient data. Consider increasing sample size.

Confidence Intervals (95%)

• Formula: Estimate ± (2 x Standard Error)
• Interpretation: 95% confident that true population value is within the limits.
• Factors impacting width: Sample size and standard deviation.
• Consistency with p-values: 95% CI includes zero, p ≥ 0.05 (lack of significant difference); CI does not include zero, p < 0.05 (significant difference).

Standard Deviation vs. Standard Error

• Standard Error (SE): Measure of precision (of an estimate); standard deviation of estimates, related to n.
• Standard Deviation (SD): Measure of spread (of the data itself).

Sample Size Estimation

• Formula: n = 1.96 x (m/SE)² where m = margin of error

Confounding vs. Effect Modification

• Confounding: Distortion of the effect of a risk factor on an outcome. Related to exposure and outcome (but not in causal pathway). Example: Age confounding smoking & lung cancer.
• Effect Modification: Different relationship between exposure and outcome depending on a third variable. Example: Gender modifying the effect of a medicine on blood pressure.
• Identification: Compare odds ratios or risk ratios to see if relationship changes with categories.

Probability Sampling

• Simple Random Sampling: Each sample has equal selection chance. Example: Random name selection. Strengths = minimizes bias. Weaknesses = underrepresentation for small populations, requires a complete list, potentially difficult to attain.
• Stratified Sampling: Randomly sampling from subgroups/strata with shared characteristics. Example: GPA study across degrees (English, Science, Engineering) . Strengths: precise representation, all subgroups. Weaknesses: Requires detailed population information, not easy to survey.
• Systematic Sampling: Select every kth unit after a random starting point. Example: Every 10th name. Strengths = even distribution/representation. Weaknesses: Sensitive to patterns within population.
• Cluster Sampling: Randomly selecting entire clusters/groups. Example: Randomly selecting schools. Strengths = large populations over wide area. Weaknesses = increased sampling error.
• Two-Stage Sampling: Two levels of selection. Example: survey streaming platform users. Strengths: cost-effective, wide-area representation. Weaknesses: biased results for uneven population distribution across clusters.

Identifying Statistical Significance with Difference in Means/Risk and Confidence Intervals

• Difference in Mean: 95% CI does not include 0 → significant difference; includes 0 → no significant difference.
• Risk: 95% CI does not include 1 → significant risk difference; includes 1 → no significant risk difference.

This quiz covers the fundamentals of descriptive statistics, including data distributions, key differences between categorical and continuous data, and visualization techniques. Understand the appropriate measures of central tendency and spread for various data types, as well as how to effectively represent data visually.