Statistics Basics for Data Analysis

Population and Samples

• Population: entire group being studied
• Census: data collected for every individual in the population
• Parameter: numerical summary of the population
• Sample: subset of the population, data is collected from
• Statistic: numerical summary of a sample, used to estimate population parameters
• Individuals: entities measured in a study

Descriptive Statistics

• Descriptive statistics: methods to summarize data
  • Tables, graphs, numerical summaries (averages, percentages) are used
  • Researcher gets an overview of data and determines appropriate statistical methods.

Inferential Statistics

• Inferential statistics: methods to extend sample results to the population
• Accuracy of generalizations always contains uncertainty
• Process includes:
  • Identifying research question
  • Collecting necessary data
  • Describing data
  • Making inferences

Types of Variables

• Distribution: information on the possible values a variable can take and how often they occur
• Quantitative Variables: numerical measures
  • Continuous: infinite values within a range (temperature, height)
  • Discrete: limited values, often counts (number of things)
• Qualitative Variables: categorical classifications
  • Nominal: names of things (colors, types of animals)
  • Ordinal: categories with order (ranking)
  • Binary: "yes" or "no"

Relative Frequency Distributions

• Relative frequency distribution: lists each category with the corresponding proportion (relative frequency) of observations
• Pie charts, bar graphs, Pareto charts

Sampling Methods

• Simple random sampling: every individual has an equal chance of selection
• Stratified sampling: population divided into strata, samples from each
• Cluster sampling: population divided into clusters, samples from some clusters.
• Systematic sampling: selecting every kth individual from the population
• Convenience sampling: sample based on convenience (self-selected individuals)

Bias in Sampling

• Sampling bias: favors one part of the population in the sample
• Nonresponse bias: individuals in the sample do not respond
• Response bias: answers don't reflect true opinions due to questions or interviewer

Data Characteristics

• Shape: symmetry, number of peaks, clusters, gaps, outliers
• Center: mean or median (median less affected by outliers)
• Spread: variability in the data (described by standard deviation, IQR, etc)

Standard Deviation

• Standard deviation: measure of spread of numerical data
• If the standard deviation equals 0, all data values are the same.

Percentiles and IQR

• Percentile (p): the pth percentile means that p% of data falls below that value.
• IQR: Interquartile range, difference between Q3 and Q1
  • Q1(25th percentile): median of the data below the median
  • Q2(50th percentile): median of the data
  • Q3(75th percentile): median of the data above the median

Outliers

• IQR Rule for Outliers:
• Lower fence = Q1 - 1.5 * IQR
• Upper fence = Q3 + 1.5 * IQR

Five-Number Summary

• Minimum
• Q1
• Median
• Q3
• Maximum

Scatter diagrams

• Response variable: measures the outcome
• Explanatory variable: provides context/influences outcome
• Linear relationship: scatter plot displays a straight-line trend.

Linear Correlation Coefficient

• Measures strength and direction of linear relationship between two quantitative variables (r)
• -1 ≤ r ≤ 1.
  • r > 0: positive correlation
  • r < 0: negative correlation
  • r close to 0: weak correlation

Correlation vs Causation

• Correlation: measures the relationship strength and direction, not causality.

Least Squares Regression

• Regression line minimizes the sum of squared residuals, finding the best-fitting line
• Regression line is used for prediction.

Residuals

• Prediction error for the value of y
• Data is better predicted when residuals (error) is small.

Probability

• Probability: likelihood of an outcome in the long run
• Probability is concerned with uncertainty of phenomenon outcomes

Small Samples

• Small samples can give misleading results; outcomes might seem unusual instead of random.

Probability Model

• A probability model describes a probability experiment by listing possible outcomes and their likelihoods.

Sample Space

• Sample space: set of all possible outcomes for a probability experiment

Tree Diagrams

• Tree diagrams: visualize the possible outcomes of a sequence of events

Replacement and No Replacement

• Replacement: outcome of each event doesn't affect subsequent events
• No Replacement: outcome of each event affects subsequent events.

Events

• Event: any collection of outcomes within the probability experiment
• Simple events: consist of only one outcome
• Capital letters represent events