Population and sample are foundational concepts in theoretical statistics. They form the basis for understanding how we can make inferences about large groups using smaller, manageable datasets.

Sampling methods, sample size considerations, and potential biases all play crucial roles in statistical analysis. These concepts help us bridge the gap between what we can observe and what we aim to understand about entire populations.

Definition of population vs sample

• Population encompasses all individuals or items of interest in a statistical study, forming the complete set from which data can be collected
• Sample represents a subset of the population, selected to make inferences about the larger group
• Understanding the relationship between population and sample is crucial for accurate statistical analysis and interpretation in theoretical statistics

Finite vs infinite populations

• Finite populations contain a countable number of elements (all students in a university)
• Infinite populations have an unlimited or uncountable number of elements (all possible outcomes of rolling a die)
• Sampling approaches differ based on population type, impacting statistical methods and inference

Complete vs incomplete samples

• Complete samples include every member of the population, providing exhaustive data
• Incomplete samples contain only a portion of the population, more common in practical research
• Sample completeness affects statistical power and generalizability of results

Sampling methods

• Various techniques exist to select representative samples from populations
• Choice of sampling method impacts the validity and reliability of statistical inferences
• Understanding different sampling approaches is essential for designing robust statistical studies

Simple random sampling

• Each member of the population has an equal probability of selection
• Utilizes random number generators or lottery methods for unbiased selection
• Provides a foundation for many statistical theories and inferential techniques

Stratified sampling

• Divides population into homogeneous subgroups (strata) before sampling
• Ensures representation from all relevant subgroups within the population
• Improves precision and reduces sampling error compared to simple random sampling

Cluster sampling

• Divides population into clusters, then randomly selects entire clusters
• Useful for geographically dispersed populations or when individual sampling is impractical
• Can be less precise than other methods but often more cost-effective

Systematic sampling

• Selects every kth element from the population after a random starting point
• Requires a sorted list of population elements
• Can introduce bias if the population has a cyclical pattern aligned with the sampling interval

Sample size considerations

• Determining appropriate sample size is crucial for balancing statistical power and resource constraints
• Larger samples generally provide more precise estimates but increase cost and time requirements
• Sample size calculations involve multiple factors and statistical formulas

Margin of error

• Represents the maximum expected difference between the true population parameter and the sample estimate
• Expressed as a percentage, typically ranging from 1% to 10%
• Inversely related to sample size: larger samples yield smaller margins of error

Confidence level

• Probability that the true population parameter falls within the confidence interval
• Common levels include 90%, 95%, and 99%
• Higher confidence levels require larger sample sizes to maintain the same margin of error

Population variability

• Degree of diversity or heterogeneity within the population
• Greater variability requires larger samples to achieve the same level of precision
• Estimated using measures like standard deviation or variance from prior studies or pilot data

Sampling distributions

• Theoretical distributions of sample statistics obtained from repeated sampling
• Form the basis for inferential statistics and hypothesis testing
• Understanding sampling distributions is crucial for estimating population parameters

Central limit theorem

• States that the sampling distribution of the mean approaches a normal distribution as sample size increases
• Applies regardless of the underlying population distribution, given a sufficiently large sample size
• Enables the use of normal distribution-based statistical techniques for many types of data

Standard error

• Measures the variability of a sample statistic across multiple samples
• Calculated as the standard deviation of the sampling distribution
• Decreases as sample size increases, improving the precision of parameter estimates

Sampling bias

• Systematic errors in the sample selection process that lead to non-representative samples
• Can significantly distort statistical inferences and conclusions
• Identifying and mitigating sampling bias is crucial for valid statistical analysis

Selection bias

• Occurs when certain members of the population are more likely to be included in the sample
• Can result from flawed sampling procedures or self-selection by participants
• Leads to overrepresentation or underrepresentation of specific population subgroups

Non-response bias

• Arises when individuals chosen for the sample do not participate or provide incomplete data
• Can occur due to refusal, inability to contact, or survey fatigue
• May introduce systematic differences between respondents and non-respondents

Voluntary response bias

• Results from samples composed of self-selected volunteers
• Often leads to overrepresentation of individuals with strong opinions or interests
• Can severely skew results, particularly in opinion polls or surveys

Parameter vs statistic

• Parameters describe characteristics of populations, while statistics describe samples
• Understanding the distinction is fundamental to inferential statistics
• Theoretical statistics focuses on using sample statistics to estimate population parameters

Population parameters

• Fixed, unknown values that describe the entire population
• Denoted by Greek letters (μ for mean, σ for standard deviation)
• Typically the target of estimation in statistical inference

Sample statistics

• Calculated values from sample data used to estimate population parameters
• Denoted by Roman letters (x̄ for sample mean, s for sample standard deviation)
• Vary from sample to sample due to random sampling variation

Estimation theory

• Branch of statistics focused on using sample data to estimate population parameters
• Involves developing and evaluating estimators for various statistical properties
• Central to many applications of theoretical statistics in real-world problems

Point estimation

• Provides a single value as the best guess for a population parameter
• Utilizes estimators like sample mean, median, or proportion
• Evaluated based on properties such as unbiasedness, consistency, and efficiency

Interval estimation

• Produces a range of values likely to contain the true population parameter
• Confidence intervals are the most common form of interval estimates
• Balances precision with the level of confidence in the estimate

Sampling frame

• List or procedure used to identify and select members of the target population
• Crucial for ensuring that the sample accurately represents the population of interest
• Imperfections in the sampling frame can lead to various types of bias

Coverage error

• Occurs when the sampling frame does not accurately represent the target population
• Can result in undercoverage (exclusion of population subgroups) or overcoverage (inclusion of ineligible units)
• Impacts the generalizability of study results to the entire population

Sampling frame bias

• Systematic differences between the sampling frame and the target population
• Can arise from outdated lists, incomplete databases, or exclusion of certain population segments
• Requires careful consideration and potential adjustments in the sampling design

Resampling techniques

• Statistical methods that involve repeatedly drawing samples from the original dataset
• Used for estimating the sampling distribution of a statistic empirically
• Particularly useful when theoretical distributions are unknown or difficult to derive

Bootstrap sampling

• Involves repeatedly sampling with replacement from the original dataset
• Generates multiple resamples of the same size as the original sample
• Used to estimate standard errors, construct confidence intervals, and perform hypothesis tests

Jackknife sampling

• Systematically leaves out one observation at a time from the original sample
• Calculates the statistic of interest for each reduced dataset
• Useful for estimating bias and variance of estimators

Sample representativeness

• Degree to which a sample accurately reflects the characteristics of the population
• Critical for making valid inferences and generalizations from sample data
• Influenced by sampling method, sample size, and potential biases

Generalizability

• Extent to which findings from a sample can be applied to the broader population
• Depends on the sampling method, sample size, and similarity between sample and population
• Crucial for applying statistical results to real-world situations or policy decisions

External validity

• Refers to the applicability of study findings beyond the specific sample and context
• Influenced by factors such as sample representativeness and study design
• Important consideration when extrapolating results to different populations or settings