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Inferential Statistics Overview

Sep 16, 2025

Overview

This lecture introduces inferential statistics, focusing on how sample data is used to estimate population parameters and the importance and calculation of confidence intervals.

Descriptive vs. Inferential Statistics

  • Descriptive statistics summarize and visualize characteristics of a sample.
  • Inferential statistics use sample data to make statements about population characteristics.

Sampling and Population Concepts

  • Research often uses samples because studying entire populations is impractical.
  • The sample (n) is a subset of the population (N).
  • The sample mean (xÌ„) estimates the population mean (μ).

Parameters, Statistics, and Variables

  • A population is all individuals of interest.
  • A parameter is a summary characteristic of a population (e.g., μ).
  • A statistic is a summary characteristic of a sample (e.g., xÌ„).
  • The variable is the measured trait, such as glucose level.

Sampling Distribution & Central Limit Theorem

  • The sampling distribution of the mean is formed by repeatedly sampling and calculating means.
  • The mean of the sampling distribution equals the population mean.
  • The standard deviation of the sampling distribution (standard error) is σ/√n.
  • As sample size increases, standard error decreases and the sampling distribution becomes more normal, regardless of population distribution.
  • Central Limit Theorem: With sufficiently large n (about 30+), the sampling distribution of the mean approximates normality.

Standard Error and Confidence Intervals

  • Standard error (SE or SEM) measures the precision of the sample mean as an estimate of the population mean.
  • A larger sample size gives a smaller standard error and more precise estimates.
  • 95% confidence interval: xÌ„ ± 2 × SEM gives the range that likely contains the population mean.
  • Wider confidence intervals indicate less certainty about the estimate; narrower intervals mean more precision.
  • Increasing the confidence level (e.g., to 99%) widens the interval.

Reporting and Visualization

  • Use standard deviation to show variability within a sample.
  • Use confidence intervals to show uncertainty in estimating the population mean.
  • Visualize mean and confidence intervals using bar graphs with error bars.

Key Terms & Definitions

  • Population — entire group of interest.
  • Sample — subset drawn from the population.
  • Parameter (μ) — numerical summary of the population.
  • Statistic (xÌ„) — numerical summary of the sample.
  • Variable — measured trait or characteristic.
  • Sampling Distribution — distribution of sample means from repeated sampling.
  • Standard Error (SEM or SE) — standard deviation of the sampling distribution.
  • Confidence Interval (CI) — range likely to contain the true population parameter.

Action Items / Next Steps

  • Practice calculating confidence intervals with different sample sizes and standard deviations.
  • Prepare bar graphs with error bars for visualizing confidence intervals.
  • Review the differences between standard deviation and standard error.

Full transcript