Introduction to Statistics

Statistics is a system of methods for extracting information from data. We collect data not merely to inspect numbers, but to answer questions about where observations concentrate, how much they vary, whether groups differ, and whether one variable can help explain or predict another.

Descriptive statistics: organizing and presenting data#

Descriptive statistics summarize, organize, and visualize the data actually observed so that their main features are easier to understand.

• Present data in tables and graphs
• Describe the center with a mean or median
• Describe variability with a range, variance, or standard deviation
• Inspect distribution shape and possible outliers

Appropriate summaries and displays depend on the type of data. Identifying the measurement type and distribution is therefore an essential step before choosing a statistical method.

Inferential statistics: learning about a population from a sample#

In most studies we cannot observe an entire population. A census may be too expensive, too slow, or even destructive. We therefore select a sample and use its information to infer features of the population. This is inferential statistics.

• What is a plausible value for the population mean?
• Is there evidence that two populations differ?
• Are two variables associated in the population?
• Could the observed result arise from sampling variation alone?

Why sampling matters#

Different random samples from the same population generally produce different means, proportions, and other statistics. Inference must therefore evaluate not only a sample estimate but also its uncertainty.

How inference handles uncertainty#

• Point estimation: estimate a population parameter with a sample statistic
• Interval estimation: give a range of plausible parameter values
• Hypothesis testing: assess compatibility between data and a statistical hypothesis
• Prediction: use observed data to predict unobserved outcomes
• Statistical modeling: describe relationships among variables

Why learn hypothesis testing?#

Many later methods—t tests, chi-square tests, analysis of variance, and regression—share the hypothesis-testing framework.

1. State a null and an alternative hypothesis
2. Compute a test statistic from the data
3. Evaluate how unusual the result would be if the null hypothesis were true
4. Interpret the p-value together with confidence intervals, effect sizes, and study context