Inferential Statistics

Abstract

In many topics within criminology and criminal justice research, we want to draw conclusions from our data that are generalizable to wider populations. These make our findings relevant to the real world, and not specific to any one study or any one dataset. This is where inferential statistics are particularly useful. There are a number of key concepts underlying inferential statistics that we can demonstrate visually within R. This chapter will review sampling from a population, standard error/confidence intervals, and how to generate data based on a distribution in R. In doing so, you will generate a synthetic dataset of intelligence quotient (IQ) scores for each of the approximately 3.6 million probationers in the United States.

Key Terms

Bell Curve

See Gaussian distribution.

Central limit theorem

A theorem that states: “If repeated independent random samples of size N are drawn from a population, as N grows large, the sampling distribution of sample means will be approximately normal.” The central limit theorem enables the researcher to make inferences about an unknown population using a normal sampling distribution.

Confidence interval

An interval of values around a statistic (usually a point estimate). If we were to draw repeated samples and calculate a 95% confidence interval for each, then in only 5 in 100 of these samples would the interval fail to include the true population parameter. In the case of a 99% confidence interval, only 1 in 100 samples would fail to include the true population parameter.

Gaussian distribution

Normal distribution or bell curve.

Inferential statistics

A broad area of statistics that provides the researcher with tools for making statements about populations on the basis of knowledge about samples. Inferential statistics allow the researcher to make inferences regarding populations from information gained in samples.

Population

The universe of cases that the researcher seeks to study. The population of cases is fixed at a particular time (e.g., the population of the United States). However, populations usually change across time.

Population distribution

The frequency distribution of a particular variable within a population.

Sample

A set of actual observations or cases drawn from a population.

Sample distribution

The frequency distribution of a particular variable within a sample drawn from a population.

Sample statistic

A characteristic of a sample—for example, the mean number of previous convictions in a random sample of 1,000 prisoners.

Sampling distribution

A distribution of all the results of a very large number of samples, each one of the same size and drawn from the same population under the same conditions. Ordinarily, sampling distributions are derived using probability theory and are based on probability distributions.

Standard error

The standard deviation of a sampling distribution.

Synthetic data

Computer-generated data.