# What is the difference between descriptive and inferential statistics?

My understanding was that descriptive statistics quantitatively described features of a data sample, while inferential statistics made inferences about the populations from which samples were drawn.

For the most part, statistical inference makes propositions about
populations, using data drawn from the population of interest via some
form of random sampling.

The “for the most part” has made me think I perhaps don’t properly understand these concepts. Are there examples of inferential statistics that don’t make propositions about populations?

Coming from a behavioural sciences background, I associate this terminology particularly with introductory statistics textbooks. In this context the distinction is that :

• Descriptive statistics are functions of the sample data that are intrinsically interesting in describing some feature of the data. Classic descriptive statistics include mean, min, max, standard deviation, median, skew, kurtosis.
• Inferential statistics are a function of the sample data that assists you to draw an inference regarding an hypothesis about a population parameter. Classic inferential statistics include z, t, $\chi^2$, F-ratio, etc.

The important point is that any statistic, inferential or descriptive, is a function of the sample data. A parameter is a function of the population, where the term population is the same as saying the underlying data generating process.

From this perspective the status of a given function of the data as a descriptive or inferential statistic depends on the purpose for which you are using it.

That said, some statistics are clearly more useful in describing relevant features of the data, and some are well suited to aiding inference.

• Inferential statistics: Standard test statistics like t and z, for a given data generating process, where the null hypothesis is false, the expected value is strongly influenced by sample size. Most researchers would not see such statistics as estimating a population parameter of intrinsic interest.
• Descriptive statistics: In contrast descriptive statistics do estimate population parameters that are typically of intrinsic interest. For example the sample mean and standard deviation provide estimates of the equivalent population parameters. Even descriptive statistics like the minimum and maximum provide information about equivalent or similar population parameters, although of course in this case, much more care is required. Furthermore, many descriptive statistics might be biased or otherwise less than ideal estimators. However, they still have some utility in estimating a population parameter of interest.

So from this perspective, the important things to understand are:

• statistic: function of the sample data
• parameter: function of the population (data generating process)
• estimator: function of the sample data used to provide an estimate of a parameter
• inference: process of reaching a conclusion about a parameter

Thus, you could either define the distinction between descriptive and inferential based on the intention of the researcher using the statistic, or you could define a statistic based on how it is typically used.