### 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?

Descriptive statistics: A coin was tossed ten times and came down heads six times. Statistical inference: The maximum likelihood estimate of the probability of Heads is $0.6$, or, This information is insufficient to reject the hypothesis that the coin is a fair coin.

Inference without the concept of "population": Assume your data are generated by some (partially) unknown random mechanism/rule. Inferential methods allow to assess properties of this mechanism based on the data. Example: You want to verify an electro-physical formula based on outcomes that can be measured only approximately or under imperfect conditions.

@Michael: Yes; or indeed *make* your data be generated by a known random mechanism - random assignment of experimental treatments.

• 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.

How is it justified to call t or F *scores* (rather than e.g. t-*tests*) inferential statistics?

@jona The t-score is the "statistic" that is used in the t-test, therefore one could describe the t-score as an inferential statistic when used as part of such an inferential process. I guess I have started with the assumption that a statistic is a function of the data. But perhaps you are alluding to the point that we often think of inferential statistics as the broader set of techniques used to do inference?

Let me phrase it differently - isn't a t-statistic a description of a sample, rather than an inferential statement (such as a p-value)?

Well yes, a function of the data is equivalent to a description of a sample. I guess I was thinking that such statistics are used in an inferential process (e.g., researchers relate the t-statistic to a t-distribution to get a p-value and then relate p to alpha to draw an inference). I've often seen textbooks use these examples. But I suppose the p-value and the binary inference itself could be seen as statistics (i.e., functions of the sample data). And the binary inference itself could be seen as most clearly aligned to the inference. Is that what you are getting at?

The definition of the p-value (probability of sample given some population) refers to the population (or alternatively, long-run frequencies), so I'd file it under inferential. The definition of *t* is phrased only in reference to the sample, isn't it?

So for example, you use the data to get to *t* which is related to a distribution, which gives you *p*, which in turn yields a binary inference about a population parameter. So from a frequentist perspective, t, p, and the binary inference are all random variables. All were involved in the inferential process. I'm not sure what the pros and cons are of labelling all or only some such statistics as inferential.

There are also many other ways of doing inference (e.g., bootstrapped confidence intervals, cut-offs on Bayesian posterior densities). So perhaps in those cases the above definitions would need to be tweaked to focus more on the final inference. That said, once I go outside the traditional frequentist test statistics, I tend to think more in terms of inferential procedures rather than needing to clearly distinguish descriptive from inferential statistics.