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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.

However, the wikipedia page for statistical inference states:

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?

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  • $\begingroup$ 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. $\endgroup$
    – Dilip Sarwate
    Oct 5, 2013 at 13:44
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    $\begingroup$ 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. $\endgroup$
    – Michael M
    Oct 5, 2013 at 17:09
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    $\begingroup$ @Michael: Yes; or indeed make your data be generated by a known random mechanism - random assignment of experimental treatments. $\endgroup$
    – Scortchi - Reinstate Monica ♦
    Oct 5, 2013 at 17:23

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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.

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  • $\begingroup$ How is it justified to call t or F scores (rather than e.g. t-tests) inferential statistics? $\endgroup$
    – jona
    Oct 5, 2013 at 13:41
  • $\begingroup$ @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? $\endgroup$
    – Jeromy Anglim
    Oct 5, 2013 at 13:47
  • $\begingroup$ 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)? $\endgroup$
    – jona
    Oct 5, 2013 at 13:56
  • $\begingroup$ 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? $\endgroup$
    – Jeromy Anglim
    Oct 5, 2013 at 14:06
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    $\begingroup$ 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. $\endgroup$
    – Jeromy Anglim
    Oct 5, 2013 at 14:36
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One form of inference is based on the random assignment of experimental treatments, & not on random sampling from a population (even hypothetically). Oscar Kempthorne was a proponent.

The first example in Edgington (1995), Randomization Tests illustrates the approach well. A researcher obtains ten subjects, divides them into two groups at random, allocates treatment $A$ to one group & $B$ to the other, measures their responses & calculates Student's t-statistic for the difference in group means. Rather than using normal sampling theory to assess significance he calculates $t$ for every possible way the treatments might have been assigned (there are 252 of them); then, noting that each permutation is equally probable under the null hypothesis of no treatment effect, he sees that nine give a higher value of $t$ than that observed & calculates a p-value of $10/252=0.04$. "Obtains" here, as very often, could mean anything at all—perhaps the first ten undergraduates at his lecture to put up their hands were picked—but with this analysis there's no need to maintain the pretence that the subjects have been randomly sampled from the population of interest (the downside is that any generalization beyond these ten is extra-statistical).

Prediction is another area where you're not necessarily formulating propositions about populations. (I don't know that everyone would want to call prediction "inference", but there's Geisser (1993), Predictive Inference: An Introduction). Often prediction follows from a fitted population model, but not always; e.g. @Matt's classification example, model averaging (Bayesian or based on Akaike weights), or forecasting algorithms such as exponential smoothing.

NB I think "inferential vs descriptive statistics" more often refers to the discipline Statistics, rather than to quantities calculated from samples. There's no essential difference between an inferential & a descriptive statistic; as @Jeremy has pointed out, it's a matter of what use you're putting it to.

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I'm not sure that classification necessarily makes a statement about the population(s) from which the data points are drawn. Classification, as you probably know, uses training data consisting of some "feature" vectors, each labelled with a specific class, to predict the class labels belonging to other unlabeled feature vectors. For example, we might use a patient's vital signs and a doctor's diagnosis to predict whether other patients are healthy or ill.

Some classifiers, called "generative classifiers", try to explicitly model the populations or data generating process that produces each class. For example, the Naive Bayes algorithm computes $P(\textrm{class}=c|\textrm{features})$ for each class $c$, assuming that the features are all independent. These models could reasonably be seen as statements about the population.

However, other classifiers look for differences between the classes without modeling the classes themselves; these are called discriminative classifiers. One classic example is the nearest neighbour classifier, which assigns an unlabeled example to the class of its closest neighbor (where close is defined in some sensible way for the problem). This doesn't seem like it contains much, if any, information about the populations from which the data points were drawn.

If you are interested in the difference between descriptive and inferential statistics, it might be more fruitful to think about the purpose of the analysis. A descriptive statistic, like the mean, might tell you how many trout are in a typical lake--they describe something. An inferential statistic, like a $t$-test, might tell you if there are typically more trout than bass in these lakes-- it lets you make a claim about a descriptive statistic.

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In one line, given the data, descriptive statistics try to summarize the content of your data with minimum loss of information ( depending on what measure do you use). You get to see the geography of the data.( Something like, see the performance graph of the class and say who is on top, the bottom and so on)

In one line, given the data, you try to estimate and infer to the properties of the hypothetical population from which the data comes from. ( Something like, understanding 7th grade students through the good sample from the class, assuming that the underlying population is large enough that you cannot take them into account in totality)

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    $\begingroup$ I don't think it's a definition or characterisation of descriptive statistics that they aim for minimum loss of information. It's entirely possible to have descriptive statistics that leave out really important detail and that's often a problem. $\endgroup$
    – Nick Cox
    Oct 18, 2013 at 19:43
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In Short

Descriptive statistics is the analysis of data that describe, show or summarize data in a meaningful; it simply a way to describe our data/talk about the whole population. some of them are Measures of central tendency and Measure of dispersion

Inferential statistics is technique that allow us to use samples to make generalizations about the populations from which the samples were drawn.example hypothesis testing and

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