# Do descriptive statistics have p-values?

I'm being asked to find the p-values for descriptive statistics. However, it's my understanding that p-values are for test statistics. If I'm not mistaken, a p-value is the probability of observing a value as extreme as the test statistic if the null hypothesis were true.

• Maybe the person was referring to testing mean differences between groups if you have several groups (e.g. gender)? Or if you have odds ratios, the test that they're not 1 in the population... something like that. – Patrick Coulombe May 5 '15 at 19:04
• What an odd question! Descriptive is an extrinsic property, viz that of not being used for inferential purposes; so descriptive statistics can't be used for inference, much as bachelors can't be married. But there's no intrinsic property of any statistic that prevents it being used for inference, much as any bachelor can get married. Do you have practical concerns about how the data were gathered that make you doubt the propriety of whichever inferential procedures you're being urged to carry out? – Scortchi - Reinstate Monica May 5 '15 at 20:22
• A statistic is just that -- a statistic, a value calculated from a sample. It doesn't have a p-value. P-values come from hypothesis tests, so to generate a p-value for some statistic it must be used in some hypothesis test. What's the hypothesis? [I'd generally suggest not testing any more hypotheses than necessary.] – Glen_b May 6 '15 at 2:20
• See also this closely related post. – Glen_b May 6 '15 at 2:22
• In medicine, it's the norm that your table 1 include some kind of comparison, usually by exposure group. However, even when lacking the need to make a comparison between groups, people (co-authors, reviewers) will insist you compare something - which often defaults to comparing males and females. We would be better off taking that space reserved for pointless tests to give fuller summaries of the data. – D L Dahly May 27 '15 at 8:50

Your are correct. Descriptive statistics characterize the data with which you are working. To generate p-values, assumptions need to be generated. Assumptions are not descriptive.

Descriptive statistics do not have p-values. Hypothesis tests, which can test whether or not a descriptive statistic equals a specific value, can have p-values. Whoever asked you to get p-values for descriptive statistics likely meant for you to get a p-value for whether or not that descriptive statistic equals 0. I recommend you follow up and clarify this.

What you can do is get a confidence interval for a descriptive statistic which tells you much of the same thing.

• Confidence intervals are essentially the same as p-values. Consider this, in non-Bayesian statistics you compute confidence at a given significance, say $\alpha=0.05$, it's not a p-value but it's its twin brother, or a sister – Aksakal May 5 '15 at 19:43
• Confidence intervals are also inferential statistics. – Alexis May 5 '15 at 20:28
• Not sure why I'm getting downvoted here. I am not claiming that confidence intervals are telling you something different than a p-value. I am saying that you don't know what to do when someone says "get me a p-value on that mean!" but you can get a a confidence interval on that mean which tells much of the same thing. – TrynnaDoStat May 6 '15 at 14:29
• Although I did not downvote, I held back from my initial reflex of upvoting because the last paragraph, confusingly, almost seems to contradict what you previously said. A confidence interval cannot be related to a p-value in the absence of a hypothesis. Moreover, despite your earlier speculation, it's not always the case that (a) a descriptive statistic corresponds naturally to some property of an underlying distribution; and (b) even if so, whether it would be meaningful to compare that property to zero; and (c) even what that distribution would be. – whuber May 6 '15 at 15:25
• It's true that a null hypothesis is a prerequisite of a p-value; what has the characterization of a statistic as "descriptive" to do with this? Whether you consider a statistic as an estimate of a population parameter & calculate a confidence interval, or as a test-statistic for a hypothesis about the population & calculate a p-value, you're no longer considering it as merely descriptive of the sample. – Scortchi - Reinstate Monica May 8 '15 at 15:27

Almost all descriptive statistics are used in hypothesis testing too. So, it's not exclusive classification into inferential and descriptive when we talk about the metrics such as the mean and standard deviation.

For instance, the sample mean is a descriptive statistic. Yet, you can obtain its p-value if you construct a hypothesis, such as $H_0: E[x]=0$, i.e. that the mean of the population is zero.

• An hypothesis test is a form of inferential statistics not descriptive statistics. – Alexis May 5 '15 at 19:03
• The point is that almost all descriptive statistics are used in hypothesis testing too. So, it's not exclusive classification into inferential and descriptive – Aksakal May 5 '15 at 19:08
• The OP is asking whether descriptive statistics have p-values. They don't. Statistics with p-values are inferential by definition; there are no non-inferential (i.e. only descriptive) statistics with p-values. (Note: there are forms of inference, such as confidence intervals, that do not use p-values). – Alexis May 5 '15 at 20:27
• @Alexis, (& downvoters): It's not clear whether the OP's asking about the de re or de dicto possibility of calculating p-values for descriptive statistics, or has thought about the distinction; so I think this example of a statistic commonly known to be used either descriptively or inferentially may be useful. – Scortchi - Reinstate Monica May 6 '15 at 8:39
• @Aksakal: I think your comment explaining what your point is should be included in your answer. And isn't it worth noting that assumptions need to be made about the sampling scheme to obtain p-values; not only from pedantry, but also because reluctance to make such assumptions can often be a reason for settling for descriptive statistics instead of performing inference? – Scortchi - Reinstate Monica May 6 '15 at 8:45