# Descriptive and Inferential vs Parametric and Non-Parametric Statistics

Are the concepts of descriptive vs inferential statistics and parametric vs non-parametric statistics orthogonal? As in, can we have a descriptive parametric statistic or a descriptive non-parametric statistic? if so what are some examples of? I know you can have an inferential parametric or non-parametric statistic.

My original thought was that parametric vs non-parametric statistics falls in a subcategory of inferential statistics, and that inferential statistics makes use of descriptive statistics. but some sources I read seem to say that you can have descriptive non-parametric stats.

Anyone can shed some light on this?

You can have any combination of nonparametric/parametric and descriptive/inferential.

In plain language:

Descriptive statistics describe a sample. Inferential statistics infer from a sample to a population.

Nonparametric vs. parametric is trickier. See http://en.wikipedia.org/wiki/Non-parametric_statistics

B.S. Everitt, in Dictionary of Statistics, defines "parametric methods" as "procedures for testing hypotheses about parameters in a population described by a specified distributional form...." and contrast them with "distribution free methods". However, I think that if you estimated an OLS regression on a sample without making any inferences to a population, that would be a parametric descriptive statistic.

• Following up a comment I made in a conversation with David Marx (elsewhere in this thread), I am curious about what kind of procedure you would consider to be at once "descriptive" and "non-parametric." Every one that comes to my mind is even better described as either parametric or is not genuinely descriptive. Incidentally, uses of OLS such as you mention can be genuinely non-parametric, but because they include some specific assumptions (linearity, e.g.) they are sometimes termed "semi-parametric."
– whuber
Commented Feb 17, 2014 at 23:25
• What about a loess curve of a distribution? Commented Feb 17, 2014 at 23:32
• Yes, that's a nice example. Unfortunately I mis-typed (in the preceding comment, but not in the other comment I referenced): I was concerned about descriptive parametric tests but reversed the words "parametric" and "non-parametric." I apologize for creating that confusion. (Almost all descriptive procedures are non-parametric, ranging from simple summaries of moments through the myriad plots described by Tukey in EDA.)
– whuber
Commented Feb 18, 2014 at 8:19
• I don't think tests can be descriptive, a test of a hypothesis is naturally inferential. But a statistic needn't be a test. What if you take a normal density plot? That certainly uses a distribution, but need involve no tests. Commented Feb 18, 2014 at 11:25
• Well done! That idea sounds good, understanding that a "normal density plot" would be something like superimposing a graph of a Normal density function on a histogram or other plot of the raw data, in effect providing a parametric visual reference for evaluating the data.
– whuber
Commented Feb 18, 2014 at 16:54

Here are some examples:

Descriptive Parametric: (I'm having trouble coming up with an example here. I guess the idea is that "descriptive" statistics, by nature, are non-parametric, hence they let us see the shape of the data prior to our making assumptions about it.)

Descriptive Non-Parametric: A histogram of the data.

Inferential Parametric: A first order ordinary least squares linear regression, which assumes a particular shape in the data (i.e. a linear fit) is an appropriate model.

Inferential Non-Parametric: Fitting the data using an ensemble of regression trees to develop a predictive model. The model does not make any assumptions about the shape of the data.

• Most of these examples don't seem quite right: there are non-parametric linear regressions, for instance, and the sample mean qua estimator can be parametric or not depending on distributional assumptions. The "analytical calculation" you refer to doesn't even fit into the context of a statistical procedure at all: it's a theoretical calculation to which these concepts of inference/description and parametric/nonparametric cannot naturally be applied (and rarely are). I believe you haven't quite captured the correct meaning of a parametric statistical procedure.
– whuber
Commented Feb 17, 2014 at 22:50
• If you prefer I could replace the gamma mode example with a more commonly used MLE, but I don't see how it doesn't fit the bill for a descriptive parametric statistic. I see your point regarding the sample mean, that wasn't a great example, but it's not wrong either: the sample mean is defined as $\frac{1}{n}\sum x_i$: there aren't any distributional assumptions in there. I'll qualify my "linear regression" example as ordinary least squares, since that's what I meant, and I believe it qualifies as a parametric technique. Regression trees are definitely a non-parametric technique. Commented Feb 17, 2014 at 23:05
• The distributional assumptions can change the interpretation of the mean (or whatever statistic you offer). The point is that "parametric" does not apply to the statistic, but to how the data are modeled. Thus, in focusing on the statistics (or, more generally, the calculations) I believe you are missing the main point. Your notation in the Gamma example did not seem to refer to any data or estimates: if it did, it might then be a valid example of a "descriptive parametric" procedure.
– whuber
Commented Feb 17, 2014 at 23:09
• The distributional assumptions still don't change how we calculate the mean, though. No matter what distributional assumptions we may have, the arithmetic mean remains the arithmetic mean, and this is how the "sample mean" is defined. The statistic is immune to our distributional assumptions. In any event, I changed that example to "histogram." You make a good point about the gamma mode example, though. I'll try to come up with a better example, unless you'd like to suggest one. Commented Feb 17, 2014 at 23:17
• Right: the fact that the distributional assumptions are unrelated to the calculation of the mean should be a clear indication that offering the mean as an example is not going to help anyone understand the distinction between parametric and non-parametric (nor between descriptive and inferential, for that matter!). "Descriptive parametric" is a tough category to describe, because a good description almost by definition should be independent of assumptions made about the data. It is hard to conceive of a procedure of this sort that wouldn't also be inferential in nature.
– whuber
Commented Feb 17, 2014 at 23:21

The "parametric" part of "parametric statistics" refers to the fact that in parametric statistics you are assuming/inferring a (descriptive) statistic has a known distribution (e.g Normal distribution). For example, you may infer that the mean is normally distributed as the sample size increases.

In non-parametric statistics, you are not making any assumptions about the distribution of the mean (or any other descriptive statistic). As an example, Chebyshev's inequality guarantees that no matter what the distribution, no more than 1/k^2 of the values can be greater than k standard deviations away from the mean.

The sticking point between parametric and non-parametric approaches is that you can do hypothesis testing quite easily if you go the parametric approach. The road is not as smooth if you go the non-parametric route. Which road you choose depends, in part, on how you view the validity of the Central Limit Theorem and convergence in general.