What's the difference between a descriptive statistic and a test statistic?

Question

In the Wikipedia article about test statistic it is written

A test statistic is a statistic used in statistical hypothesis testing. A hypothesis test is typically specified in terms of a test statistic, considered as a numerical summary of a data-set that reduces the data to one value that can be used to perform the hypothesis test.

while the Wikipedia article about descriptive statistics states

A descriptive statistic is a summary statistic that quantitatively describes or summarizes features of a collection of information.

It seems to me that both statistics are a means to summarize the given data (e.g. a sample of a population), but, depending on the context (i.e. hypothesis testing or descriptive statistics), we use one expression rather than another. Is this correct? What are the differences between them?

Answer 1

Both descriptive statistics and test statistics are functions of the data, or in other words, numerical summaries calculated from the data. So the difference is only in your purposes, goals, and uses of them.

Descriptive statistics are used for informal summary of the data:!different statistics tells you about different aspects of the data.

When you are interested in some formal hypothesis testing, some data summary is chosen for the test, as a test statistic, from criteria such as maximizing power, good robustness, and others, which are not relevant (or meaningful) when only used as descriptives.

Answer 2

Next to what Kjetil says, hypothesis testing is a form of statistical inference, i.e., you do not only describe a sample, but try to draw inferences about features of an underlying population.

E.g., a sample average may tell you that average height in a dataset of men is 181 cm, which is a descriptive statistic. You summarize the dataset and in so doing do not remember or report each data point.

You might use that sample average to test the hypothesis that the true mean height of men in the population from which you consider the sample to be drawn (what that is will depend on the problem - e.g., the height of all European men, all European male basketball players, etc.) is, say, at most $\mu=180$ cm.

Example:

> x <- rnorm(10) # generate some data for which we know that true mean mu is 0

> mean(x)        # sample mean will be some value likely somewhat close to but different from 0
[1] 0.2188956

> t.test(x)      # performs a test of whether mu=0 vs. that it is not. A rejection of the null would be a type-I error

    One Sample t-test

data:  x
t = 0.60125, df = 9, p-value = 0.5625
alternative hypothesis: true mean is not equal to 0
95 percent confidence interval:
 -0.6046832  1.0424744
sample estimates:
mean of x 
0.2188956