# What are some of the application of order statistics in engineering/data science /statistics? [closed]

I am looking for an interesting application of order statistics in engineering/data science /statistics.

I know many applications where order statistics is used as a mathematical tool.

However, I would like to see some application-oriented examples.

• Can you be more specific and elaborate on what you mean by "order statistics"? Commented Apr 7, 2020 at 2:51
• @MarkWhite Does this help: en.wikipedia.org/wiki/Order_statistic?
– Carl
Commented Apr 7, 2020 at 3:07
• Rank is the basis for many nonparametric statistical tests. That is a usage of order statistics. Example Spearman's correlation coefficient is sometimes called rank-order correlation
– Carl
Commented Apr 7, 2020 at 3:13
• what makes it difficult to answer this Q is not what is order statistics, but what is data science exactly? usually it means either IT guys who prepare data for statisticians or simply statisticians. If it's the latter then question makes no sense. If it's the former, then who knows what they are doing Commented Apr 9, 2020 at 17:44
• @Boby: as these comments suggest, people around here, myself included, would put "order statistics" in the realm of statistics and not of data science. Statistics is a branch of mathematics, and as you can see in my example below, figuring out the max in the population is a mathematical exercise. Is there a reason you said "data science" and not "statistics"? If no particular reason, I would suggest changing your question to ask for examples in "statistics" rather than in "data science". Commented Apr 9, 2020 at 21:52

There was one real-life application of order statistics during World War II, which is not only an interesting application of order statistics but is a testament to the power of the statistical inference. Credit goes to Richard J. Larsen and Morris L. Marx for popularizing this study.

In the 1940s, during World War II, the Allied forces have found themselves in possession of a collection of Nazi weapons, each with a serial number written on it. It was reasonable to assume that the numbers indicated the order in which these weapons were produced. If that were true, which turned out to be the case, then figuring out the highest serial number in use would have amounted to figuring out the size of the Nazi arsenal. Without a doubt, a valuable piece of information to the Allied forces!

Imagine that the Allies captured a total of 6 weapons and the serial numbers inscribed on these were 4, 12, 292, 332, 443, 793. We then want to estimate the maximum serial number produced by the Nazis using our sample of size 6 and the sample maximum value of 793.

Knowing the distribution of the maximum order statistic to be:

it is straightforward to estimate the maximum serial number in the population, and thus the size of the Nazi arsenal. Legend has it that the estimate obtained by the statisticians turned out to be much more accurate than that provided by the spies, whose estimate was far too high as it was effectively inflated by Nazi propaganda.

• Interesting! Thank you!
– Boby
Commented Apr 9, 2020 at 20:05
• How did they figure out $F_Y$ and $f_Y$?
– Boby
Commented Apr 9, 2020 at 20:38
• en.wikipedia.org/wiki/German_tank_problem
– whuber
Commented Apr 9, 2020 at 20:56
• You're welcome. Since these are serial numbers, each of them would occur exactly once in the population, so we have a discrete random variable with a uniform pdf. When you roll a fair die, numbered 1 through 6, each outcome has probability 1/6. When you have serial numbers going from 1 through t, then each outcome has probability 1/t. Commented Apr 9, 2020 at 20:56
• @whuber: thank you for the link. I didn't know it also goes under the name "German tank problem". Commented Apr 9, 2020 at 20:59

Using "robust scaling" in the sense we use the median and the interquartile range to centre and scale a variable before being used within a model is a very common use of order statistics. Similarly the MAD (Median Absolute Deviation) is relatively commonly used robust estimator of a sample's standard deviation $$\sigma$$. Similarly, using min-max scaling where we rescale a variable to be in the range of $$[0, 1]$$ is a classical case of using a sample's minimum and maximum values.

(Strictly speaking any sample quantile is not a order statistic statistic unless it relates to a pre-existing sample value and is not a function of averaging some sample values (e.g. when calculating the median of sample with an even numbered sample size, that median is not an order statistics, while it would if the sample size is an odd number.))