Sensitivity

Suppose you have some imperfect test function $T$: is this cat picture cute? For n-many cat pictures $p\in P$ (where $P$ is the collection of cat pictures under test) our test will produce many results, some true, some false, as to to the cutest of the cat in the picture

$$R=\left\{T\right(p)|p\in P\}$$

Now, it's easy to imagine that some cats are truly cute (effectively no question), truly not-cute (or falsely cute). Yet we have a test that is imperfect. What is to stop our test from incorrectly rate "not-cute" a cute cat or incorrectly rate "cute" a not-cute cat!

Sensitivity is the measure of how many truly cute cat pictures are among all cat pictures rated "cute".

Allow for a change in notation:

$TP=$ a truly cute cat
$TN=$ a truly not-cute cat
$FP=$ a not-cute cat rated as "cute"
$FN=$ a cute cat rated as "not-cute"

Sensitivity then is given by the following equation:

$$Sensitivity=\frac{TP}{TP+FN}$$

Alternately, the plain English of our example so far:

Sensitivity is the ratio of cats evaluated as "cute" to all of the cats that were truly cute.

Applications

Sensitivity is a common measure in applied statistics, as well as modern machine learning, though it often goes by another name: recall.

In either case, we may often want to "improve our test sensitivity" or "live with false positives, but we can't have a false negative". In both cases, the above measurement is what's being referenced: by decreasing the number of false-negative results produces a shrinking denominator, thereby improving the fraction ever towards $1$.