Truncation (statistics)

In statistics, truncation results in values that are limited above or below, resulting in a truncated sample.^[1] A random variable ${\displaystyle y}$ is said to be truncated from below if, for some threshold value ${\displaystyle c}$, the exact value of ${\displaystyle y}$ is known for all cases ${\displaystyle y>c}$, but unknown for all cases ${\displaystyle y\leq c}$. Similarly, truncation from above means the exact value of ${\displaystyle y}$ is known in cases where ${\displaystyle y<c}$, but unknown when ${\displaystyle y\geq c}$.^[2]

Truncation is similar to but distinct from the concept of statistical censoring. A truncated sample can be thought of as being equivalent to an underlying sample with all values outside the bounds entirely omitted, with not even a count of those omitted being kept. With statistical censoring, a note would be recorded documenting which bound (upper or lower) had been exceeded and the value of that bound. With truncated sampling, no note is recorded.