Every score is a noisy estimate
A student takes a test and scores 75. That is not her true ability, it is an observation. If she took the test again tomorrow, she might score 77 or 73. The Standard Error of Measurement quantifies this noise. SEM of 3 points means her true score probably lies between 72 and 78 with 68% confidence, and between 69 and 81 with 95% confidence.
The noise comes from many sources: testing conditions (tired vs rested), item wording (ambiguous questions), luck (guessing), and measurement error in the test itself. A reliable test has low SEM; an unreliable one has high SEM.
Using SEM to interpret pass-fail decisions
If a cutoff score is 70 and a student scores 71, the SEM tells you the confidence in that borderline decision. With SEM of 5, she could easily be below the cutoff on a retest. Many high-stakes testing programs build in a confidence band: if your score is too close to the cutoff, you retake to be sure.
SEM also informs how much change matters. If someone scores 75 on a pretest and 78 on a posttest, but SEM is 4, that 3-point gain might be noise, not growth. The bands overlap; you cannot call it a real improvement with confidence.