For the five readings 3, 8, 11, 15 and 23, the quantity $(3-a{)}^2+(8-a{)}^2+(11-a{)}^2+(15-a{)}^2+(23-a{)}^2$ is as small as possible when a equals which value?

The sum of squared deviations is least at the MEAN, which is $(3+8+11+15+23)/5=60/5=12$. Checking the candidates: at $a=12$ the sum is 81 + 16 + 1 + 9 + 121 = 228, at $a=11$ it is 64 + 9 + 0 + 16 + 144 = 233 and at $a=13$ it is 100 + 25 + 4 + 4 + 100 = 233. The value 11 is the median, which is the value that minimises the sum of the deviations taken without squaring, that is Σ|$x$ − a| — squaring changes which centre wins, because it penalises the far readings much more heavily and so pulls the best point towards them. The value 13 is the midrange, halfway between the least and greatest readings, and 23 is simply the largest reading. This property is why the mean sits at the heart of variance and standard deviation, both of which are built from squared deviations.