Let Y be a nonnegative random variable with mean µ and finite positive variance σ 2 , and let Y s , defined on the same space as Y , have the Y size biased distribution, that is, the distribution characterized byfor all functions f for which these expectations exist.Under a variety of conditions on the coupling of Y and Y s , including combinations of boundedness and monotonicity, concentration of measure inequalities such asfor all t ≥ 0 hold for some explicit A and B. Examples include the number of relatively ordered subsequences of a random permutation, sliding window statistics including the number of m-runs in a sequence of coin tosses, the number of local maximum of a random function on a lattice, the number of urns containing exactly one ball in an urn allocation model, the volume covered by the union of n balls placed uniformly over a volume n subset of R d , the number of bulbs switched on at the terminal time in the so called lightbulb process, and the infinitely divisible and compound Poisson distributions that satisfy a bounded moment generating function condition.