Turn one number — the correlation between a pre-experiment covariate and your metric — into the variance reduction, effective sample-size gain, and runtime savings CUPED buys you.
CUPED (Controlled-experiment Using Pre-Experiment Data), introduced by Deng, Xu, Kohavi and Walker at Microsoft, removes the part of your metric that was already predictable before the experiment started. If each user has a covariate X from the pre-period, CUPED forms an adjusted metric:
Because θ is the least-squares slope, the adjusted metric keeps the same expected treatment effect — it is unbiased — but its variance drops to:
where ρ is the correlation between X and Y. That single ρ² term is the whole story. Variance in an A/B test drives how many samples you need, and required sample size scales linearly with variance. So the effective sample-size multiplier is 1 / (1 − ρ²), and — assuming traffic is roughly constant — the required test duration shrinks by the same factor. A covariate correlated at ρ = 0.7 cuts variance by 49%, nearly doubling your effective users and halving runtime; at ρ = 0.3 the payoff is a modest 9%.
The best covariate is usually the same metric measured on the same users in the period just before the test. Notice the reduction depends on ρ², so weak covariates barely help while strong ones pay off super-linearly. The chart above plots 1 − ρ² across the full correlation range and marks your current value, so you can see whether wiring up the pre-experiment join is worth the engineering before you build it. CUPED changes nothing about your randomization, false-positive rate, or the estimand — it is pure variance reduction, which is why it has become a default in mature experimentation platforms.