---
title: CUPED
description: "How Statsig uses CUPED variance reduction to improve experiment sensitivity by adjusting for pre-experiment user behavior on metric values."
product: general
lang: en
last_updated: 2025-09-18
token_estimate: 1165
---
# CUPED

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CUPED (Controlled-experiment Using Pre-Existing Data) reduces a metric's variance by adjusting each user's metric value with that user's pre-exposure data. Pre-exposure data is the same metric measured over the 7 days before the user's exposure. Statsig defines this window per user, not as a fixed window before the experiment starts for all users. The adjustment increases confidence in experiment metrics and reduces pre-exposure bias, for example when groups were randomly different before you applied any treatment. Use CUPED when a metric is autocorrelated over time. Use [CURE](https://docs.statsig.com/statsig-warehouse-native/cure/introduction), which extends CUPED on Warehouse Native, when you need covariates beyond the metric's own pre-exposure history, such as in new-user experiments or for metrics that aren't autocorrelated.

Statsig Cloud always uses the 7-day pre-exposure window. On Warehouse Native, Statsig recommends the 7-day window, but you can set any length. For how CUPED fits with Statsig's other variance reduction options, refer to [Variance reduction](https://docs.statsig.com/experiments/statistical-methods/variance-reduction).

## CUPED for simple aggregations

The original [Microsoft paper](https://www.exp-platform.com/Documents/2013-02-CUPED-ImprovingSensitivityOfControlledExperiments.pdf) describes the methodology for simple aggregations, as does Statsig's [in-depth CUPED article](https://www.statsig.com/blog/cuped).

Statsig Cloud uses stratification alongside CUPED to account for users who have no pre-exposure data. Statsig groups users into strata based on the pre-exposure data available, estimates treatment and control effects within each stratum, then aggregates the strata to produce an overall result. Statsig then applies the standard difference-in-means and variance estimation. This approach retains users with missing pre-exposure data while still reducing variance where pre-exposure data exists.

## CUPED for ratio metrics

The Microsoft paper also describes how to implement CUPED for metrics with a different analysis unit (Appendix B). Statsig extends this methodology to ratio metrics, where a numerator and a denominator represent each experiment unit. The variance reduction process finds the variance of the experiment data, the variance of the pre-exposure data, and the covariance between the two.

Denote the numerator, denominator, pre-exposure numerator, and pre-exposure denominator of a unit as $Y$, $N$, $X$, and $M$, respectively. Using the CUPED-reduced variance formula,

$$
Var(\frac{Y_{cv}}{N_{cv}})=Var(\frac{Y}{N})+\theta^2 Var(\frac{X}{M})-2\theta Cov(\frac{Y}{N}, \frac{X}{M})
$$

where Statsig finds optimal $\theta$ as

$$
\frac{Cov(\frac{Y}{N}, \frac{X}{M})}{Var(\frac{X}{M})}
$$

expanded to

$$
\frac{Cov(\frac{Y}{\mu_N}-\frac{\mu_Y N}{\mu^2_N}, \frac{X}{\mu_M}-\frac{\mu_X M}{\mu^2_M})}{Var(\frac{X}{\mu_M}-\frac{\mu_X M}{\mu^2_M})}
$$

This gives:

$$
\frac{\hat{Y_{c}}}{\hat{N_{c}}}=\frac{Y_{c}}{N_{c}}-\theta( \frac{X_{c}}{M_{c}} - \mathbb{E}[R])
$$

$$
\frac{\hat{Y_{t}}}{\hat{N_{t}}}=\frac{Y_{t}}{N_{t}}-\theta( \frac{X_{t}}{M_{t}} - \mathbb{E}[R])
$$

Because $\mathbb{E}[R]$ is hard to derive, the expectation term is the same for both groups. Substituting $\mathbb{E}[R]$ with $\frac{X_{c}}{M_{c}}$ transforms these two formulas into the following:

$$
\frac{Y_{cv}(control)}{N_{cv}(control)}=\frac{Y(control)}{N(control)}
$$

$$
\frac{Y_{cv}(test)}{N_{cv}(test)}
:=\frac{Y(control)}{N(control)} - (\frac{Y(control)}{N(control)} - \theta \frac{X(control)}{M(control)}) + (\frac{Y(test)}{N(test)} - \theta\frac{X(test)}{M(test)})
:=\frac{Y(test)}{N(test)} - \theta\frac{X(test)}{M(test)} + \theta \frac{X(control)}{M(control)}
$$

Using the optimal $\theta$, Statsig reduces group-level variance by applying the parameter to calculate the adjustment. An across-group $\theta$ doesn't necessarily reduce the variance for one group, or the sum of variances of all groups, but in most cases it does. Simulations show that 98.3% of metrics saw a decrease through CUPED.

Statsig applies the CUPED-reduced variance when all of the following are true:

- The core assumptions of the CUPED model hold. Rounding error or other data artifacts can violate them:

  - $E(\hat{X}) = E(X)$
  - The pooled variance of the adjusted population across groups is less than the variance of the unadjusted population.
- More than 100 units have pre-exposure values.
- More than 5% of units have pre-exposure values.