pooled rate

Also called: pooled conversion, mean of rates, pooled proportion

All successes divided by all trials across groups: (40 + 30) ÷ (1,000 + 200) = 5.83%. It is a weighted mean of the group rates, unlike the plain mean of rates.

When groups differ in size, there are two ways to "average" their rates. The mean of rates adds the rates and divides by the number of groups, so a group of 200 counts as much as a group of 1,000. The pooled rate adds all successes and all trials first:

$\bar{p} = \frac{x_1 + x_2}{n_1 + n_2} = \frac{\sum n_i p_i}{\sum n_i}$

The second form shows that the pooled rate is a weighted mean of the rates, with group sizes as weights. It leans toward the bigger group.

Most business questions ("how does a typical trial convert?") need the pooled rate. The mean of rates answers a different question: how does a typical group or channel convert. The pooled p̄ also appears in the two-proportion z-test, where it estimates the common rate under "no difference".

Example

Ads: 40 payers of 1,000 trials (4%). Referrals: 30 of 200 (15%).

Mean of rates: (4% + 15%) ÷ 2 = 9.5%.
Pooled rate: 70 ÷ 1,200 = 5.83%, close to the bigger channel.

Common mistakes

Averaging rates of unequal groups and calling it overall conversion.
Comparing pooled rates across months when the mix of groups changed: fix the weights first.
Forgetting that equal groups make both methods agree, so the gap only shows when sizes differ.

Learn it in the course

Weighted mean and mix · Σwx/Σw, why a pooled rate is not the mean of rates, mix-adjusted rates, pooled variance and pooled p̄, the cost of capital, PERT estimates and weighted scoring.
Pythagoras in statistics: distances and errors in quadrature · Euclidean and Manhattan distance, why independent errors add as squares, the pooled SE of a z-test vs the unpooled SE of an interval, and how to read the Wilson and Newcombe interval formulas.