e should by now be familiar with the idea of a vaccine’s efficacy, but last Monday three analyses of vaccine effectiveness were published. Despite the confusing similarity, these are different concepts: efficacy is measured in tightly controlled clinical trials, effectiveness is how well a vaccine works in the messy real world. In trials, healthy volunteers are put in vaccinated and control groups at random – this ensures the groups are comparable and differences in outcomes must be due to the vaccine. If we simply compare people who have been jabbed with those who have not, they will differ in all sorts of ways: older and other higher-risk people will be first in the queue, while communities that are hesitant to be vaccinated may also be at higher risk. These confounders can lead to systemic bias in estimating effectiveness. So, studies use elaborate statistical analysis to make fair comparisons. For example, a remarkable study of the entire adult population from Scotland compared hospital cases with Covid-19 in 1.1 million vaccinated people with 3.2 million who were not vaccinated, taking into account differences in age, sex, deprivation and other factors. Effectiveness peaked four weeks after one dose, at 85% for the Pfizer/BioNTech and 94% for the Oxford/AstraZeneca vaccine. It is tempting to claim the Oxford jab was better but the overlapping, plausible ranges around these estimates (76% to 91% and 73% to 99%) show we can’t conclude they are really different. Encouragingly, the combined effectiveness for over-80s was 81% (range 65% to 90%). Reduced hospital admissions are vital but we must also stop infection and transmission. Pfizer’s effectiveness against infections in healthcare workers, accounting for age, ethnicity and other factors, was around 70% three weeks after one dose, and 85% a week after the second dose. This reassuring conclusion also held for the new B.1.1.7 variant. These studies show that both vaccines give high levels of protection after one dose to all ages, vindicating the bold decisions the UK made and giving grounds for optimism. David Spiegelhalter is chair of the Winton Centre for Risk and Evidence Communication at Cambridge. Anthony Masters is statistical ambassador for the Royal Statistical Society