For the claim, as represented by the image:
Do 90% of rapes go unreported in the UK?
Rigorously correct wording doesn't usually translate well to social media memes, but there is some quality data on sexual assault and rape in England and Wales that support a claim that 83% of individuals 15-59 who answered yes to a survey question about whether they were raped in the past year did not report it to the police. Interestingly, as the report mentions, the number of people who endorse the question about having been raped in the past 12 months has been pretty stable (i.e., the survey's estimate of total rapes is stable), but the proportion who report it has been increasing. This is also consistent with police reports. From these data, it seems 90% may be too high for a current estimate, but the proportion of individuals who report to police seems to be increasing.
There are a few caveats, of course, but here are some relevant details about the methodology:
The Crime Survey for England and Wales includes this statistic: of the individuals in their sample who, in a face-to-face interview with a researcher, answered yes to a question about experiencing rape or assault by penetration in the last 12 months, 17% also answered yes to a question about reporting to the police. These questions were only asked to the subset of the sample that were aged 16-59, and the most recent report of this question comes from the 12 months ending in March 2017. The exact number of individuals who were asked this question can't be calculated using the tables in the report, but the overall adult sample for this 12 month period includes 35,420 individuals. As analyzed by the office of national statistics, these data are consistent with data from police reports.
The survey in question collects data using face-to-face interviews of households in England and Wales. It is administered by the office of national statistics. Both the data collection and statistical analysis are sound (general information on quality and methodology here, more specific information on sampling can be found here.). These data provide a reasonably good estimate of statistics for the population they sampled.
Caveats:
The sample doesn't include Scotland or Northern Ireland (and doesn't attempt to generalize to the entire UK). Application of results of this sample to the UK in general may not be warranted.
The sample doesn't include individuals living in group residences (for example, care homes or student resident halls), as you can see from the report on the sampling methods. This may be a significant caveat.
The sample doesn't include individuals younger than 16 and older than 59. This may also be a significant caveat. It is difficult to assume that the excluded demographics have the same reporting behavior as the included demographics.
Though the methodology describes the use of inferential statistics to generalize to the population of England and Wales, the particular statistic of interest is not reported with a confidence interval. Overall summary statistics of crime are reported with a confidence interval, but not the statistic we're interested in.
I would hope it would go without saying, but it might not. We are unable to measure the extent to which the number of individuals endorsing a survey question about having been raped is a good estimate of the number of individuals who have been raped. There are useful measures of the reliability of survey questions in general, but not for this, very particular, very different question. There are a number of personal and cognitive biases that you can use to decide how to interpret a statistic measured in this way. I choose to just report the actual measurement (of the individuals who endorsed this question, this proportion endorsed this other question).
