This "answer" is too long for a comment, and is definitely incomplete. It is meant to supplement @BradC's answer, which is currently marked as a work in progress.

The analysis RAINN is doing appears to stitch together data from a variety of different sources. They list their sources, but do not include any indication of  how they picked and processed their data. Although this is better than the average completely unsourced infographic, it is still frustratingly opaque. When the question was first posted, I spent a few frustrating hours researching this, and came to some of the same conclusions that @BradC did. I also found the following.

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**NCVS Data related to claim 1**

Data from the National Crime Victimization Survey (NCVS) lumps rape and sexual assault together, and defines sexual assault as:

> A wide range of victimizations, separate from rape or attempted rape. These crimes include attacks or attempted attacks generally involving unwanted sexual contact between victim and offender. Sexual assaults may or may not involve force and include such things as grabbing or fondling. Sexual assault also includes verbal threats.

RAINN takes these two together and reports it in their infographic simply as "rape." This seems seriously misleading; Rape includes things far more heinous than grabbing or fondling. I find it baffling that the NCVS does not report these separately.

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>**Claim 4**: 64% of prosecutions lead to a felony conviction (7 of 11)

[This government report][3] presents data on felony defendants in America's 75 largest counties. These counties contain large cities accounting for 37% of the total population, and are not nationally representative.

In 2009 in the 75 largest urban counties there were 412 felony rape defendants, of which 68% were convicted of something, 57% (235) were convicted of a felony, the rest a misdemeanor. The infographic claims that of the 11 cases referred to a prosecutor, 7 of them will result in a felony conviction. The two sources have a similar felony conviction rate; The difference could be explained by rounding errors in the 7 and 11 or it could be that RAINN took the average of several years of data. ([Source: Table 21][3])

> **Claim 5:** 85% of felony convictions are incarcerated (6 of 7)

According to the same source, 89% of convicted rapists are incarcerated. (Source: Table 24) This is very similar to the numbers presented by RAINN.


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**BJP Data related to claims 4 and 5**
These statistics describe only residents of the biggest cities, 37% of the total population. Big cities and tiny towns have different culture, different police forces, and different patterns of crime. Although extrapolating data from big cities to the whole country is terrible analysis, I can't be sure that RAINN actually did this. The source is called out in RAINN's sources, and the numbers approximately match, but that is not conclusive.

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**Lack of a conclusion**

After looking at all of this data, I still do not know what to think. The data sources RAINN looked through are credible and authoritative, but the sources use a variety of definitions, methodologies and sample populations. The infographic was an attempt to weave these threads of evidence into a clear cut story. After reviewing the evidence, I still believe the broad strokes of the story, but I am skeptical of details and specifics. I believe that a **distressingly low number** of rapes end in incarceration, but the final number could be very different from 6 out of 1000.

  [1]: https://www.bjs.gov/content/pub/pdf/cv15.pdf
  [2]: https://www.rainn.org/about-rainns-statistics
  [3]: https://www.bjs.gov/content/pub/pdf/fdluc09.pdf