How many Roman emperors ruled during the 50 year period covered by this chart? How many
died a natural death? What was the most frequent cause of death of these men?

Explanation:

On the chart (as seen in the attachment) we can see there are 19 rules from 235 to 285. CE, but if we include those who were co-ruler, there have been 23 emperors in total.  

The line starts with Maximinus Thrax (235 - 238). Some, like his successors, Gordian I and Gordian II, father, and son, had short reign - the mentioned pair only ruled for 21 days.

The last emperor on the chart is Carinus, who ruled from 283. to 285. and was succeeded by the famous Diocletian.

The only one died a natural death, possibly two, though it is not sure if Hostilian died of the plague or other reasons.  

The one who was confirmed to have died of natural causes is Claudius Gothicus, the emperor who is said to have died from so-called "pestilence", fatal desire at the time which historians have found possibly to be the plague. At the time there was a pandemic of the plague of Cyprian that lasted from 249. to 262. and is in symptoms very similar to smallpox and ebola.  

We can see on the chart his rule was the longest one in that time period. Historians believe that he named Aurelian as his successor, but Quintillus seized the power, which made Aurelian come only after him.

The most frequent cause of the end of the ruler was the assassination, with 8 confirmed assassinations and 5 possible assassinations. Considering two of those were the pair of emperors who ruled together, that makes 15 out of total 23 emperors who have died of assassinations.

The second most numerous cause of death was killed in the battle, while two causes are considered to be by their own hand. Previously it has been mentioned only two died of natural causes, possibly plague, though one is not confirmed and the cause of death is seemingly unknown.  

the first question's answer is a.