You could just visit the page to view the source, or even look at the 'Our World in Data' logo on the chart, to get some idea.
Here's where the chart is from: http://ourworldindata.org/data/population-growth-vital-statistics/life-expectancy/
There's a section at the bottom of that page detailing the data sources. Knock yourself out.*
People have to die from something. More people dying from things like cancer and heart disease is partly a consequence of people living longer, due to better healthcare following the rise of modern medicine. Your argument is essentially: "Modern medicine prevented this person from dying at 5yo from a preventable childhood disease, but instead they died at 90yo from cancer. Therefore medicine is useless".
Most of modern drug usage on patients is simply a method of slowly poisoning them. That's the reason for so many deaths in hospitals with major diseases.
When you mix good hygiene with poison, and call the whole thing health care, you are misleading the people into their own deaths, mostly sooner.
As far as your example of the person, we don't know what would have happened if the child had not been treated. You can't try it both ways to see what might happen the other way. The child might easily have lived without treatment, or with non-medical treatment, and might still be alive at 110.
* But beforehand, please arrange to be brought back to consciousness by a medical professional (who knows what they're doing) rather than a priest (who doesn't).
Edit, second reply:
It's quite amusing that just in the space of a couple of sentences, you suggest that I don't understand what life expectancy means, and then proceed to wildly misunderstand it. You're not stupid, so I assume you're simply being disingenuous. A life expectancy of, for example, 40, certainly does not suggest that a large number of people die at age 40, as I'm sure you know. It also doesn't say very much about the chances of someone who is 70 being able to survive to the age of 80. The biggest factor, historically, has been infant mortality. If you are genuinely arguing that modern medicine has not reduced infant mortality, and that average lifespan has not increased, even in the face of the data, then you may as well go back to reading the bible with BADecker, because I'm not going to be able to shake your faith with something as mundane as evidence.
Again, health care regarding infant mortality that saves lives is different than medicine health care. Health care being used by the doctors of 100 years ago, was much different than the health care being used today. Simple hygienic and mechanical health care is something that can be taught to mid-wives or anybody; often it involves simple common sense. But that is what saves babies; not something that requires years of schooling and medical experience as things are today in modern medicine, and, of course, all kinds of medicine poisons.
Who is going to tell you when a modern technique kills the baby? Certainly not the doctor. He may not even realize that's what it was. His report might be lies, because he did his best, even though the baby might still be alive if he hadn't done anything.
You can't do it over a different way once you have done it it one way.