I've suggested this reference when similar questions come up. Here's an excerpt explaining the two cases it is used:
The Past Perfect expresses the idea that something occurred before another action in the past. It can also show that something happened before a specific time in the past.
With Non-Continuous Verbs and some non-continuous uses of Mixed Verbs, we use the Past Perfect to show that something started in the past and continued up until another action in the past.
This is a bit tricky, but the way I'm understanding it, this sentence:
The chart shows the percentage of people aged 65 and over in the United States between 1900 and 2000.
establishes a context where we are talking about the population from 1900 to 2000.
So, the "before" action or specific time that this sentence references
However, by 1960 this figure had doubled.
is the percentage of people aged 65 and over in the United States in 2000 (the end of the implied progressive action of the population growing from 1900 to 2000).
The "before" action or specific time may not always be explicit. In cases like this where the connection is rather weak and the time is explicitly specified I don't believe it's wrong to just use the simple past tense.
However, by 1960 this figure doubled.
This doesn't sound wrong to me.