Average out of which number? There has not been enough empires in human history to get any kind of valid statistical conclusion.
Also, the ancient egyptian empire lasted over 3k years, for you to get an average of 250y with such outlier you would need to include what, several 10y "empires", or divide empires by ruler. Which would then make the conversation moot since each US president would be a new "empire".
The claim comes from John Glubb, and he used this chart to make the average out of... 11 data points!?! While missing tons of other ancient empires that lasted thousands of years?!
So to answer your comment, yeah math is easy. Impossible to reach such average number with all the data though, given that it was made with a wildly incomplete and incorrect data...
It's not really that straightforward though, is it? Firstly is it a mean or a median average? What counts as an empire? When do we date the rise and fall of specific empires? These are not questions with straightforwards answers. Would Hitler's Germany count as an empire? How many Roman empires were there?
What bit did you worry I hadn't read? Was it an answer you posted in another thread, perhaps, that you're talking about? I don't believe there was anything in my reply that suggested I hadn't comprehended anything relevant up to that point...
Do you count the Byzantine as separate or the same as Rome?
Your talking about structures comprising huge numbers of people across multiple generations. There is no clear "death". Just the gradual shifting from one set of conditions to another. Pick any line in the sand, declare it to be the "end" of an empire, and you'll still find people living under its rules, speaking the language, and using the currency well afterward.
Hell, look at Britain. No longer the globe-strangling power that they were, but it's still the same country with the same rules and government and money.