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Math Help - Concept of average

  1. #1
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    Concept of average

    Hello, its my first time posting in this forum.

    In all my life i always thought that the average concept was really natural to me, we have many values, with many ocurrences for each value, we sum each value multiplied by his number of ocurrences, and divide this sum by the total number of ocorrences of values.

    This always seemed to be fine to me. Some other day i was talking about leap years, and each year heaving an average of 365.25 days. My question is, we know that in each 4 years 1 of them is a leap year, so (365*3+366)/4 = 365.25

    i understand this result, but there's something that seems weird to me, the years dont stop counting, so whats the sense in calculating the average days of all years that existed and will existed basing ourselves in only 4 days?

    If the universe only had 5 years, we wouldn't be able to conclude that the average of days by year was 365.25, since we would get (4*365+366)/5 != 365.25

    Can someone help me understanding the concept?

    Thanks in advance
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  2. #2
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    Re: Concept of average

    #1 - You need to do a little more homework Every 4 is not quite the rule. Check out the centuries divisible by 400. :-)
    #2 - You are having a limit crisis!! Congratulations. This is how many great thinkers came to be great thinkers. "What if..." "But what about..." "Why ..."

    Limit:

    Take 4 years (not including a century). Average is 365.25
    Take 5 years, with leap year #4. Average is 365.2
    Take 6 years, with leap year #4. Average is 365.16666...

    Continue this process and see how you feel about 365.25 (just ignore the 400 problem for now.)
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  3. #3
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    Re: Concept of average

    I understand, it starts to converge to 365,25. So in fact, this average is never a final average, its the limit of the average when time goes to +oo, which means a year doesn't has an average of 365.25 days, only in a subset of 4 years, is this right?
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  4. #4
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    Re: Concept of average

    There are a couple of other important points.

    #3 - A year cannot have an average. Each year has either 365 or 366.

    #4 - The average length of a year, given some number of consecutive years, does approach 365.25 as the number of consecutive years increases without bound. Please be careful with the concept that something "goes to infinity". There may be times when infinity will be a place, but generally it should be known as a goal or a concept of unboundedness.
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  5. #5
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    Re: Concept of average

    I understand, so what i may conclude, is that the average length of a year converges to 365,25 days, but can't be considered exactly 365,25 , right?
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  6. #6
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    Re: Concept of average

    The limit of the average IS 365.25 as we have defined it.

    Okay, now we have to start talking about those centuries divisible by 400.
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