Calculate average of a set numbers with reported standard errors

I have 365 daily measurements that all have standard errors associated with them.

    Date        | Prediction | Standard Error
-----------------------------------------
Jan-01-2003 | 24.8574    | 10.6407
Jan-02-2003 | 10.8658    | 3.8237
Jan-03-2003 | 12.1917    | 5.7988
Jan-04-2003 | 11.1783    | 4.3016
Jan-05-2003 | 16.713     | 5.3177
etc ...


What is the statistically appropriate way of getting the yearly average with a 95% Confidence Interval around it ? I am assuming that the errors must be propagating somehow and need to be accounted for.

Google returns mostly information on how to calculate the average or standard deviation of a set of numbers, not a set of numbers with errors.

I would also appreciate some type of internet reference so I can refer to it later.

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Do you know if the data normally distributed? –  ahoffer Jan 13 '12 at 22:06
I do not. For sake of argument we can say it is but it is likely Poisson because much of the other data I work with usually is. –  user918967 Jan 14 '12 at 5:15
The Poisson distribution is used for discrete data whereas your data seems to be continuous. What I would like to know is how the standard errors were obtained. Are they related to the measrements themselves or were they somehow obtained separately? –  MånsT Jan 15 '12 at 9:11
An average is just a the sum of each item times its proportion. In the case of a normal average these would just be equal for each item (summing to 1 of course). So Is it appropriate to just use normal addition error propagation after multiplying by the proportion? –  KennyPeanuts Jan 15 '12 at 17:18
MånsT- Sorry, I've not tested it and realized that as well. It would likely be a log-normal with a very high peak near the Y axis and a long tail. Onur - the "Practical Example" is not relevant as it is a standard example of working from a known distribution with a known SD. In my case, each measurement has its own SE associated with it and its own Confidence Interval. What I actually want to do with my 365 numbers is say this: At a 95% Confidence Interval, the mean is above a certain standard, say 35. ../.. –  user8559 Jan 16 '12 at 14:45
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migrated from stackoverflow.comJan 15 '12 at 5:03

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"The variance of the sums is the sum of the variances". So:

Square each of the 365 standard errors so they become variances. Add them together; this will give you the variance of the annual total. Divide that variance by 365^2; this will give you the variance of the annual average. Take the square root of that variance; this will give you the standard error of your annual average.

From there, I suspect your sample size is big enough (bigger than 500 in total, right?) it doesn't matter too much what the underlying population is (log normal etc) as your estimate is probably roughly normally distributed due to the central limit theorem. So multiply the standard error calculated above by 1.96 to give the +/- of your 95 percent confidence interval.