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I have a time series containing n values (numbers) $t = v1, v2, v3... vn$.

For example, the time series could be: $50,6,7,8,90,10,25$

For a value $v_i$ ($1 \leq v_i \leq n$) from the time series, I would like to know if that value is significantly greater than the other values in the time series.

For example, is the value 90 significantly greater than the other values?

I have thought about a simple solution which is to check if the value v_i is greater than the sum of the average plus the standard deviation. This solution tell me that a value is quite large compare to the other values. But it does not tell me if this is statistically significant. Thus, my question is which statistical tests or statistical method could be used to determine if a value is significantly greater than other values?

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  • $\begingroup$ What can you say about the distribution of timeseries values? Are all values positive? It looks like integer only values - is that correct? Would you expect it to be unimodal? $\endgroup$
    – Eumenedies
    Apr 19 '17 at 8:29
  • $\begingroup$ Yes, only positive and integers. The data could have more than one mode. $\endgroup$
    – Phil
    Apr 20 '17 at 12:52
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I would like to know if that value is significantly greater than the other values

From your question, I am interpreting that the order of your values is not important to you. As such, you have a sample of $n$ values from which you want to construct a sample of $n-1$ values and a single value $k$. The question is:

Is it statistically significantly likely that $k$ originated from the same distribution as the other $n-1$ values?

What you want, therefore, is a prediction interval: i.e. a prediction interval is constructed from $n-1$ values such that, under the joint distribution of all $n$ values, the probability that the $k$th value lies within the prediction interval is $p$ (probably 95%). If $k$ falls outside the prediction interval, then you can say that there was only a 5% chance of that happening if it originated from the same distribution which would mean that it is "significantly different".

If you want a 95% prediction interval for a unimodal distribution then you can use the sample mean $\bar{x} \pm 3s$ as your interval.This will provide a rough prediction interval. Then you check if your value lies outside of this interval.

EDIT: Since the data may have more than one mode, you know less about its distribution. the $3s$ limit comes from the Vysochanskij–Petunin inequality, which assumes a unimodal distribution. If that assumption cannot be made then the less restrictive Chebyshev's inequality can be used, to get a rough estimate of an upper bound on the confidence. So to say that a value is significantly different from the others (at 95% confidence) it would need to be greater than the $\bar{x}+4.5s$.

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