# Defining confidence intervals. I have a set of n observations O. I want to calculate the mean of these observations and their confidence interval. I can do this when the observations are iid, by looking up the t-table for the required confidence value and then calculate the interval as -- 2t sigma/(sqrtn)

Defining confidence intervals
I have a set of n observations O. I want to calculate the mean of these observations and their confidence interval. I can do this when the observations are iid, by looking up the t-table for the required confidence value and then calculate the interval as --
$2t\frac{\sigma }{\sqrt{n}}$
However, if the ${i}^{th}$ observation of this sequence is related to $\left(i-1{\right)}^{th}$ observation, how do I now define the confidence interval. I believe the iid assumptions are no longer valid.
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Kaitlyn Levine
Step 1
This is very similar to a time-series problem, where you have Lag-1 correlation among data points. You first need to fit a model to this relationship and then remove it from each data point so you can analyze the residuals, which will hopefully be iid. For example, if ${x}_{2}={x}_{1}+{ϵ}_{1}$ where ${ϵ}_{1}\sim F\left(ϵ\right),\left(F\left(ϵ\right)$ is arbitrary) then you need to model this as a random walk and then you can calculate the mean value of $ϵ$, since the residuals will be iid.
Step 2
In other words, correlations between successive points make your data non-stationary, and hence not iid. You need to compare apples to apples by adjuting for the correlation via some model and examining the residuals of the model using iid-based statistics.