Clarification - Central limit theorem using sample means

Just some clarification regarding the central limit theorem. I understand that the theorem says

"Given a distribution with mean $\mu$ and variance $\sigma^2$, the sampling distribution of the mean approaches a normal distribution with mean $\mu$ and variance $\sigma^2/n$, as the sample size increases"

I am unclear as to how this extends to some real data examples.

For instance, say we are trying to estimate the average monthly electricity use for a household in London.

• The population is therefore households in London.
• The sample would be a subset of London properties.

Since we wish to estimate a monthly average, the data collected from each household would need to be a monthly figure. Therefore, looking at the distribution of household means, does the CLT say that this would approximate a normal distribution?

I have a sample of this data, which contains around 2000 properties. However, due to the nature of the data (cannot have electricity usage <0, and no upper limit), the distribution is heavily skewed. Therefore, I cannot see how the CLT can be applied?

The problem, is that I wish to obtain a confidence interval around the mean, and compare this with different sample sizes. I was planning on using a simple t-test, however this assumes normality.

Have I got some of my assumptions/logic wrong?

• CLT talks about the distribution of "mean of the distribution". Think of it in this way: what is the probability that you get a specific sample-mean value for random samples from the population. Jan 13 '17 at 12:38
• This is a complicated issue partly because there are many forms of the central limit theorem. In your case I worry that the sequence of variables that you are looking at could be highly dependent. I am sure that there is a version of the theorem that allows for weak dependence but not strong forms of dependence. Also with asymptotics the question of how large is "sufficiently large" arises. This can vary depending on what assumptions are realistic for your data where the central limit theorem may hold. Jan 13 '17 at 14:41