# 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. – Ujjwal Kumar 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. – Michael Chernick Jan 13 '17 at 14:41

## 1 Answer

You have a distribution of "Average Monthly Elec. Usage" for London. Each point in your distribution represents a unique household in London. CLT can only be applied in two ways:

• CLT on a specific household's usage: If all days were same, and you randomly sampled N days to get mean of their usage values for a single household, the distribution of these mean values would be a normal distribution, OR
• CLT on average-usage in London-households: if you randomly sample N households from all the 2000 you have, then, the distribution of sample-mean would follow said normal distribution.

Now using the SD value for the distribution of sample-mean, and N (sample-size), you can estimate the true distribution mean and confidence interval.

• However, since each of my 2000 households has a mean monthly elec. reading, then doesn't the CLT say that the distribution of these means would approximate a normal as sample size increases? I can tell that I am getting confused here, but I can't seem to work out why my statement isn't true. – sym246 Jan 13 '17 at 12:44
• For CLT's statement, "mean-usage of a random sample of houses" is a single observation, and then, the distribution of these "sample-mean" values should be normal. – Ujjwal Kumar Jan 13 '17 at 12:54
• Ok thanks. So since I have effectively, samples of size 1 (a single households mean value), albeit, lots of them, the CLT wouldn't apply here, unless I re-sample from my pool of 2000 households, using a large enough n. – sym246 Jan 13 '17 at 12:56
• Yes, you can conduct and experiment to get more clarity. Generate random values from any distribution, now calculate sample-means for many random samples of these values. You should see that the sample-mean is normally distributed. There are multiple examples of CLT experiments all over the internet – Ujjwal Kumar Jan 13 '17 at 12:58
• As a final point, the notes on this link, have example data which is similar to mine, yet section 3.6 uses the CLT. They have one sample mean (using n=40), but determine the confidence interval using the central limit theorem. Based on the previous comments, shouldn't they have taken multiple samples of size n=40, then used the CLT on the mean of the means, as it were? – sym246 Jan 13 '17 at 13:20