The central limit theorem and its importance to statistical estimation

Hi guys

This is precisely why essentially all statistics taken on small populations have large variances. The CLT is key for getting estimations which don't have really large variances. It tells you with fairly small variance where the mean of a distribution lies, *whatever* the distribution (as long as it does have a mean and variance). As you say, it does require taking lots of samples if the underlying distribution isn't well-behaved; however, in the cases where CLT is not useful, it's often true that statistics is simply not something that helps. For instance, if you have only two samples, there's very little you can say about the distribution.

Actually, I'm not sure you understand the statement of the CLT. It says that if you repeatedly take samples of a fixed size (the larger, the better), then the means of those samples will approximate a normal distribution centred on the original distribution's mean. It doesn't say that if the sample means all lie between 1m and 3m then the normal distribution will have mean 2m: consider the case that all humans but one are 2.5m tall, so nearly all the samples are simply lists of 2.5, so all the sample means are all very close to 2.5, so the normal distribution is centred on 2.5.

Thanks for your reply

I think I'm going to have to go over things a bit more as I'm still really lost. When I think of normal distribution I see the bell shaped curve. But does the bell shape curve always peak in the middle of the two measurements in this case 1m and 3m?

I can see that your explanation says no but I can't see how the distribution can't be bell shaped unless it peaks in the middle of the two measurements... Can it start its rise a little bit in? (oh dear that last sentence makes me sound so simple)


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Consider the standard normal distribution: mean 0, variance 1. Imagine I take two samples from it, and get as my measurements 0.5 and 1.6. Now, that's told me very little about the mean: in particular, it hasn't told me enough to say that it's halfway between 0.5 and 1.6. All I can say is that the mean is unlikely to be horribly far away from 1.05.

If I do this again, I might get the measurements 0.1 and -4. Again, the mean I'd predict from that is -1.95, which isn't very close to the true mean of 0.

Suppose I did this "pick pairs, take the mean of the pair" procedure several times and got 1.05, -1.95, 0.2, 0.5, -0.3. It becomes clear that the distribution mean is approximately 0 (in fact, the mean of these sample means is -0.1). It's the Central Limit Theorem that has allowed me to say that. Without the CLT, I can't go from "I drew lots of samples, and their means clustered around 0" to "the distribution mean is around 0", and that's a crucial part of statistics: without the CLT, it's really hard to say anything about the distribution's mean even if we have taken lots of samples and know the means of the samples.

Consider the standard normal distribution: mean 0, variance 1. Imagine I take two samples from it, and get as my measurements 0.5 and 1.6. Now, that's told me very little about the mean: in particular, it hasn't told me enough to say that it's halfway between 0.5 and 1.6. All I can say is that the mean is unlikely to be horribly far away from 1.05.

If I do this again, I might get the measurements 0.1 and -4. Again, the mean I'd predict from that is -1.95, which isn't very close to the true mean of 0.

Suppose I did this "pick pairs, take the mean of the pair" procedure several times and got 1.05, -1.95, 0.2, 0.5, -0.3. It becomes clear that the distribution mean is approximately 0 (in fact, the mean of these sample means is -0.1). It's the Central Limit Theorem that has allowed me to say that. Without the CLT, I can't go from "I drew lots of samples, and their means clustered around 0" to "the distribution mean is around 0", and that's a crucial part of statistics: without the CLT, it's really hard to say anything about the distribution's mean even if we have taken lots of samples and know the means of the samples.

So basically , if I'm correct, the CLT makes an estimation on the population parameters from sampling distribution value?

If the true population parameter is known then what is the point of sampling ? This is what I'm confused about. If the CLT is so useful in predictions regarding the samples taken from a population, then why sample?

Using an example of a population of teachers within a city. I choose to sample the teachers who teach maths and wish to find out how many of these drive a red car. Can someone please explain how the CLT would be any good here?

Thanks mate

So it's more of a statement than a calculation that is practically used in statistics?

Or

It is only used when the population distribution is not known...?

Mmm I'm starting to get it, thanks

Put simply, when we take a sample the results we have will only be reflective of that sample, eg. Brown haired people in London who live in a flat with a cat. The more we sample, the theorem willbe able to tell us that the mean of these samples will be close to the true value of the population.

Although it would be very hard to sample all the brown haired people etc etc, the CLT enables us to make a goo estimate without doing a census..?

Am I starting to get the gist?

Yep, exactly.

Once again Smaug, thanks for your help.

Hopefully stats will start to get a bit easier soon as at the moment I'm so stressed out lol


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