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I had been studying statistics, I have a doubt that I couldn't find the answer of. Its related to estimating population parameters using statistic.

Suppose we have a population size of 10000, we want to estimate mean for it since it is too costly to collect data for all the observations. As far as I know standard process is to arrive on a sample size, say, 100 find the mean for this sample, iterate these steps over some time and then arrive at population mean and its standard deviation assuming normal distribution.

Now, doing this we are not estimating population mean but we are estimating the mean of samples for which we have the observation data. So, if we have the observations for 100 data points, we are estimating mean of this sample of 100 rather than the population of 10000. Then why do we call it estimating population mean?

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"As far as I know standard process is to arrive on a sample size, say, 100 find the mean for this sample, iterate these steps over some time and then arrive at population mean and its standard deviation assuming normal distribution."

I think this is your fundamental misunderstanding. It is standard to take a single sample and use statistics calculated from the sample to estimate the required parameters.

Your point of confusion seems to be how we estimate the uncertainty in our parameter estimates. We do so by considering the "sampling distribution" of the estimator - imagining how it would be distributed if we took hundreds of samples, but not actually doing so. Looking at how widely spread out the sampling distribution is gives us a measure of uncertainty for our estimate - the standard error of the sampling distribution is called the "standard error". Larger samples have a smaller standard error, so less uncertainty. If we could afford to take multiple samples, we are actually better off to take one large sample!

In the special but common case where the parameter of interest is the mean, we estimate it using the sample mean (usually), and if the original data is normally distributed then the sampling distribution of the sample mean is too. Even if the sample is not drawn from a normal distribution, for large samples we can appeal to the Central Limit Theorem and say the sample mean is approximately normally distributed. The meaning of a "large" sample depends just how far from normality the original population is; if it is symmetric and unimodal a fairly small sample may suffice, but a very skewed population may require a fairly substantial sample for the sample mean to be reasonably close to normally distributed.

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  • $\begingroup$ So, we use sampling distribution to find out confidence interval for population mean. This makes sense. Thanks. $\endgroup$
    – Swastik
    Commented Nov 23, 2014 at 9:08
  • $\begingroup$ The sampling distribution of a statistic has other purposes too, for instance if we want to see whether an estimator is biased. In the case of the sample mean it turns out not to be, but to find an unbiased estimate of the population variance we find that it's necessary to apply the Bessel correction. Again the key here is that we can't see if there is a bias based on the one value of the statistic we calculate from the sample we took, but instead we imagine what would happen if we took hundreds of samples - on average would the estimates we calculate be overestimates, underestimates, or okay? $\endgroup$
    – Silverfish
    Commented Nov 24, 2014 at 12:45
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See, the very word estimation implies that you are not necessarily getting the exact value, but reasonably close enough.

In sampling theory, the sample mean is an unbiased estimate of the population mean. Therefore, if you have a set of randomly selected 100 data points, you can be pretty much sure that the mean of this sample will not be far off from the mean of the population. If the number of samples are increased, the accuracy goes up, but never touches 100% in general. You should refer to the error of estimation. But increasing the number of samples beyond a point is not efficient, and the whole point of using samples to determine the population properties is lost. Therefore, you decide an acceptable error bar, and then find the required sample size.

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  • $\begingroup$ Thanks for answering but my point is that we can directly calculate the mean of the 1000 samples and call it an estimate of population. Is there any increase in accuracy by splitting 1000 data points to sample of 100s an then take average of these samples? $\endgroup$
    – Swastik
    Commented Nov 23, 2014 at 8:17
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    $\begingroup$ Swastik -- The average of averages (with the same sample sizes in each small sample) is the same as one large average. Consider the smaller case of 4 observations (say 3.5, 4.1, 3.2, 3.4) - it makes no difference to the estimate whether we calculate $\frac{3.5+4.1+3.2+3.4}{4}$ or we compute two sample means $\frac{3.5+4.1}{2}$ and $\frac{3.2+3.4}{2}$ and then average those. The estimate of the population mean is the same. $\endgroup$
    – Glen_b
    Commented Nov 23, 2014 at 8:24
  • $\begingroup$ In addition to that, see whether you need 1000 samples for the given accuracy requirement. This [wiki][1] page will give you an idea. [1]: en.wikipedia.org/wiki/Sample_size_determination $\endgroup$
    – Ayyappadas
    Commented Nov 23, 2014 at 8:38
  • $\begingroup$ Exactly that's what I am saying. Since we have data of 1000 samples, we can directly take mean of this data. Why do we further divide this into 100 samples then apply CLT and then find out SE of this. It is a more complicated process and I don't think it increases any accuracy since this estimate will be biased toward mean of the 1000 sample for which we have data rather than population. $\endgroup$
    – Swastik
    Commented Nov 23, 2014 at 8:43
  • $\begingroup$ I do not see any reason why it should be done. Could you post the context of the problem? Is it a Simple Random Sampling With/Without Replacement (SRSWR or SRSWOR)? $\endgroup$
    – Ayyappadas
    Commented Nov 23, 2014 at 9:12
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We estimate population parameters because we don't have exactly values or completed data in any population example when we want to estimate the mean and the standard deviation of 10000 sample size became difficult as why

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    Commented Jan 14, 2023 at 12:50

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