I'm not sure why sometimes in problems leave out version of estimators uses. for example when we want standard the data instead of dividing numbers on whole of the sample's standard deviation, sometimes dividing on SD of whole data except the current one.
Any help will be appreciated.
The SD is a bad example to start with, so it makes more sense to think in terms of variance. If we are just interested in describing the sample the variance can be calculated as the sum of the squared deviations of each observation from the sample mean divided by n (i.e., the average squared deviation). The problem only arises if we want to estimate the population variance from the sample. Now we discover that the sample (on average) underestimates the population variance. The underestimation decreases with n and specifically is the proportion (n-1)/n so we can correct the variance by multiplying by n/(n-1) to get the unbiased estimator.
This gives us two perfectly valid formulae - one that describes a sample and one that uses a sample to estimate the population variance.
For the SD the correction factor is more complex but people usually just calculate it from the unbiased variance estimate (as its square root). This quantity is biased but the degree of bias is negligible for most purposes.
The SD is a bad example to start with, so it makes more sense to think in terms of variance. If we are just interested in describing the sample the variance can be calculated as the sum of the squared deviations of each observation from the sample mean divided by n (i.e., the average squared deviation). The problem only arises if we want to estimate the population variance from the sample. Now we discover that the sample (on average) underestimates the population variance. The underestimation decreases with n and specifically is the proportion (n-1)/n so we can correct the variance by multiplying by n/(n-1) to get the unbiased estimator.
This gives us two perfectly valid formulae - one that describes a sample and one that uses a sample to estimate the population variance.
For the SD the correction factor is more complex but people usually just calculate it from the unbiased variance estimate (as its square root). This quantity is biased but the degree of bias is negligible for most purposes.
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