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I am reading a book about Statistics and I have encountered a text line:

...except in the case where standard deviation of the basic set is unknown...

I am not really sure what it means, could you please help me. For what types of data can't we define the standard deviation?

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What they are talking about is the value of the standard deviation. When the value is assumed to be known you can divide the sample mean by it and if the samples have a normal distribution the sample mean minus the population mean divided by the "known" standard deviation divided by the square root of the sample size n has a standard normal distribution (a normal distribution with mean 0 and variance 1). This fact is then used to construct confidence intervals for the population mean based on teh standard normal distribution.

However in most prectical situations the value of the standard deviation is unknown and we use a sample estimate to scale since we can't use a known value. Under the same assumptions about the sample distribution the sample mean- the population mean divided by its estimated standard error has a t distribution with n-1 degrees of freedom. So when the population variance is unknown the t distribution is used to construct confidence intervals for the population mean. The standard error of the mean is the sample standard deviation divided by the square root of n.

I hope this makes it clear. Statistics text categorize the estimate of the mean into two case (1) standard deviation known where the standard normal is used and the case where the standard deviation is unknown and the t distribution is used.

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  • $\begingroup$ Very good explained! @Michael Chernick $\endgroup$ – Takarakaka Jul 6 '12 at 2:23
  • $\begingroup$ @MichaelChernick What is needed is an sufficiency criterion for what constitutes known. For example, when are the confidence intervals of the mean small enough that we can say that the value of the mean is known, and for what data is the mean defined (at all) and thus even theoretically knowable? $\endgroup$ – Carl Apr 17 '17 at 13:03

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