Sample Standard Deviation vs. Population Standard DeviationI have an HP 50g graphing calculator and I am using it to calculate the standard deviation of some data. In the statistics calculation there is a type which can have two values:Sample PopulationI didn't change it, but I kept getting the wrong results for the standard deviation. When I changed it to "Population" type, I started getting correct results!Why is that? As far as I know, there is only one type of standard deviation which is to calculate the root-mean-square of the values!Did I miss something?

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Sample Standard Deviation vs. Population Standard DeviationI have an HP 50g graphing calculator and I am using it to calculate the standard deviation of some data. In the statistics calculation there is a type which can have two values:Sample PopulationI didn&#039;t change it, but I kept getting the wrong results for the standard deviation. When I changed it to &quot;Population&quot; type, I started getting correct results!Why is that? As far as I know, there is only one type of standard deviation which is to calculate the root-mean-square of the values!Did I miss something?

Ostrakodec3 · Accepted Answer

Step 1There are, in fact, two different formulas for standard deviation here: The population standard deviation   σ and the sample standard deviation s.If       x    1    ,      x    2    ,  …  ,      x    N   denote all N values from a population, then the (population) standard deviation is  σ  =            1      N              ∑              i        =        1            N        (          x      i        −    μ          )      2        ,where   μ is the mean of the population.If       x    1    ,      x    2    ,  …  ,      x    N   denote N values from a sample, however, then the (sample) standard deviation is  s  =            1              N        −        1                    ∑              i        =        1            N        (          x      i        −                  x        ¯                    )      2        ,where             x      ¯       is the mean of the sample.Step 2The reason for the change in formula with the sample is this: When you&#039;re calculating s you are normally using       s    2   (the sample variance) to estimate       σ    2   (the population variance). The problem, though, is that if you don&#039;t know   σ you generally don&#039;t know the population mean   μ, either, and so you have to use             x      ¯       in the place in the formula where you normally would use   μ. Doing so introduces a slight bias into the calculation: Since             x      ¯       is calculated from the sample, the values of xi are on average closer to       x    i   than they would be to   μ, and so the sum of squares       ∑          i      =      1        N    (      x    i    −            x      ¯            )    2   turns out to be smaller on average than       ∑          i      =      1        N    (      x    i    −  μ      )    2  . It just so happens that that bias can be corrected by dividing by   N  −  1 instead of N. (Proving this is a standard exercise in an advanced undergraduate or beginning graduate course in statistical theory.) The technical term here is that       s    2   (because of the division by   N  −  1) is an unbiased estimator of       σ    2  .Another way to think about it is that with a sample you have N independent pieces of information. However, since             x      ¯       is the average of those N pieces, if you know       x    1    −            x      ¯        ,      x    2    −            x      ¯        ,  …  ,      x          N      −      1        −            x      ¯      , you can figure out what       x    N    −            x      ¯       is. So when you&#039;re squaring and adding up the residuals       x    i    −            x      ¯      , there are really only   N  −  1independent pieces of information there. So in that sense perhaps dividing by   N  −  1 rather than N makes sense. The technical term here is that there are   N  −  1 degrees of freedom in the residuals       x    i    −            x      ¯      .

Sample Standard Deviation vs. Population Standard Deviation. I have an HP 50g graphing calculator and I am using it to calculate the standard deviation of some data. In the statistics calculation there is a type which can have two values: Sample Population

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