When I calculate Standard Deviation manually, I get that the value of Standard Deviation is 4.604, but when I calculate in SPSS program, the value of Standard Deviation is 5.14780.
Value Used for Calculation: 8, 12, 13, 15, 22
both results are correct, the question is what you are looking for.
4.604 (the quare root of the sum of squares divided by the number of cases (5)) is the correct result for the sample standard deviation (s). The problem with this parameter is that the SD is no unbiased estimator for the population value. It is out by n/(n-1).
Therefore, you can estimate the population SD by the quare root of the sum of quares divided by the number of degrees of freedom (instead of cases) (this is the estimated sigma). This is the 5.14780 which is given to you by SPSS. It is the value you should use for your analyses, because you want to draw conclusions about the underlying population and not about your specific sample.
the "real" population SD is unknown unless you sample the whole population.
SPSS is correct: s=5.14782. The value you obtained is the population sigma=4.6043. If you are dealing with a sample you have to use the degrees of freedom (n-1=4) in the denominator, to calculate the standard deviation.
Bibliography:
SNEDECOR, G.W. and COCHRAN, W.G. 1989. Statistical Methods. Blackwell Publishing.
STEEL, R.G.D. and TORRIE, J.H. 1997. Principles and Procedures of Statistics: A Biometrical Approach. McGraw and Hill.
both results are correct, the question is what you are looking for.
4.604 (the quare root of the sum of squares divided by the number of cases (5)) is the correct result for the sample standard deviation (s). The problem with this parameter is that the SD is no unbiased estimator for the population value. It is out by n/(n-1).
Therefore, you can estimate the population SD by the quare root of the sum of quares divided by the number of degrees of freedom (instead of cases) (this is the estimated sigma). This is the 5.14780 which is given to you by SPSS. It is the value you should use for your analyses, because you want to draw conclusions about the underlying population and not about your specific sample.
the "real" population SD is unknown unless you sample the whole population.
There should be no difference. however, the chances of error as well as the time consuming when calculating SD manually does not support this approach.
In the sample standard deviation the denominator is (n-1) while in he population sd the denominator is N.
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