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In this study, we will explain the meaning of “n” and “n-1” in the standard deviation. There are two ways to determine the variance and standard deviation.
When the number of data is n.
(1)
sampling variance= sum of squares of deviations÷n,
standard deviation=√sum of squares of deviations÷n
(2)
unbiased variance = sum of squares of deviations÷n-1,
standard deviation=√sum of squares of deviations÷n-1
What is the proper use of “n” and “n-1” in the standard deviation formula ?
Equation (1) is used to determine the variation in the observed data.
Equation (2) is used when estimating the whole from the extracted data, such as tests randomly sampled from a large number of patients, or when looking at variations in the extracted data.
For example, it is impossible to determine the standard deviation in height for all Japanese people. In this case, we must guess from some data; therefore, we use Formula (2). The variance obtained in (2) is called the unbiased variance.
Notes on unbiased variance
The standard deviation estimated from the data is slightly smaller than that of the actual population. Therefore, a good guess is to divide by “n-1” to obtain a value slightly larger than the standard deviation that is inferred from the data.
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