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Table of Contents
Have you ever wondered why, when calculating the standard deviation, we sometimes divide by đť“· and sometimes by đť’Ź-1? We are happy to explain this using the formula PDF.
Population standard deviation
If you know all the data of a population, you calculate the standard deviation from all deviations from the true mean and divide by the number of elements.
Problem with a sample
In practice, we usually only have a sample. We do not know the true mean, so we use the sample mean. This adapts to the data, so the sum of all deviations is always zero. As a result, the deviations are smaller on average, and the standard deviation would be systematically underestimated if you simply divided by đť“·.
Degrees of freedom
Because the mean is estimated from the data, we lose one degree of freedom. Of đť’Ź independent deviations, only đť’Ź-1 remain, because the last one is determined by the condition that the sum of the deviations must be zero.
The correction
The so-called Bessel’s correction therefore does not divide by đť’Ź, but by đť’Ź-1. This makes the estimate of the standard deviation unbiased—that is, on average it equals the true population standard deviation.
Why this is so important
Without this correction, we underestimate the dispersion of the data. This leads to confidence intervals that are too narrow, test results that are falsely significant too often, and a systematic underestimation of risks in statistics, research, and financial models.
Who was Bessel?
Friedrich Wilhelm Bessel (1784–1846) was a German astronomer and mathematician. He was the first to calculate the parallax of a star and developed methods for estimating measurement errors—hence the correction bears his name.
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