Bessel’s correction is a statistical adjustment made to the calculation of the sample variance (and by extension, sample standard deviation). It involves subtracting 1 from the number of observations (n) in the calculation. Named after the German mathematician Friedrich Bessel, the correction addresses bias in the estimation of the population variance.
The formulas for the sample variance and standard deviation with Bessel’s correction are as follows:
Sample Variance: s² = Σ ( xi – x̄ )² / (n – 1)
Sample Standard Deviation: s = sqrt[ Σ ( xi – x̄ )² / (n – 1) ]
where:
– Σ ( xi – x̄ )² is the sum of the squared differences from the mean
– n is the number of observations
– xi is the i-th observation
– x̄ is the sample mean
– sqrt denotes the square root
Why do we use Bessel’s correction?
In statistics, we often have to estimate population parameters (like population mean or population variance) using a sample. However, the sample mean tends to be a more accurate estimate of the population mean than the sample variance is of the population variance. This is because the variance is calculated using the mean, and if the mean is off even slightly, it can have a large effect on the variance.
When calculating the sample variance, if we divide by n, we’re likely to get an underestimate of the actual population variance. This is called a bias in estimation. To correct this bias, instead of dividing by n, we divide by n – 1. This tends to produce a slight overestimate of the population variance, but on average it’s a better estimate.
This is the essence of Bessel’s correction. It’s used when we’re working with a sample of data and trying to estimate the population variance or standard deviation. However, it’s not used when calculating the variance or standard deviation of an entire population, where we know all the data and are not estimating any parameters.