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Bessel's Correction

When working with sample data in statistics, one of the challenges is to make accurate inferences about the population from which the sample is drawn. Bessel's correction is a method used to get a better estimate of the population variance and standard deviation based on sample data.

Why is Bessel's Correction Needed?

When we calculate the sample variance, using the sample mean as a reference point, we tend to underestimate the actual population variance. This is because the sample mean is closer to the individual data points of the sample than the population mean. As a result, the squared differences between the data points and the sample mean are generally smaller than they would be if compared to the actual population mean.

To correct this bias, we use Bessel's correction by subtracting 1 from the number of data points in the denominator when calculating the sample variance.

Formula

Sample Variance with Bessel's Correction:

[ s^2 = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2 ]

Where:

  • ( s^2 ) is the sample variance.
  • ( n ) is the number of sample data points.
  • ( x_i ) represents each data point in the sample.
  • ( \bar{x} ) is the sample mean.

Population Variance (for comparison):

[ \sigma^2 = \frac{1}{n} \sum_{i=1}^{n} (x_i - \mu)^2 ]

Where:

  • ( \sigma^2 ) is the population variance.
  • ( n ) is the number of population data points.
  • ( x_i ) represents each data point in the population.
  • ( \mu ) is the population mean.

Implications

  • Unbiased Estimate: By applying Bessel's correction, the sample variance becomes an unbiased estimator of the population variance.

  • Standard Deviation: When calculating the sample standard deviation, one should use the square root of the unbiased sample variance. It's crucial to remember this, especially when working with small sample sizes.

Example

Suppose we have a sample of scores: [5, 7, 9, 10]. The sample mean is 7.75. Using Bessel's correction, the sample variance is calculated as:

[ s^2 = \frac{1}{3} ((5-7.75)^2 + (7-7.75)^2 + (9-7.75)^2 + (10-7.75)^2) ]

Conclusion

Bessel's correction is an essential adjustment when working with sample data. It ensures that the sample variance and, consequently, the sample standard deviation are unbiased estimators of their respective population metrics, providing more accurate insights into the variability of the data.

References: