Variance: Why divide by n-1? Part 1: Empirical

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In this video we will learn empirically why the sample variance equation uses n-1 in the denominator.
3:07 Average
4:33 Sample Variance using Population Average and dividing by n
6:36 Sample Variance using Sample Average and dividing by n
7:45 Sample Variance using Sample Average and dividing by n-1
8:50 Sample Variance using Sample Average and dividing by n-2
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In statistics, the variance is a measure of how spread out the values in a data set are around the mean. It quantifies the degree of dispersion within the data set. When calculating the variance, it's crucial to understand the rationale behind dividing by n-1 rather than simply by n.
The formula for variance involves finding the average of the squared differences between each data point and the mean. Intuitively, dividing by n seems logical because it represents the total number of data points in the sample. However, using n as the denominator would result in what's known as the biased sample variance.
The reason for dividing by n-1 lies in the concept of degrees of freedom. Degrees of freedom represent the number of independent observations in a data set that can vary without violating any constraints. In the context of variance calculation, using n-1 as the denominator provides an unbiased estimate of the population variance from a sample.
Dividing by n-1 adjusts for the fact that when calculating the sample mean, we use the sample itself to estimate the population mean. This process introduces a degree of dependency among the data points. By dividing by n-1 instead of n, we account for this loss of one degree of freedom, resulting in an unbiased estimate of the population variance.
Using n-1 as the denominator ensures that the calculated sample variance is an unbiased estimator of the population variance. This adjustment helps to minimize the bias that would otherwise be present if we were to divide by n, making the variance calculation more accurate and reflective of the true dispersion within the population.