For a given set of data, which type of variance is always the largest?

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The correct choice is the population variance.

When discussing variance in statistics, the population variance is calculated using all data points from the entire population, while the sample variance uses data points from a subset of the population. The formula for the sample variance includes a correction factor (Bessel's correction), dividing by ( n-1 ) instead of ( n ), where ( n ) is the number of data points in the sample. This adjustment is made to account for the fact that the sample might not fully represent the population, leading to an underestimation of variability.

Consequently, since the population variance is based on all data and does not apply this correction factor, it will always be greater than or equal to the sample variance. The population standard deviation, which is the square root of the population variance, also does not change this relationship, as it is simply a transformed version of the variance.

In summary, the population variance is the largest because it encompasses all data points without any bias adjustment, capturing the complete variability of the dataset.