Final answer:
The function r₂(x) is used to calculate the sum of squared deviations, which is foundational for determining the variance and standard deviation of a dataset. These statistical measures give insight into the data's dispersion around the mean.
Step-by-step explanation:
Understanding Variance and Standard Deviation
The function r2(x) = Σᵢ(xᵢ - x)² is a representation of how to calculate the sum of squared deviations of a dataset from its mean. To compute the sum of squared deviations, you subtract the mean (x) from each data point (xᵢ), square the result, and sum all the squared differences together. This is a crucial step in calculating both the variance and the standard deviation of the population.
The population standard deviation (σ) is the square root of the population variance, which is the average of the squared deviations (when the mean of the population is used, denoted as μ). For a sample, the variance (s²) and standard deviation (s) are calculated similarly but with n-1 in the denominator, where n is the sample size. This adjustment for the sample standard deviation is known as Bessel's correction.
The importance of variance and standard deviation cannot be overstated in statistics as they measure the dispersion or spread of a dataset around its mean. The greater the variance or standard deviation, the more spread out the data points are.