Final answer:
The standard deviation measures the dispersion of a data set and is different for population and sample data. Population standard deviation uses the population mean (μ) and the number of observations (N), while sample standard deviation uses the sample mean (x) and the sample size minus one (n-1). Calculators or software are typically used to perform the calculations in practice.
Step-by-step explanation:
The standard deviation is a measure of the amount of variation or dispersion in a set of values. It is important to distinguish between a population and a sample when calculating standard deviation. For population data, the population mean (μ) is used, and the formula for population standard deviation (σ) is σ = √[Σ(x-μ)² / N], where N is the number of observations in the population.
Conversely, for sample data, the sample mean (x) is used and the sample standard deviation (s) is calculated with s = √[Σ(x-μ)² / (n-1)], where n is the number of observations in the sample. The denominator is n-1 because it corrects the bias in the estimation of the population standard deviation; this is known as Bessel's correction.
In practice, most people use a calculator or software to compute the standard deviation. The deviation of a single data point is represented as x - x for the sample or x - μ for the population. The variance is the mean of these squared deviations. Understanding these concepts is critical for interpreting how much the data is spread out about the mean.