Final answer:
To compute variance, use σ0² for a population and s² for a sample. Standard deviation is the square root of variance, denoted as σ for a population and s for a sample. Use calculators or software for accurate calculations.
Step-by-step explanation:
To calculate the variance, you need to first determine whether you're working with a sample or a population. For a population, the formula for variance (σ0²) is Σ(x-μ)² / N. This means that you sum the squared deviations of each data point from the population mean (μ), and then divide by the number of items (N) in the population. For a sample, you use a similar approach but with a slight adjustment: s² = Σ(x-ξ)² / (n - 1), where ξ is the sample mean and n is the number of items in the sample. The subtraction of 1 from n accounts for the degrees of freedom in a sample.
Once you have calculated the variance, you find the standard deviation by taking the square root of the variance. If calculating for a population, the population standard deviation (σ) is the square root of the population variance. Similarly, the sample standard deviation (s) is the square root of the sample variance (s²).
Remember to use a calculator or computer software for these calculations to minimize the potential for error, especially for large data sets. Devices like the TI-83, 83+, or 84+ calculators have functions to directly calculate standard deviation. Always verify the outputs and use the correct formula depending on whether you're working with a sample or a population.
To interpret the standard deviation, recognize that it measures how spread out the data are from the mean. A larger standard deviation indicates that the data points are more widespread.