Final answer:
The standard deviation is calculated using a calculator or software by adding data to the summary statistics. The mean is found by dividing the total of the values by their count, and values a number of standard deviations from the mean are found using a specific formula. Standard deviation provides a measure of the spread of data around the mean.
Step-by-step explanation:
The standard deviation is a measure of how spread out numbers are within a data set. When using a calculator or computer software like a TI-83, 83+, or 84+ calculator, you can find the standard deviation typically represented as σx or sx for a population or a sample, respectively. To calculate standard deviation using a calculator, follow the summary statistics method which usually involves entering the data set into the calculator, choosing the statistical analysis function, and then selecting the standard deviation option to get the result.
To find the mean, add up all the numbers and divide by the total count of numbers. For example, if you have a population with a mean (μ) of 125 and a standard deviation (σ) of seven, and you draw a sample of size 40, the software does the arithmetic to provide you the mean and standard deviation for that specific sample. It's important not to round intermediate results to maintain accuracy.
To find a value that is a certain number of standard deviations away from the mean, use the formula x = mean + (number of standard deviations)(standard deviation). For instance, to find a value that is two standard deviations above the mean, if the mean is 180 and the standard deviation is 20, we'd calculate x = 180 + (2)(20), which equals 220.
The empirical rule allows you to estimate the percentage of values that fall within a certain range of the mean in a normally distributed data set. For example, approximately 95% of values should fall within two standard deviations of the mean. Using the Central Limit Theorem, as the sample size increases, the sample mean tends to converge towards the population mean. This is important for interpreting standard deviation in relation to larger data sets.