Final answer:
For grouped data, we approximate the mean using the formula Σ fm / Σf. Standard deviation and mean are essential for understanding data distributions, especially in large samples which can be approximated by a normal distribution.
Step-by-step explanation:
When dealing with grouped data, such as in a frequency table, it is important to acknowledge that individual data values are not known, meaning we cannot determine the exact mean, median, or mode. However, we can provide a best estimate of these measures of central tendency. The formula for estimating the mean of a frequency table is Mean of Frequency Table = Σ fm / Σf, where f is the frequency and m is the midpoint of each group.
Understanding the properties of the standard deviation within a set of data and its relationship with the mean is crucial. For instance, having a distribution where the sample mean is higher in probability close to the population mean implies that the data is likely to be normally distributed, especially when the sample size is considerable (greater than 30). This holds true for different scenarios, such as those presented in the exercises with varying cholesterol levels, sums, and z-scores.
The normal distribution is particularly helpful in predicting the likelihood of certain outcomes, as seen with the provided probabilities for intervals of sums within a given range around a sample mean. Taking the distribution with a mean of 180 and standard deviation of 20 for example, we can calculate sums that are specific standard deviations from the mean, as well as the probabilities of these sums occurring.