Final answer:
The standard deviation measures the spread of data around the mean in a normal distribution, which is symmetrical and bell-shaped. Z-scores standardize values across different datasets and are crucial for comparing and finding probabilities. The Empirical Rule estimates the distribution of data within specific intervals around the mean.
Step-by-step explanation:
The standard deviation is a measure that indicates how each value in a dataset varies from the mean. When we talk about the standard deviation as a ruler, we refer to its role in measuring the spread of a dataset in a normal distribution, which is a bell-shaped curve symmetrical about the mean. This mean is also the median and mode in a normal distribution. The curve becomes wider (higher standard deviation) or narrower (lower standard deviation) based on the value of the standard deviation, while the mean shifts the entire curve left or right without changing its shape.
Z-scores are standardized scores that transform a normal distribution into a standard normal distribution, denoted as N(0, 1), with a mean of 0 and a standard deviation of 1. Z-scores help compare values from different datasets by placing them on a common scale. They are calculated with the formula z = (x - µ) / σ, where x is the value in the dataset, µ is the mean, and σ is the standard deviation of the original distribution. These scores can be used to find probabilities associated with normal distributions, using a Z-Table, for various means and standard deviations.
The Empirical Rule provides a quick way to estimate the probability of a value occurring within certain number of standard deviations from the mean in a normal distribution. It states that approximately 68% of the data falls within one standard deviation, 95% within two standard deviations, and 99.7% within three standard deviations from the mean.