Final answer:
Standard deviation is a measure of data dispersion around the mean, with examples showing how scores compare to averages in classes or how spatially on a number line values deviate based on multiples of standard deviation.
Step-by-step explanation:
The standard deviation is a statistic that measures the dispersion of a dataset relative to its mean. It is calculated as the square root of the variance. To understand the concept better, consider the following examples:
- In a statistics class, the final exam scores were normally distributed with a mean of 63 and a standard deviation of five. This means that most scores lie within five points above or below the average score of 63. A score of 68 would be one standard deviation above the mean.
- Rachel's math exam had a mean grade of 74 with a standard deviation of five. Rachel scored an 80, which is one standard deviation above the mean. Similarly, for Becca's class, with a mean of 47 and a standard deviation of two, her score of 51 is two standard deviations above the mean.
- For the number line example, if we place the numbers five and seven on a line, seven is one standard deviation to the right of five since 5 + (1×2) = 7. Conversely, the number one would be two standard deviations to the left of five because 5 + (−2×2) = 1.
Your concentration should be on what the standard deviation reveals about the data. It indicates the degree to which the data points spread out or cluster around the mean. The calculation is typically done using a calculator or computer software, like a TI-83/84 calculator, to obtain either sample standard deviation (sx) or population standard deviation (σ).