Final answer:
A very small standard deviation in a data set indicates little variation among the data points, suggesting they are close to the mean. A very large standard deviation shows that the data points are spread out, exhibiting high variability. The standard deviation is a key metric for assessing data spread and comparison to the data set mean.
Step-by-step explanation:
If a data set has a very small standard deviation, it indicates that the data points are very close to the mean, showing little variation or spread. In other words, most of the values are similar to each other and the mean is a good representation of the data. Conversely, a very large standard deviation means that the data points are widely scattered around the mean, suggesting a large degree of variability or spread within the data set. This could mean that the data points vary significantly from each other and from the mean, making the mean less representative of the individual data points.
The standard deviation can be used to determine if a specific data point is close to or far from the mean, and helps us understand the overall variation within a data set. It can be especially insightful for symmetrical distributions, but for skewed distributions, additional measures such as quartiles may provide better insights. The standard deviation is essential for comparing individual data or classes to the mean of the data set numerically.