How is the standard deviation of a data set defined?

The standard deviation of a data set is defined as a measure of the amount of variation or dispersion of a set of values. Specifically, it quantifies how much the individual data points deviate from the mean (average) of the dataset. A low standard deviation indicates that the data points tend to be close to the mean, while a high standard deviation indicates that the data points are spread out over a wider range of values.

In practical terms, calculating the standard deviation involves finding the mean of the data set, subtracting the mean from each data point to determine the deviations, squaring these deviations to eliminate negative values, averaging these squared deviations, and then taking the square root of that average. This process highlights the degree to which individual values in the data set differ from the mean, providing valuable insight into the variability of the dataset.

Other choices do not accurately define standard deviation. The average of all values refers to the mean, while central tendency describes a different aspect of data characteristics, and calculating the range merely accounts for the difference between the highest and lowest values in the data set without considering how values cluster around the mean. Thus, the definition provided in the correct answer is crucial for understanding the concept of variability in statistics.