Final answer:
The standard deviation measures the spread of data around the mean, with low values indicating little variation and high values indicating greater dispersion. In a normal distribution, the standard deviation shapes the curve, while the mean determines its central location.
Step-by-step explanation:
The standard deviation is a fundamental statistical measure that assesses the amount of variability or spread in a set of data. When data points are closely packed around the mean, indicating little variability, the standard deviation is small. Conversely, when data points are widely dispersed from the mean, the standard deviation is large. This measure reflects how much each data point deviates from the mean of the dataset.
For a perfectly uniform set of data where all values are equal, the standard deviation would be zero, showing no spread at all. In practical scenarios, the mean represents the center of the distribution, and the standard deviation quantifies the extent to which the values diverge from that central value.
In the context of a normal distribution, changes to the standard deviation affect the shape of the distribution curve, while changes to the mean shift the entire curve without altering its shape. This demonstrates the inseparable relationship between the mean and the standard deviation in defining the characteristics of a distribution.