Under what circumstance will the standard deviation of a data set remain unchanged?

The standard deviation measures the dispersion or spread of a data set around its mean. When a constant value is added to each data point in a data set, the entire distribution is shifted up or down on the number line, but the relative distances between the data points remain the same. This is crucial because standard deviation is concerned with how data points vary from the mean, not with the actual values of the data points themselves.

For example, if you have a data set consisting of the numbers 2, 4, and 6, the mean is 4, and the standard deviation measures how far each number is from the mean. If you add a constant value, like 3, to each number, the new data set becomes 5, 7, and 9. The mean shifts to 7, but the distances from the mean (2, 0, and -2) remain consistent. Therefore, the standard deviation remains unchanged because it's still measuring the same relative distances among the data points.

In contrast, removing a data point alters the overall data set, potentially changing both the mean and the spread. Multiplying each data point by two expands the distances between them, thus increasing the standard deviation. Dividing each data point by