Explanation:

When the variables are increased or decreased by the same amount, the standard deviation remains unchanged. This can be understood by considering the formula for calculating the standard deviation.

The standard deviation is a measure of the spread or dispersion of a set of values. It is calculated by taking the square root of the variance. The variance is the average of the squared deviations from the mean.

When the variables are increased or decreased by the same amount, the mean of the data set also increases or decreases by the same amount. However, the deviations from the mean remain the same.

Example:

Let's consider a simple example to illustrate this concept. Suppose we have a set of numbers: 1, 2, 3, 4, 5. The mean of this data set is 3.

If we increase each number by 2, we get: 3, 4, 5, 6, 7. The mean of this new data set is 5.

If we calculate the deviations from the mean for both data sets, we get the following:

Original data set: -2, -1, 0, 1, 2
New data set: -2, -1, 0, 1, 2

As we can see, the deviations from the mean are the same for both data sets. Therefore, the standard deviation remains unchanged.

Conclusion:

When the variables are increased or decreased by the same amount, the standard deviation remains unchanged. This is because the spread or dispersion of the data set is not affected by a constant shift in all values. The standard deviation only changes when there is a change in the variability or distribution of the data.