Question

Suppose that everyone in a company receives an annual pay raises of 5%. How does it affect the standard deviation of annual pay?

Answer 1

A 5% across-the-board annual pay raise in a company does not alter the standard deviation of annual pay since it maintains the relative distances between salaries.

Step-by-step explanation:

When everyone in a company receives an annual pay raise of 5%, the standard deviation of annual pay does not change. The reason for this is that a percentage increase changes all salaries by the same proportion, which means the relative distance between any two salaries remains constant.

Standard deviation is a measure of the spread of values around the mean in a dataset. It calculates the average distance of each data point from the mean. Hence, if each data point (salary, in this case) is increased by the same percentage, the spread or dispersion around the mean is unaffected.

The scenario can be compared to stretching a rubber band uniformly by 5% at every point; while the rubber band gets longer, the relative distances between any points marked on the band do not change.

Answer 2

the standard deviation should also increase by 5%

Step-by-step explanation:

If all the wages had increased by a fixed amount, then the mean would have increased by a fixed amount and the standard deviation would remain the same. e.g. data = 2,4,6 with a mean of 4 and a standard deviation of 1.33. If you add 4 to all the numbers: 6,8,10, you get a new mean of 8, but the standard deviation remains at 1.33

But if you increase all the wages by 5%, some wages will increase a little (the lowest) and others will increase a lot (the highest). If the same set of numbers is increased by 50%: 3,6,9 with a mean of 6 (mean increased by 50%) and a standard deviation of (|3 - 6| + |6 - 6| + |9 - 6|) / 3 = (3 + 0 + 3) / 3 = 2, which is 50% higher