Final answer:
The measure that does not change when every number in a data set increases by a constant amount, like 3, is the standard deviation. This measure of dispersion remains constant because the increase does not affect the relative spacing between data points.
Step-by-step explanation:
If Every Number in a Data Set Increases by 3, Which Measure Does Not Change?
When every number in a data set is increased by a constant amount, such as 3, there is a specific statistical measure that does not change. The measure that remains unaffected is the measure of dispersion known as the standard deviation.
This can be understood using graphical methods to solve the problem. Imagine a scatter plot of a data set on a graph; the individual points will all move to the right by 3 units if each number in the set is increased by 3. The shape of the distribution does not change; it simply shifts. However, because standard deviation is a measure of how spread out the numbers in a set are relative to the mean, this shift does not affect the relative spacing between the data points.
If we look at specific examples: Suppose our data set is {5,8,11}. The mean would be 8, and let's say the standard deviation is 3. If each number is increased by 3, we get a new set {8,11,14}. The new mean is 11, but the spread is unchanged, just shifted. Therefore, the standard deviation remains 3, illustrating that while central tendency measures like mean, median, and mode shift, the measure of variability such as standard deviation remains consistent.
It should also be noted that other measures of spread such as the range and interquartile range also do not change with a shift. This is because these measures are based on the difference between data points, which remains constant when all values are increased uniformly.