Question

multiple choice
a shoelace company makes shoelaces of differing lengths.
the production team is ordered to increase the length of every shoelace they
make by 3 centimeters.
which statement about the lengths of the shoelaces the company will produce
after the change is true?
remember:
adding same number to every value in a data set increases the measures of center
for the data set by the same value. the measures of dispersion are unchanged.
a the median length of the shoelaces produced will increase by 3
centimeters.
b the average length of the shoelaces produced will increase by 6
centimeters.
c the range of lengths of the shoelaces produced will increase by 3
centimeters.

Explanation:

• For option A:
• Median is a measure of central tendency. When we add a constant value (in this case, \(3\) centimeters) to every data - point in a data set, the median (the middle value when the data is ordered) increases by that constant value.
• For option B:
• The formula for the mean (average) \(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}\). If we change each \(x_{i}\) to \(x_{i}+3\), then the new mean \(\bar{y}=\frac{\sum_{i = 1}^{n}(x_{i}+3)}{n}=\frac{\sum_{i = 1}^{n}x_{i}+3n}{n}=\frac{\sum_{i = 1}^{n}x_{i}}{n}+3=\bar{x}+3\). So the average increases by \(3\) centimeters, not \(6\) centimeters.
• For option C:
• The range is a measure of dispersion. The formula for the range is \(R=\text{Max}-\text{Min}\). If we add \(3\) to both the maximum (\(\text{Max}\)) and the minimum (\(\text{Min}\)) value of the data set, the new range \(R'=(\text{Max}+3)-(\text{Min}+3)=\text{Max}-\text{Min}=R\). So the range does not change.

Answer:

A. The median length of the shoelaces produced will increase by 3 centimeters.