Question

mean: 150 median: 162 mode: 163 range: 60 iqr: 35 after the measurements were recorded, the team saw that the tool was broken. it had not been calibrated (set up) correctly and each measurement (data value) in the set should actually be 3.75 feet greater than the recorded width. how will adding 3.75 to each data point change the 5 measures above? a the mean, median and mode need to add 3.75. the range and iqr stay the same. b each data point and each of the 5 measures need to add 3.75. c only the range and the iqr need to add 3.75. all other numbers stay the same.

Explanation:

Step1: Effect on Mean, Median, Mode

If we have a set of data \(x_1,x_2,\cdots,x_n\), the mean \(\bar{x}=\frac{\sum_{i = 1}^{n}x_i}{n}\). If we add a constant \(c\) to each data point, the new mean \(\bar{y}=\frac{\sum_{i=1}^{n}(x_i + c)}{n}=\frac{\sum_{i = 1}^{n}x_i+nc}{n}=\bar{x}+c\).
The median is the middle - value of the ordered data set. If we add \(c\) to each data point, the order of the data points remains the same, but each value is increased by \(c\). So the median of the new data set is the median of the old data set plus \(c\).
The mode is the most frequently - occurring value. If we add \(c\) to each data point, the most frequently - occurring value of the new data set is the most frequently - occurring value of the old data set plus \(c\).

Step2: Effect on Range and IQR

The range is \(R=x_{\text{max}}-x_{\text{min}}\). If we add \(c\) to each data point, the new range \(R'=(x_{\text{max}} + c)-(x_{\text{min}}+c)=x_{\text{max}}-x_{\text{min}}=R\).
The inter - quartile range (IQR) is \(IQR = Q_3 - Q_1\) (where \(Q_1\) is the first quartile and \(Q_3\) is the third quartile). If we add \(c\) to each data point, the new \(IQR'=(Q_3 + c)-(Q_1 + c)=Q_3 - Q_1=IQR\)

Answer:

A. The Mean, Median and Mode need to add 3.75. The Range and IQR stay the same.