Question

the data set and some summary statistics are listed. 11.5, 12.3, 13.5, 15.6, 16.7, 17.2, 18.4, 19, 19.5, 21.5
mean: 16.52
median: 16.95
standard deviation: 3.11
iqr: 5.5
how does adding 5 to each of the values in the data set impact the shape of the distribution?
how does adding 5 to each of the values in the data set impact the measures of center?
how does adding 5 to each of the values in the data set impact the measures of variability?
both will go up by 5. adding 5 has no impact. does not change; the values are all shifted 5 units to the right.

Explanation:

Step1: Analyze shape of distribution

Adding a constant to each data - point is a translation. It does not change the relative distances between data - points. So, the shape of the distribution does not change; the values are all shifted 5 units to the right.

Step2: Analyze measures of center

The mean and median are measures of center. If we have a data set \(x_1,x_2,\cdots,x_n\) with mean \(\bar{x}=\frac{\sum_{i = 1}^{n}x_i}{n}\) and median \(M\). When we transform the data set to \(y_i=x_i + 5\) for \(i = 1,2,\cdots,n\), the new mean \(\bar{y}=\frac{\sum_{i = 1}^{n}(x_i + 5)}{n}=\frac{\sum_{i = 1}^{n}x_i+5n}{n}=\bar{x}+5\), and the new median is \(M + 5\). So both will go up by 5.

Step3: Analyze measures of variability

The standard deviation \(s=\sqrt{\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n - 1}}\) and the inter - quartile range (IQR) are measures of variability. For the transformed data set \(y_i=x_i + 5\), the new standard deviation \(s_y=\sqrt{\frac{\sum_{i = 1}^{n}((x_i + 5)-(\bar{x}+5))^2}{n - 1}}=\sqrt{\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n - 1}}=s\), and the IQR also remains the same because the relative spread of the middle 50% of the data does not change. So adding 5 has no impact.

Answer:

How does adding 5 to each of the values in the data set impact the shape of the distribution? - Does not change; the values are all shifted 5 units to the right.
How does adding 5 to each of the values in the data set impact the measures of center? - Both will go up by 5.
How does adding 5 to each of the values in the data set impact the measures of variability? - Adding 5 has no impact.