Question

use the data 1, 2, 3, 4, 4, 5. add 3 to each of the numbers. how does this affect the mean? how does this affect the standard deviation?
choose the correct answer below.
○ a. the standard deviation will be increased by 3, and the mean will remain the same.
○ b. both of the mean and the standard deviation will remain the same.
○ c. the mean will be increased by 3, and the standard deviation will remain the same.
○ d. both of the mean and the standard deviation will be increased

Explanation:

Step1: Recall the formula for the mean

The mean of a data set \(x_1,x_2,\cdots,x_n\) is \(\bar{x}=\frac{\sum_{i = 1}^{n}x_i}{n}\). If we transform each data point \(x_i\) to \(y_i=x_i + 3\), then the new mean \(\bar{y}=\frac{\sum_{i=1}^{n}(x_i + 3)}{n}=\frac{\sum_{i = 1}^{n}x_i+3n}{n}=\frac{\sum_{i = 1}^{n}x_i}{n}+3=\bar{x}+3\).

Step2: Recall the formula for the standard deviation

The standard deviation \(s=\sqrt{\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n - 1}}\). For the new data set \(y_i=x_i + 3\), the new mean \(\bar{y}=\bar{x}+3\). Then \(y_i-\bar{y}=(x_i + 3)-(\bar{x}+3)=x_i-\bar{x}\). So the standard deviation of the new data set \(s_y=\sqrt{\frac{\sum_{i = 1}^{n}(y_i-\bar{y})^2}{n - 1}}=\sqrt{\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n - 1}}=s_x\)

Answer:

C. The mean will be increased by 3, and the standard deviation will remain the same.