Question

a set of data is found to have a sample standard deviation of 29. suppose 5 was added to each of the numbers in the dataset. the standard deviation of the resulting dataset

would be 31.5.

would be 34.

wouldnt change.

cannot be determined from this information.

Explanation:

Step1: Recall the property of standard deviation

The standard deviation measures the spread of data. When a constant \(c\) is added to each data point \(x_i\) (i.e., \(y_i=x_i + c\)), the mean \(\bar{x}\) changes to \(\bar{y}=\bar{x}+c\).
The formula for the sample standard deviation is \(s=\sqrt{\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n - 1}}\). For the new data set \(y_i=x_i + 5\), the new standard deviation \(s_y=\sqrt{\frac{\sum_{i=1}^{n}[(x_i + 5)-(\bar{x}+ 5)]^2}{n - 1}}\).

Step2: Simplify the formula for the new standard deviation

Expand \((x_i + 5)-(\bar{x}+ 5)\) inside the square - root. We get \((x_i + 5)-(\bar{x}+ 5)=x_i-\bar{x}\). So, \(s_y=\sqrt{\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n - 1}}\), which is the same as the formula for the original standard deviation \(s_x\).

Answer:

wouldn't change.