Questions: A sample of 81 objects had the following results: The sample mean (x̄) = 181 The median = 182 The sample standard deviation (S) = 4. Suppose 4 was added to each of the numbers in the dataset. What is the x̄ of the resulting dataset? What is the median of the resulting dataset? What is the S of the resulting dataset? What is the S^2 (the variance) of the resulting dataset?

Solution Steps

When 4 is added to each number in the dataset, the new mean \( \bar{x}' \) can be calculated as follows:

\[ \bar{x}' = \bar{x} + 4 = 181 + 4 = 185 \]

Similarly, the new median \( \text{median}' \) is calculated by adding 4 to the original median:

\[ \text{median}' = \text{median} + 4 = 182 + 4 = 186 \]

The standard deviation \( S' \) remains unchanged when a constant is added to each value in the dataset:

\[ S' = S = 4 \]

The variance \( S'^2 \) is calculated as follows:

\[ S'^2 = (S')^2 = 4^2 = 16 \]

Final Answer