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?

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?
Transcript text: A sample of 81 objects had the following results: $\bar{x}$ (The sample mean) $=181$ The median $=182$ $S$ (The sample standard deviation) $=4$. Suppose 4 was added to each of the numbers in the dataset. What is the $\bar{x}$ 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?
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Solution

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Solution Steps

Step 1: New Mean Calculation

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

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

Step 2: New Median Calculation

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

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

Step 3: New Standard Deviation and Variance Calculation

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

\[ \boxed{ \begin{align_} \text{New Mean} & : 185 \\ \text{New Median} & : 186 \\ \text{New Standard Deviation} & : 4 \\ \text{New Variance} & : 16 \end{align_} } \]

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