In this problem, we explore the effect on the mean, median, and mode of multiplying each data value by the same number. Consider the following data set \[ 6,6,7,10,14 \] (a) Compute the mode, median, and mean. (Enter your answers to one (1) decimal places.) Mean value \( = \) \( \square \) Median \( = \) \( \square \) Mode \( = \) \( \square \) (b) Multiply 2 to each of the data values. Compute the mode, median, and mean. (Enter your answers to one (1) decimal places.) Mean value \( = \) \( \square \) Median \( = \) \( \square \) Mode \( = \) \( \square \) (c) Compare the results of parts (a) and (b). In general, how do you think the mode, median, and mean are affected when the same constant is added to each data value in a set? Multiplying each data value by the same constant c results in the mode, median, and mean increasing by a factor of c . Multiplying each data value by the same constant c results in the mode, median, and mean remaining the same. There is no distint pattern when each data value is multiplied by the same constant. Multiplying each data value by the same constant c results in the mode, median, and mean decreasing by a factor of c .

First, let’s break down the initial data set \(6, 6, 7, 10, 14\): - **Mean value**: To find the mean, add all the numbers together: \(6 + 6 + 7 + 10 + 14 = 43\). Then divide by the count of numbers, which is \(5\): \(43/5 = 8.6\). - **Median**: The median is the middle value when arranged in order. With five numbers, the middle one is the third number: \(7\). - **Mode**: Mode is the value that appears most frequently. Here, \(6\) appears most often, so the mode is \(6\). Now, for part (b), we multiply each data point by \(2\): The new data set is \(12, 12, 14, 20, 28\). - **Mean value**: Add \(12 + 12 + 14 + 20 + 28 = 86\) and divide by \(5\): \(86/5 = 17.2\). - **Median**: The middle number in the sorted new set \(12, 12, 14, 20, 28\) remains \(14\). - **Mode**: The mode also changes to \(12\) because it had the highest frequency in the new set. So to summarize: (a) - Mean value \( = 8.6\) - Median \( = 7\) - Mode \( = 6\) (b) - Mean value \( = 17.2\) - Median \( = 14\) - Mode \( = 12\) In general, when you multiply each data value by the same constant \(c\), all measures of central tendency (mean, median, and mode) are also multiplied by that constant \(c\). So if \(c\) is greater than \(1\), they increase in value, while if \(c\) is less than \(1\), they will decrease proportionally.