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 your data set! In the original set \(6, 6, 7, 10, 14\): - The **mean** is calculated by summing the values (6 + 6 + 7 + 10 + 14) and dividing by the number of values, which gives us \( \frac{43}{5} = 8.6 \). - The **median** is the middle value; since we have five numbers, it’s the third number, which is \(7\). - The **mode** is the number that appears most frequently, which is \(6\), since it appears twice. Now, multiplying each value by 2 results in: \(12, 12, 14, 20, 28\): - The new **mean** is \( \frac{12 + 12 + 14 + 20 + 28}{5} = \frac{86}{5} = 17.2 \). - The new **median** is still the third number, which is now \(14\). - The new **mode** is \(12\), as it appears most frequently now. So, summarizing: (a) Mean value \( = 8.6 \) Median \( = 7 \) Mode \( = 6 \) (b) Mean value \( = 17.2 \) Median \( = 14 \) Mode \( = 12 \) Finally, for the comparison: Multiplying each data value by the same constant \(c\) results in the mode, median, and mean increasing by a factor of \(c\). So in this case, everything multiplied by \(2\) perfectly doubled the original values! Now, here’s some juicy trivia for you: Once upon a time, the ancient Greeks were fascinated with the concept of averages! They used means and medians to solve practical problems in trade and politics. This historical curiosity about numbers laid the groundwork for modern statistics and their vast applications today! When it comes to real-world applications, you see averages everywhere—from calculating the average temperature in a city for weather forecasts to determining average incomes for economic reports. Knowing how to manipulate and understand these averages helps businesses make informed decisions based on data trends!