The mean is the "most sensitive" average because it is affected by any change in the data.
a) Determine the mean, median, mode, and midrange for $2,4,6,7,7,8,14$.
b) Change the 8 to 11 in part (a). Determine the mean, median, mode, and midrange.
c) Which averages were affected by changing the 8 to 11 ?
d) Which averages will be affected by changing the 14 to 11 in part (a)?
Answer
\(\boxed{The mean and midrange were affected by changing the 8 to 11, and the mean and midrange were affected by changing the 14 to 11.}\)
Step 1 ：First, we need to calculate the mean, median, mode, and midrange for the given set of numbers $2,4,6,7,7,8,14$.
Step 2 ：The mean is calculated by adding all the numbers and dividing by the count of numbers. So, the mean is \(\frac{2+4+6+7+7+8+14}{7} = \frac{48}{7} = 6.857142857142857\).
Step 3 ：The median is the middle number when the numbers are arranged in ascending order. Since we have 7 numbers, the median is the 4th number, which is 7.
Step 4 ：The mode is the number that appears most frequently. In this case, the mode is 7 because it appears twice.
Step 5 ：The midrange is the average of the smallest and largest numbers. So, the midrange is \(\frac{2+14}{2} = 8\).
Step 6 ：Next, we change the 8 to 11 and recalculate the mean, median, mode, and midrange.
Step 7 ：The new mean is \(\frac{2+4+6+7+7+11+14}{7} = \frac{51}{7} = 7.285714285714286\).
Step 8 ：The new median, with the numbers arranged in ascending order, is still the 4th number, which is now 7.
Step 9 ：The mode remains the same, which is 7.
Step 10 ：The new midrange is \(\frac{2+14}{2} = 8\).
Step 11 ：Comparing the new values with the original ones, we see that the mean and midrange were affected by changing the 8 to 11.
Step 12 ：Finally, we change the 14 to 11 and determine which averages will be affected.
Step 13 ：The new mean is \(\frac{2+4+6+7+7+11+11}{7} = \frac{48}{7} = 6.857142857142857\).
Step 14 ：The new median remains the same, which is 7.
Step 15 ：The mode remains the same, which is 7.
Step 16 ：The new midrange is \(\frac{2+11}{2} = 6.5\).
Step 17 ：Comparing these new values with the original ones, we see that the mean and midrange were affected by changing the 14 to 11.
Step 18 ：\(\boxed{The mean and midrange were affected by changing the 8 to 11, and the mean and midrange were affected by changing the 14 to 11.}\)
