Eight sellers list the same used bass guitar at 699, 599, 699, 680, 590, 720, 650 and 800 dollars. Find the mean price to the nearest cent, the median price, and the mode price.

Step-by-step solution

1. 1

   Add the eight prices carefully. Running total:

   $699+599=1298,\ +699=1997,\ +680=2677,\ +590=3267,$
   $+720=3987,\ +650=4637,\ +800=5437.$

   The total is \5437$.

2. 2

   Divide by the count for the mean.

   $\bar{x}=\frac{5437}{8}=679.625\ \Longrightarrow\ \$679.63.$

   A value of \670.63$sometimes appears on this exercise; it is a digit transposition. Any mean must lie between the smallest and largest data values, and it should sit near the middle of the cluster$590$–$800$ — a quick way to catch such an error.

3. 3

   Sort the data before finding the median. This step cannot be skipped:

   $590,\ 599,\ 650,\ 680,\ 699,\ 699,\ 720,\ 800.$

4. 4

   Average the two middle values. With $n=8$ (even), the median is the mean of the 4th and 5th entries:

   $\text{median}=\frac{680+699}{2}=\frac{1379}{2}=\$689.50.$

5. 5

   Find the most frequent value for the mode. Counting: $699$ appears twice; every other price appears once. So

   $\text{mode}=\$699.00.$

6. 6

   Compare the three measures. Mean \679.63<$median$$689.50<$mode$$699$. The mean is pulled down by the two low prices$590$and$599$, which is exactly the sensitivity to outliers that makes the median the more robust summary of a price list.