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# The table above lists the lengths, to the nearest inch, of a random sample of...

### Question

The table above lists the lengths, to the nearest inch, of a random sample of 21 brown bullhead fish. The outlier measurement of 24 inches is an error. Of the mean, median, and range of the values listed, which will change the most if the 24-inch measurement is removed from the data?

### Options

A)
Mean
B)
Median
C)
Range
D)
They will all change by the same amount.

Choice A is incorrect. Let $$m$$ be the mean of the lengths, in inches, of the 21 fish. Then the sum of lengths, in inches, of the 21 fish is 21$$m$$. After the 24-inch measurement is removed, the sum of the lengths, in inches, of the remaining 20 fish is 21$$m$$ - 24, and the mean length, in inches, of these 20 fish is $$\frac{21m - 24}{20}$$, which is a change of $$\frac{24 - m}{20}$$ inches. Since $$m$$ must be the smallest and largest measurements of the 21 fish, it follows that 8 < m < 24, from which it can be seen that the change in the mean, in inches, is between $$\frac{24 - 24}{20} = 0$$ and $$\frac{24 - 8}{20} = \frac{4}{5}$$, and so must be less than the change in the range, 8 inches. Choice B is incorrect because the median length of the 21 fish represented in the table is 12, and after the 24-inch measurement is removed, the median of the remaining 20 lengths is also 12. Therefore, the change in the median (0) is less than the change in the range (8). Choice D is incorrect because the changes in mean, median, and range of the measurements are different.