File sizes How much storage space does your music use? Here is a dotplot of the file sizes (to the nearest tenth of a megabyte) for 18 randomly selected files on Nathaniel’s mp3 player:

(a) The distribution of file size has a mean of $\overline{x}=3.2$megabytes and a standard deviation of$s_x=1.9$ megabytes. Interpret the standard deviation.

(b) Suppose the music file that takes up 7.5 megabytes of storage space is replaced with another version of the file that only takes up 4 megabytes. How would this affect the mean and the standard deviation? Justify your answer.

The given mean and standard deviation are 3.2 and 1.9 respectively.

The standard deviation is a measure of how much the data values deviate from the mean.

The storage space of the 18 files in the sample is expected to vary by 1.9 megabytes on average, compared to the mean of 3.2 megabytes.

The most significant data value, 7.5, is replaced with 4.

The data value of 7.5 is the greatest and looks to be an outlier since it is separated from the other dots in the dotplot by a considerable gap. Outliers have a substantial influence on the mean, so if we reduce an outlier, we can expect the mean to drop as well.

We also see that the new number 4 is closer to the mean than the previous value of 7.5. We expect the standard deviation to drop (as 4 is closer to the mean than 7.5) since the standard deviation indicates the average deviation of the data values from the mean.