This exercise is designed to help you explore how different measures of central tendency can help us determine the "normality" of a data set. We will use the 10%-trimmed mean for this exercise.

The definition in the book would suggest removing the upper 10% and the lower 10% (see the exercise section of §3-2). However, the definition in Excel is that to get the 10% trimmed mean, you trim a total of 10%, thus 5% from the top and 5% from the bottom. Thus, for a data set with 20 values, the 10% trimmed mean would be calculated by removing the minimum and maximum value (5% of 20 is 1, so 1 value from the top and 1 value from the bottom) and then finding the mean of the remaining 18 values. We will use the definition used by Excel.

Here is a data set ($\displaystyle {n}={40}$) that is nearly normal, but appears to have two outliers (as can be seen in the histogram provided after the data set).

41.4	43.5	43.6	45.9	46.8	47.6	50.6	51
51.7	52	52.4	52.7	53.8	54.1	54.7	55.3
55.6	57.6	57.6	58.2	58.8	59.8	60.8	61.6
62	62.2	64.2	64.8	65	65.7	66.7	67.3
69	69	69.8	70.3	72	77.7	99	115.8

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Use Excel or StatDisk to find the requested measures of central tendency for this data set. You should be able to select the table and copy it directly to Excel or other such programs. Report the mean using the rounding rules suggested in the book. Report the 10%-trimmed mean using the same rounding rules for the mean.

mean =

median =

Find the relative difference between the mean and the median using this formula:

$\displaystyle \frac{{{m}{e}{a}{n}-{m}{e}{d}{i}{a}{n}}}{{{m}{e}{d}{i}{a}{n}}}\cdot{100}\%$

Report your answer as a percentage accurate to 3 decimal places (the % symbol is provided for you). Remember: You reported the mean after rounding, but future calculations—like this one—require using as many decimal places of accuracy as possible!

relative difference = %

Now, calculate the 10%-trimmed mean for this data set.

10%-trimmed mean =

Also report the relative difference for the 10%-trimmed mean and the median.

relative difference = %

Briefly describe how the trimmed mean helps us understand the effect of (possible) outliers in the data set. (This part of the exercise is not automatically graded by the computer.)