Because the mean is very sensitive to extreme values, we say...

Because the mean is very sensitive to extreme values, we say that it is not a resistant measure of center. By deleting some low values and high values, the trimmed mean is more resistant. To find the $10 \%$ trimmed mean for a data set, first arrange the data in order. then delete the bottom $10 \%$ of the values and delete the top $10 \%$ of the values, then calculate the mean of the remaining values. Use the axial loads (pounds) of aluminum cans listed below (from Data Set 30 "Aluminum Cans" in Appendix B) for cans that are $0.0111$ in. thick. An axial load is the force at which the top of a can collapses. Identify any outliers, then compare the median, mean, $10 \%$ trimmed mean, and $20 \%$ trimmed mean. $\begin{array}{llllllllllll}247 & 260 & 268 & 273 & 276 & 279 & 281 & 283 & 284 & 285 & 286 & 288\end{array}$ $\begin{array}{llllllll}289 & 291 & 293 & 295 & 296 & 299 & 310 & 504\end{array}$

Because the mean is very sensitive to extreme values, we say that it is not a resistant measure of center. By deleting some low values and high values, the trimmed mean is more resistant. To find the $10 \%$ trimmed mean for a data set, first arrange the data in order. then delete the bottom $10 \%$ of the values and delete the top $10 \%$ of the values, then calculate the mean of the remaining values. Use the axial loads (pounds) of aluminum cans listed below (from Data Set 30 "Aluminum Cans" in Appendix B) for cans that are $0.0111$ in. thick. An axial load is the force at which the top of a can collapses. Identify any outliers, then compare the median, mean, $10 \%$ trimmed mean, and $20 \%$ trimmed mean.
$\begin{array}{llllllllllll}247 & 260 & 268 & 273 & 276 & 279 & 281 & 283 & 284 & 285 & 286 & 288\end{array}$
$\begin{array}{llllllll}289 & 291 & 293 & 295 & 296 & 299 & 310 & 504\end{array}$

Key Concepts

Arithmetic Mean

The arithmetic mean is calculated by summing all the values in a dataset and dividing by the number of values. It is a common measure of central tendency but can be significantly influenced by extreme values or outliers.

Median

The median is the middle value when the data are arranged in order. It is a resistant measure of center because it is not affected greatly by outliers or skewed data, making it a preferred measure when the data contain extreme values.

Trimmed Mean

A trimmed mean is obtained by removing a specified percentage of the lowest and highest values from a dataset and then calculating the mean of the remaining data. This method reduces the effect of outliers, making it a more robust measure of central tendency compared to the arithmetic mean.

Outliers

Outliers are data points that differ significantly from other observations and can distort the analysis by pulling measures such as the mean away from the center of the bulk of the data. Identifying outliers is important for understanding the distribution of data and for choosing appropriate statistical measures.

Resistance to Extreme Values

Resistance, or robustness, refers to the ability of a statistical measure to remain relatively unaffected by extreme or unusual values. Measures like the median and trimmed means are more resistant compared to the arithmetic mean, making them useful in datasets with potential outliers.

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