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}{rrrrrrrrrrr}247 & 260 & 268 & 273 & 276 & 279 & 281 & 283 & 284 & 285 & 286 & 288\end{array}$20$\begin{array}{rrrrrrrr}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}{rrrrrrrrrrr}247 & 260 & 268 & 273 & 276 & 279 & 281 & 283 & 284 & 285 & 286 & 288\end{array}$20$\begin{array}{rrrrrrrr}289 & 291 & 293 & 295 & 296 & 299 & 310 & 504\end{array}$

Key Concepts

Outliers

Outliers are data points that are significantly different from the majority of the data. They can skew measures like the mean, which is why resistant measures such as the median and trimmed mean are used to provide a more reliable indication of central tendency in the presence of outliers.

Mean

The mean is the arithmetic average of a dataset, found by summing all the values and dividing by the count of values. It is sensitive to extreme values or outliers, which can disproportionately affect the result and misrepresent the data's central location.

Median

The median is the middle value in an ordered dataset, or the average of the two middle values if the dataset has an even number of data points. It is a resistant measure of center because it is not significantly impacted by extreme values.

Trimmed Mean

The trimmed mean is calculated by first sorting the data and then removing a certain percentage of the smallest and largest values before calculating the mean of the remaining data. This technique reduces the impact of outliers, providing a more robust estimate of the center.

Resistant Measures

Resistant measures are statistical tools that are not heavily affected by extreme values. Both the median and trimmed mean are resistant measures, making them useful for summarizing data sets that may contain outliers or non-normal distributions.

Transcript

00:01 Okay, so in this problem we have 20 different values and we're going to compute the mean as we have done in that section of the book very often, except we're going to compute different means and see how extreme values can really influence and modify the mean of a dataset.

00:24 So if we look at those 20 values, we notice quite easily that there's one outlier and it's 404.

00:33 So every value is within the range of 247 and 310.

00:41 So 63, there's a difference of 63 whatever units that data set use.

00:50 But this one is there's a jump of 200 units when going from 5.

00:55 300 ,310 to 400 and 404.

01:00 So that value will clearly influence the mean if we compute it normally.

01:06 So if we compute the mean, so if we sum every value and we divide by the number of value, that means if we have 247 plus 260 plus everything, well, i'm gonna write a bit more actually.

01:25 2608 plus 273 plus 276 plus 276 plus 276 plus 279 plus 281 plus and then we get on the later value so 293 295 296, 299, 310, and 504.

01:58 So if we take all of that divided by 20, we are going to get a mean of 294.

02:12 Point 35.

02:15 So this mean is absolutely not representative of the data sense since it's it's right here.

02:26 Let me write it in orange.

02:29 The mean is here.

02:33 So it's most of our values are not close to 294 .35.

02:40 Most of our values are probably more around the values in the middle.

02:46 So this does not really measure.

02:48 It's not a good central measure because most of our values pretty far from 294 .35.

02:58 The reason it's so high is because we have one very high value that doesn't really fit with the rest of the data set.

03:04 So we use the 10 % trimmed means.

03:08 So i'm going to just use a tiny 10 here.

03:11 And what we'll do is we will cut 20 % of our data.

03:18 So for the 20 % mean, we will cut the bottom 10%.

03:24 So remember, 10 % is 0 .1, and 10 % of 20 is 0 .1 times 20, which is 2.

03:37 So we will cut the bottom 10%, so the first two values, the lowest two values, and we are also going to cut the two greatest values in our data set.

03:52 So now the 10 % trimmed mean, we're only going to use 16 observations starting, of the reservations starting from 268 plus 273 plus 276.

04:08 Plus 279 everything in the middle is left on change and then we have we reach the end so 293 295 296 and 299 and 299 and we divide by 16 because we have now when we trim the number of values we only have 16 values and that leaves us with a new mean of 285 .375...

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