Question

248
259
268
272
277
280
282
283
285
285
286
289
290
292
294
295
295
298
310
501

Because the mean is very sensitive to extreme​ values, 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 for cans that are 0.0111 in. thick. Identify any​ outliers, then compare the​ median, mean,​ 10% trimmed​ mean, and​ 20% trimmed mean.

Answer 1

The scatterplot for the untrimmed data is shown below,

The scatter plot shows that there is one potential outlier valued 501 pounds.

The mean of the data values is obtained using the formula,

1	248
2	259
3	268
4	272
5	277
6	280
7	282
8	283
9	285
10	285
11	286
12	289
13	290
14	292
15	294
16	295
17	295
18	298
19	310
20	501
Mean	294.45
Median	285.5

Since n = 20 (even), median is the average of two middle values,

10% trimmed

After sorting the data values and then removing 10% (20*10%=2) of the upper and 10% of the lower values, the mean is,

1	268
2	272
3	277
4	280
5	282
6	283
7	285
8	285
9	286
10	289
11	290
12	292
13	294
14	295
15	295
16	298
Mean	285.6875
Median	285.5

Since n = 16 (even), median is the average of two middle values,

20% trimmed

ABy removing 20% (20*20%=2) of the upper and 20% of the lower values, the mean is,

1	277
2	280
3	282
4	283
5	285
6	285
7	286
8	289
9	290
10	292
11	294
12	295
Mean	286.5
Median	285.5

Since n = 12 (even), median is the average of two middle values,

Comparing median, mean,​ 10% trimmed​ mean, and​ 20% trimmed mean

	Mean	Median
untrimmed	294.45	285.5
10% trimmed	285.6875	285.5
20% trimmed	286.5	285.5

The median for each data set is same. since median is middle value of the data set, it is calculated irrespective of the data values. Hence it will be unchanged

Since there is a potential outlier present in the untrimmed data set with highest value, the mean is larger compare to mean of trimmed data set (such that mean is calculate after removing the outlier in trimmed data).