Trimmed and Winsorized Measures Introduction When outliers are

Ø Trimmed and Winsorized Measures

Ø Introduction: ü When outliers are present in the data , trimming and Winsorization are methods for reducing the effects of extreme values in the sample. ü A trimmed mean (similar to an adjusted mean) is a method of averaging that removes a small designated percentage of the largest and smallest values before calculating the mean. ü Winsorization began as a way to "robustify" the sample mean, which is sensitive to extreme values. To obtain the Winsorized mean, you sort the data and replace the smallest k values by the (k +1) smallest value. You do the same for the largest values, replacing the k largest values with the (k+1) largest value.

Ø Example: Calculate the Trimmed and Winsorized means and standard deviations for the data given below. 45, 32, 37, 46, 39, 36, 41, 48, 36 Solution: Arrange the data 32, 36, 37, 39, 41, 45, 46, 48

Ø Trimmed Mean : 32, 36, 37, 39, 41, 45, 46, 48 Q 1 Q 2 Q 3 Q 1 = 36 Q 3 = 45 To find the trimmed mean and standard deviation, we remove the two observations 32 and 36 below the first quartile and the two observations 46 and 48 above third quartile. So, we have five observations 36, 37, 39, 41, 45 as trimmed data set.

Ø Winsorized Mean: To find the Winsorized mean and standard deviation, we replace the two values 32, 36, below the first quartile with 36, and the two values 46, 48 above third quartile with 45 to get the Winsorized data set as 36, 36, 37, 39, 41, 45 and 45. 32, 36, 37, 39, 41, 45, 46, 48 Q 1 = 36 Q 3 = 45 Q 2 Q 3

X 36 36 36 37 39 41 45 45 45 Total 360 X 2 1296 1369 1521 1681 2025 14534

Ø Assignment: Question no : 1 Calculate the Trimmed and Winsorized means and standard deviations for the data given below. 80, 75, 42, 63, 65, 43, 78, 96, 82, 5879, 72, 67, 75, 68.