Consider the heights of a group of students: 165, 171, 173, 180, 180, 181

Suppose that one of the "180"s was accidentally recorded as "18". Would the mean or median be more affected by the change?

Question

Consider the heights of a group of students: 165, 171, 173, 180, 180, 181

Suppose that one of the "180"s was accidentally recorded as "18". Would the mean or median be more affected by the change?

BlackTom AI · Accepted Answer

Let's restate the scenario in clear terms: we have the heights 165, 171, 173, 180, 180, 181, and one of the 180s was recorded as 18 by mistake. The question asks whether the mean or the median is more affected by this change.
First, observe the data set before any error: 165, 171, 173, 180, 180, 181. The mean is the sum divided by count, and the median is the middle value when ordered. With six data points, the median is the average of the third and fourth values in order, which are 173 and 180, giving a median of (173 + 180)/2 = 176.5. The mean would be computed as (165 + 171 + 173 + 180 + 180 + 181)/6, which equals 1030/6 ≈ 171.67.
Now consider the change: one of the 180s becomes 18. The new data set (in order) is 165, 171, 173, 180, 18, 181. When sorted, this becomes 18, 165, 171, 173, 180, 181. The new median is the average of the 3rd and 4th values: (171 + 173)/2 = 172.0. The new mean is (18 + 165 + 171 + 173 + 180 + 181)/6 = 1,, (sum) 1,0? Let me compute: 18 + 165 = 183; 183 + 171 = 354; 354 + 173 = 527; 527 + 180 = 707; 707 + 181 = 888. Then 888/6 = 148.
Comparing the before-and-after values: the original mean was about 171.67 and the original median was 176.5. After the change, the mean drops to 148, while the median drops only to 172.0. The mean has a larger absolute change than the median, reflecting that the mean is sensitive to extreme values or outliers, whereas the median depends on the middle positions and is more robust to a single aberrant value.
In essence, introducing a drastic change to one data point (from 180 to 18) pulls the arithmetic average downward significantly because the mean incorporates every value in the sum, while the median only depends on the middle order statistics and ends up less affected by a single outlier in a small data set.

Consider the heights of a group of students: 165, 171, 173, 180, 180, 181 Suppose that one of the "180"s was accidentally recorded as "18". Would the mean or median be more affected by the change?单项选择题

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