Question

select the outlier in the data set.
if the outlier were removed from the data set, would the mean increase or decrease?
10
31
58
21
37
36
39
33
25
32
30
90
16
increase
decrease

Explanation:

Step1: Identify the outlier

An outlier is a data point that is significantly different from other data points. In the data set \(10,31,58,21,37,36,39,33,25,32,30,90,16\), the number \(90\) is much larger than the other numbers.

Step2: Analyze the effect of removing the outlier on the mean

The mean of a data set is calculated as \(\text{Mean}=\frac{\sum_{i = 1}^{n}x_{i}}{n}\), where \(x_{i}\) are the data points and \(n\) is the number of data points. The outlier \(90\) is a large value. When we remove a large - valued outlier from the data set, the sum of the data points \(\sum_{i = 1}^{n}x_{i}\) will decrease by a relatively large amount, and the number of data points \(n\) will decrease by \(1\). Since the outlier is larger than most of the other data points, the proportionate decrease in the sum is larger than the proportionate decrease in \(n\). Mathematically, let the original sum be \(S\) and the original number of data points be \(n\), and the outlier be \(x\) (\(x = 90\)). The original mean is \(\bar{x}_{1}=\frac{S}{n}\), and the new mean is \(\bar{x}_{2}=\frac{S - x}{n - 1}\). Since \(x\) is large, \(\bar{x}_{2}<\bar{x}_{1}\)

Answer:

The outlier is \(90\). If the outlier were removed from the data set, the mean would decrease.