Question

select the outlier in the data set.
881 999 997 811 780 91 979 754
if the outlier were removed from the data set, would the mean increase or decrease?
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 \(881,999,997,811,780,91,979,754\), \(91\) is much smaller than the other values.

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

The formula for the mean \(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}\). Let \(S=\sum_{i = 1}^{8}x_{i}=881 + 999+997+811+780+91+979+754\). The mean with \(n = 8\) is \(\bar{x}_{1}=\frac{S}{8}\). When we remove \(91\) (the outlier), the new sum \(S'=S - 91\) and the new number of data points \(n'=7\). The new mean \(\bar{x}_{2}=\frac{S'}{7}=\frac{S - 91}{7}\). We can also think about it conceptually. The outlier \(91\) is pulling the mean down. When we remove it, since we are dividing a larger - relative - to - the - number - of - elements sum (\(S-91\) with \(n = 7\) instead of \(S\) with \(n = 8\) where \(91\) was reducing the sum more per element) the mean will increase.

Answer:

The outlier is \(91\). If the outlier were removed from the data set, the mean would increase.