Question

select the outlier in the data set.
714 889 978 94 888 947 953 880
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 \(714,889,978,94,888,947,953,880\), \(94\) is much smaller than the other values.

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. Let the sum of the original data set be \(S\) and the number of data points be \(n\). The original mean is \(\bar{x}_{1}=\frac{S}{n}\). When we remove the outlier (which is smaller than the other values), the new sum is \(S - 94\) and the new number of data points is \(n- 1\). Since the outlier is a relatively small value, removing it will make the sum decrease by a relatively small amount compared to the decrease in the number of data points. For example, assume \(S=a + 94\) (where \(a\) is the sum of the other 7 data points). The original mean \(\bar{x}_{1}=\frac{a + 94}{8}\), and the new mean \(\bar{x}_{2}=\frac{a}{7}\). Cross - multiply: \(7(a + 94)=7a+658\) and \(8a\). We know that for positive \(a\), when \(a>658\) (and \(a\) is the sum of numbers like \(714,889,\cdots\) so \(a>658\)), \(\frac{a}{7}>\frac{a + 94}{8}\)

Answer:

The outlier is \(94\). If the outlier were removed from the data set, the mean would increase.