Question

a survey of 20 middleton residents was done to determine the number of people in each household. the numerical frequencies are shown in the dot plot. what is the outlier of this distribution? what is the mean of this data set? what is the range of this data set?

Explanation:

Step1: Identify the outlier

An outlier is a data point that is significantly different from the other data points. Looking at the dot - plot, the value \(9\) is far from the cluster of other values (which are mostly between \(1\) and \(5\)).

Step2: Calculate the mean

The formula for the mean \(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}f_{i}}{\sum_{i = 1}^{n}f_{i}}\).
The data points \(x\) and their frequencies \(f\):

• \(x = 1,f=1\)
• \(x = 2,f = 4\)
• \(x=3,f = 4\)
• \(x = 4,f=4\)
• \(x=5,f = 3\)
• \(x=9,f = 1\)

\(\sum_{i = 1}^{n}x_{i}f_{i}=1\times1 + 2\times4+3\times4 + 4\times4+5\times3+9\times1\)
\(=1+8 + 12+16+15 + 9\)
\(=61\)

\(\sum_{i = 1}^{n}f_{i}=1 + 4+4+4+3+1=17\)

\(\bar{x}=\frac{61}{17}\approx3.59\approx3.5\) (rounded to one decimal place)

Step3: Calculate the range

The formula for the range \(R=\text{Max}-\text{Min}\). The maximum value \(x_{\text{max}}=9\) and the minimum value \(x_{\text{min}} = 1\). So \(R=9 - 1=8\)

Answer:

Outlier: \(9\); Mean: \(3.5\); Range: \(8\)