Question

compare the mean and median of the following data. is the data symmetrical? use the image to answer the question

Explanation:

Step1: List the data points

The data points are \(80,80,81,81,83,84,84,84,84,85,85,85\).

Step2: Calculate the mean

The formula for the mean \(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}\). Here \(n = 12\), \(\sum_{i=1}^{12}x_{i}=80\times2 + 81\times2+83+84\times4+85\times3=960\). So \(\bar{x}=\frac{960}{12}=84\).

Step3: Calculate the median

Since \(n = 12\) (even), the median \(M=\frac{x_{\frac{n}{2}}+x_{\frac{n}{2}+1}}{2}\). \(x_{6}=84\), \(x_{7}=84\), so \(M=\frac{84 + 84}{2}=84\).

Step4: Check for symmetry

A data - set is symmetric if the mean and median are equal. Here, mean \(= 84\) and median \(= 84\).

Answer:

The mean and median of the data are both \(84\), so the data is symmetric.