Question

the average january surface water temperatures (°c) of lake michigan from 2000 to 2009 were 5.07, 3.57, 5.32, 3.19, 3.49, 4.25, 4.76, 5.19, 3.94, and 4.34. the mean value of these temperatures is 4.312. what is the variance of this data set?

Explanation:

Step1: Recall the formula for variance

The formula for variance \(s^{2}=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n}\), where \(n\) is the number of data points, \(x_{i}\) are the data values, and \(\bar{x}\) is the mean.

Step2: Sum up the \((x_{i}-\bar{x})^{2}\) values

From the table:
\(\sum_{i = 1}^{10}(x_{i}-\bar{x})^{2}=0.575 + 0.551+1.016 + 1.259+0.676+0.004+0.201+0.771+0.138+0.001\)
\(\sum_{i = 1}^{10}(x_{i}-\bar{x})^{2}=5.192\)

Step3: Calculate the variance

Here \(n = 10\). Using the formula \(s^{2}=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n}\), we substitute \(\sum_{i = 1}^{10}(x_{i}-\bar{x})^{2}=5.192\) and \(n = 10\)
\(s^{2}=\frac{5.192}{10}=0.5192\)

Answer:

\(0.5192\)