calculating variance\nthe average january surface water temperatures (°c) of lake michigan from 2000 to 2009…

Explanation:

Step1: Recall variance formula

The formula for the variance $s^{2}$ of a sample is $s^{2}=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}$, where $n$ is the number of data - points, $x_{i}$ are the individual data - points, and $\bar{x}$ is the mean. Here, $n = 10$.

Step2: Sum the squared differences

We sum the values in the $(x_{i}-\bar{x})^{2}$ column: $0.575+0.551 + 1.016+1.259+0.676+0.004+0.201+0.771+0.138+0.001$. [ \begin{align*} \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\ &=(0.575 + 0.551)+(1.016+1.259)+(0.676+0.004)+(0.201+0.771)+(0.138+0.001)\ &=1.126+2.275+0.68+0.972+0.139\ &=5.192 \end{align*} ]

Step3: Calculate the variance

Using the formula $s^{2}=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}$, with $n = 10$ and $\sum_{i = 1}^{10}(x_{i}-\bar{x})^{2}=5.192$, we have $s^{2}=\frac{5.192}{10 - 1}=\frac{5.192}{9}\approx0.577$.

Answer:

$0.577$