1. **Problem Statement:** Calculate the population mean and the sample mean separately given a data set. 2. **Formulas:** - Population mean formula: $$\mu = \frac{\sum_{i=1}^N x_i}{N}$$ where $N$ is the total number of data points in the population. - Sample mean formula: $$\bar{x} = \frac{\sum_{i=1}^n x_i}{n}$$ where $n$ is the number of data points in the sample. 3. **Explanation:** - The population mean ($\mu$) is the average of all data points in the entire population. - The sample mean ($\bar{x}$) is the average of data points in a subset (sample) of the population. 4. **Steps to calculate:** - Sum all the values in the population data set and divide by $N$ to get the population mean. - Sum all the values in the sample data set and divide by $n$ to get the sample mean. 5. **Example:** - Suppose population data: $\{2, 4, 6, 8, 10\}$, then $N=5$. - Population mean: $$\mu = \frac{2+4+6+8+10}{5} = \frac{30}{5} = 6$$ - Suppose sample data: $\{4, 6, 8\}$, then $n=3$. - Sample mean: $$\bar{x} = \frac{4+6+8}{3} = \frac{18}{3} = 6$$ Thus, both population mean and sample mean are 6 in this example.