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The sample mean is a fundamental concept in statistics that represents the average value of a dataset. Calculating it involves two main steps. First, you sum up all the data points. For the observations 2781, 2900, 3013, 2856, and 2888 given in the exercise, you add them together to get a sum of 14438.

Next, you divide this total by the number of observations, which in this case is 5. This division gives you the sample mean. So, the calculation is \( \bar{x} = \frac{14438}{5} = 2887.6 \).

#### Why It Matters
The sample mean provides a central value for the dataset, offering a quick preview of the data's general magnitude. It is a simple yet powerful indicator of the overall behavior of the data and plays a crucial role in other statistical calculations and analyses.

The sample median is essential for understanding the position of the data. To find the median, you first need to organize the data points in ascending order, as done in the exercise: 2781, 2856, 2888, 2900, and 3013. Once sorted, you identify the middle value of the dataset.

Since there are 5 values, the median is the third one, which is 2888.

#### Importance of the Median
The median is particularly useful because it is not affected by extreme values. While the mean can shift significantly with very high or low values, the median stays stable, providing a more robust center for the data in situations with outliers.

It's an excellent metric for understanding where the "middle" of the data truly lies, especially in skewed distributions.

Sample variance measures the spread or dispersion of the data points in a dataset. To find the sample variance, begin by using the formula: \[ s^2 = \frac{\sum{(x_i - \bar{x})^2}}{n-1} \] where \(x_i\) are data points, \(\bar{x}\) is the sample mean, and \(n\) is the number of observations.

In this exercise, simplifying the calculation by temporarily subtracting 2800 from each observation helps manage large numbers. Adjustments included values such as -19 for 2781 and 100 for 2900. After finding these adjusted values, compute their squares and sum them up for 66610.

Understanding variance helps to see how much the data points deviate from that central value, which is the sample mean.

#### Interpreting Variance
A higher sample variance indicates greater variability within the data, while a lower variance signals that the data points are more clustered around the mean. Variance forms the basis for standard deviation, a key descriptive statistic, making it crucial for data interpretation overall.