Agenda

<aside> 📌 Reminders

• Quiz today! </aside>

Sample Variance

<aside> 💡

The sample variance measures the dispersion of a set of sample data points around their sample mean.

$$ s^2 = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2 $$

Where:

• $x_i$ is each data point
• $\bar{x}$ is the sample mean
• $n$ is the sample size </aside>

Key Points:

• Unbiased Estimator: The sample variance provides an unbiased estimate of the population variance.
• Interpretation: Higher variance indicates more spread out data, while lower variance indicates that data points are closer to the mean.

Distribution of the Sample Variance (iid Normals)

If the data $X_1, X_2, \ldots, X_n$ are drawn from a normal distribution $N(\mu, \sigma^2)$, then the sample variance $s^2$ follows a known distribution. Specifically, the scaled sample variance is related to a chi-squared distribution.

The result is:

$$ \frac{(n-1)s^2}{\sigma^2} \sim \chi^2_{n-1}  $$

Mean Squared Error (MSE)

The mean squared error (MSE) is a metric used to measure the average of the squares of the errors, where an error is the difference between an estimator and the true value.

The formula for MSE is:

$$ \text{MSE} = \frac{1}{n} \sum_{i=1}^{n} ( \hat{y}_i - y_i )^2 $$

Where:

• $\hat{y}_i$ is the predicted value
• $y_i$ is the true value
• $n$ is the number of data points

MSE in Estimation