Understanding Variance and Its Properties

Variance is a statistical measure that quantifies the dispersion or spread of a set of data points around their mean. A low variance indicates that data points are generally close to the mean, while a high variance indicates that data points are widely spread out from the mean.

How Change of Origin Affects Variance

Let's consider a set of data points $x_1, x_2, \dots, x_n$ with mean $\mu_x$ and variance $\sigma_x^2$. The variance is calculated using the formula:

$$\sigma_x^2 = \frac{1}{n} \sum_{i=1}^n (x_i - \mu_x)^2$$

Now, suppose we change the origin by adding a constant value $c$ to each data point. The new data points are $y_i = x_i + c$. The mean of the new data set will be $\mu_y = \mu_x + c$.

Let's calculate the variance of the new data set, $\sigma_y^2$:

$$\sigma_y^2 = \frac{1}{n} \sum_{i=1}^n (y_i - \mu_y)^2$$

Substitute $y_i = x_i + c$ and $\mu_y = \mu_x + c$ into the formula:

$$\sigma_y^2 = \frac{1}{n} \sum_{i=1}^n ((x_i + c) - (\mu_x + c))^2$$

Simplifying the term inside the parenthesis:

$$ (x_i + c) - (\mu_x + c) = x_i + c - \mu_x - c = x_i - \mu_x $$

So, the variance becomes:

$$\sigma_y^2 = \frac{1}{n} \sum_{i=1}^n (x_i - \mu_x)^2$$

This is exactly the formula for the original variance, $\sigma_x^2$. Thus, $\sigma_y^2 = \sigma_x^2$. This shows that adding or subtracting a constant from each data point (changing the origin) does not change the variance.

How Change of Scale Affects Variance

Now, let's see what happens when we change the scale by multiplying each data point by a constant value $a$ ($a \neq 0$). The new data points are $z_i = a \cdot x_i$. The mean of the new data set will be $\mu_z = a \cdot \mu_x$.

Let's calculate the variance of the new data set, $\sigma_z^2$:

$$\sigma_z^2 = \frac{1}{n} \sum_{i=1}^n (z_i - \mu_z)^2$$

Substitute $z_i = a \cdot x_i$ and $\mu_z = a \cdot \mu_x$ into the formula:

$$\sigma_z^2 = \frac{1}{n} \sum_{i=1}^n (a \cdot x_i - a \cdot \mu_x)^2$$

Factor out $a$ from the term inside the parenthesis:

$$ (a \cdot x_i - a \cdot \mu_x) = a(x_i - \mu_x) $$

So, the variance becomes:

$$\sigma_z^2 = \frac{1}{n} \sum_{i=1}^n (a(x_i - \mu_x))^2$$

$$ \sigma_z^2 = \frac{1}{n} \sum_{i=1}^n a^2 (x_i - \mu_x)^2 $$

Factor out $a^2$ from the summation (since $a$ is a constant):

$$ \sigma_z^2 = a^2 \frac{1}{n} \sum_{i=1}^n (x_i - \mu_x)^2 $$

The summation part is the original variance, $\sigma_x^2$. Thus, $\sigma_z^2 = a^2 \sigma_x^2$. This shows that multiplying each data point by a constant $a$ multiplies the variance by $a^2$. Therefore, changing the scale *does* affect the variance.

Summary of Effects

We can summarize the effect of changing origin and scale on variance:

Based on this analysis, variance is independent of the change of origin only. It is dependent on the change of scale.