changing a variable
Used when we want to examen a new dataset that is based on a previous dataset
Can be used to easily determine the average (by subtracting a value somewhat in the middle of the datapoints)
→ positive and negative values generally cancel out
original data:
$\bar{x}$ and variance $\sigma^2(x)$
# Mean
$$\bar{y}=\frac1n \sum_{i=1}^nax_i+\sum_{i=1}^nb=$$
$$=a\cdot \sum_{i=1}^nx_i+b=$$
$$\bar{y}=a\cdot \bar{x} +b$$
# Variance
$$\sigma^2=\frac{1}{n}\sum_{i=1}^{n}(y_i-\bar{y})^2=$$
$$=\frac{1}{n}\sum_{i=1}^{n}[(ax_i+b)-(a\bar{x}+b)]^2=\frac{1}{n}\sum_{i=1}^{n}[a(x_i-\bar{x})]^2$$
$$=\frac{1}{n}\sum_{i=1}^{n}a^2\cdot (x_i-\bar{x})^2=$$
$$\sigma^2(y)=a^2\cdot \sigma^2 (x)$$