## 3.10 Properties of AM

• $$\sum (x_i-\bar x)=0$$; can you prove it?
• $$\sum (x_i-\bar x)^2 \le \sum (x_i-a)^2, \space a \ne \bar x$$
• Depends on change of origin and scale?
• $$\bar x + \bar y =\frac{\sum x+\sum y}{n_x+n_y}$$
• Combined mean: $$\bar x_c=\frac{n_1 \bar x_1+n_2 \bar x_2+...+n_k \bar x_k}{n_1+n_2+...+n_k}$$
• $$AM\ge GM \ge HM$$ & $$AM \times HM = (GM)^2$$
• AM of first n natural numbers = $$\frac{n+1}{2}$$