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 = \(\frac{n+1}{2}\)