## 3.33 Theorems

1. $\sum_{i=0}^n (x_i-\bar x)=0$

2. $\sum_{i=0}^n f_i(x_i-\bar x)=0$

3. $\sum_{i=1}^n (x_i-\bar x)^2 \lt \sum_{i=1}^n (x_i-a)^2; a\ne\bar x$

4. $\sum_{i=1}^n f_i(x_i-\bar x)^2 \lt \sum_{i=1}^n f_i(x_i-a)^2; a\ne\bar x$

5. (AM~origin & scale)

Prove that AM depends on origin and scale Use frequency as well i.e,

• $\bar x=\frac {\sum_{i=1}^nx_i}{n}$
• $\bar x=\frac {\sum_{i=1}^n f_ix_i}{n}$
6. If the GM of $$n_1$$ va;ues is $$G_1$$, and of $$n_2$$ values is $$G_2$$, show GM of $$n_1+n_2$$ values is $$G=\sqrt{G_1G_2}$$

7. For two non-zero positive numbers, prove $$AM \ge GM \ge HM$$

Let, the numbers be a, b

$$\therefore AM = \frac{a+b}{2}$$

$$GM = \sqrt{ab}$$

$$HM = \frac{2}{\frac 1 a +\frac 1 b}$$
We know, $\begin{eqnarray} & &(a-b)^2\ge 0 \nonumber \\ & \Rightarrow & (a+b)^2-4ab \ge 0 \nonumber \\ & \Rightarrow & (a+b)^2 \ge 4ab \nonumber \\ & \Rightarrow & (a+b) \ge 2 \sqrt{ab} \nonumber \\ & \Rightarrow & \frac{a+b} 2 \ge \sqrt{ab} \nonumber \\ & \Rightarrow & AM \ge GM \nonumber \\ \end{eqnarray}$
Similarly, $\begin{eqnarray} & &(\frac{1}{a}-\frac{1}{b})^2\ge 0 \nonumber \\ & \Rightarrow & (\frac{1}{a}+\frac{1}{b})^2 -4 \cdot \frac 1 a \cdot \frac 1 b\ge 0 \nonumber \\ & \Rightarrow & (\frac{1}{a}+\frac{1}{b})^2\ge \frac 4 {ab} \nonumber \\ & \Rightarrow & (\frac{1}{a}+\frac{1}{b}) \ge \frac 2 {\sqrt{ab}} \nonumber \\ & \Rightarrow & \sqrt{ab}(\frac{1}{a}+\frac{1}{b}) \ge 2 \nonumber \\ & \Rightarrow & \sqrt{ab} \ge \frac{2}{(\frac{1}{a}+\frac{1}{b})} \nonumber \\ & \Rightarrow & GM \ge HM \nonumber \\ \end{eqnarray}$

8. For two non-zero positive numbers, $$AM \times HM =(GM)^2$$

9. Mean and Median of first n natural numbers are $$\frac {n+1} 2$$

10. If $$\bar x_1$$ and $$\bar x_2$$ are means of 2 data sets of sizes $$n_1$$ and $$n_2$$, respectively, the combined mean is $$\bar x_c=\frac{n_1 \bar x_1+n_2 \bar x_2}{n_1+n_2}$$

11. If $$u=x+y, \bar u=\bar x + \bar y$$; if $$n_1=n_2=n$$

Given $$u=x+y$$

$\begin{eqnarray} \bar u &=& \frac{\sum u}{n} \nonumber \\ &=& \frac{\sum (x+y)}{n} \nonumber \\ &=& \frac{\sum x}{n}+ \frac{\sum y}{n} \nonumber \\ &=& \bar x + \bar y \nonumber \\ \end{eqnarray}$

12. For equal number of observations, GM of two variables is equal to the product of their individual means.

13. $$GM=Antilog(\frac{\sum \log x_i}{n})$$ or $$Antilog(\frac{\sum f_i \log x_i}{\sum f_i})$$ 14.If $$y = a + bx, \bar y = a + b \bar x$$

14. If $$z_i=ax_i+by_i, \bar z=a \bar x + b \bar y$$