Statistics

Statistics

• Mean ($\bar{x}$):
  

  $$\bar{x}=\frac{x_1+x_2+x_3+\dots +x_n}{n}=\frac{1}{n}\sum _{i=1}^n{x}_i$$

• Deviation from the Mean:
  

  $$x_i-\bar{x}$$

• Deviation of :
  

  $$Dev(x_i)=x_i-\bar{x}$$

Median / Mode

Median (Even $n$):

$Median=\frac{18+22}{2}=20$

Mode: The number with the highest occurrence.

Example:

15, 19, 16, 15, 13, 15, 12, 10, 5 = 15

Variance

• Deviation of $x_i$:
  $Dev(x_i)=x_i-\bar{x}$

• Sum of squares of Deviation (SS):
  $SS(x)={\sum }_{i=1}^n(x_i-\bar{x}{)}^2$

• Population Variance:
  $Var(x)=\frac{1}{n}{\sum }_{i=1}^n(x_i-\bar{x}{)}^2$

• Sample Variance:
  $Var(x)=\frac{SS(x)}{n-1}$

Simplified Variance Formulas

Expanded variance formula:

$SS(x)={\sum }_{i=1}^n(x_i-\bar{x}{)}^2$

$SS(x)={\sum }_{i=1}^n\{{x_i}^2+{\bar{x}}^2-2x_i\bar{x}\}$

$SS(x)={\sum }_{i=1}^n{x_i}^2+{\bar{x}}^2{\sum }_{i=1}^n.1-2\bar{x}{\sum }_{i=1}^n{x}_i$

$SS(x)={\sum }_{i=1}^n{x_i}^2+n{\bar{x}}^2-2\bar{x}{\sum }_{i=1}^n{x}_i$

$Var(x)=\frac{SS(x)}{n}$

$Var(x)=\frac{1}{n}\{{\sum }_{i=1}^n{x_i}^2+n{\bar{x}}^2-2\bar{x}{\sum }_{i=1}^n{x}_i\}$

$Var(x)=\frac{1}{n}{\sum }_{i=1}^n{x_i}^2+{\bar{x}}^2-2\bar{x}{\sum }_{i=1}^n\frac{x_i}{n}$

${\sum }_{i=1}^n\frac{x_i}{n}=\frac{1}{n}{\sum }_{i=1}^n{x}_i=\bar{x}$

$Var(x)=\frac{1}{n}{\sum }_{i=1}^n{x_i}^2+{\bar{x}}^2-2\bar{x}.\bar{x}$

$Var(x)=\frac{1}{n}{\sum }_{i=1}^n{x_i}^2+{\bar{x}}^2-2{\bar{x}}^2$

$Var(x)=\frac{1}{n}{\sum }_{i=1}^n{x_i}^2+{\bar{x}}^2(1-2)$