Statistics

Mean, Median, Mode

Mean ( $\bar{x}$ ) - Average of all data
Median - Middle number in the data
Mode - Most frequent number

Variance ( $σ^{2}$ or $s^{2}$ )

Description

Measure relative amount of 'scattering' of data.

Formula

σ^{2} = \frac{\sum (x_{i} - \bar{x})^{2}}{n}

Explanation

$σ^{2}$ - Variance is (standard deviation) $^{2}$
$\sum (x_{i} - \bar{x})^{2}$
- $x_{i}$ - Loop through all elements, element[i]
- $\bar{x}$ - Mean
- $(x_{1} - \bar{x})^{2} + (x_{2} - \bar{x})^{2} + (x_{3} - \bar{x})^{2} + \dots (x_{n} - \bar{x})^{2}$
Divided by number of elements

Population variance (

σ^{2}

) vs Sample variance (

s^{2}

)

There are 2 types of variance - Population and sample

Population: Use all data points to calculate it
Sample: Have an unbiased pick of a few data points and calculate variance using that

So, population variance will always be more accurate as it uses all the data points.

Why use sample then?
Might not have all data points so use sample instead.
For example, to test the battery of a new phone model. It's impossible to test every phone (as millions exists).
So test 50 randomly selected phones. As the sample count increases, it'll get closer to the true value.

Formula for sample deviation
$s^{2} = \frac{\sum (x_{i} - μ)^{2}}{n - 1}$
Difference:

For the symbol, use $s^{2}$ instead of $σ^{2}$
Use $μ$ instead of $\bar{x}$
Divide by $n - 1$ instead of $n$ . Wikipedia, Bessel's correction

Standard Deviation

Standard deviation ( $σ$ ) = $\sqrt{Variance}$ = $\sqrt{\frac{\sum (x_{i} - \bar{x})^{2}}{n}}$

Uses

Finding probability given mean and standard deviation

NOTE: This is only for normal distribution (Bell curve) (I think)

Common values, from the video (Timestamp: 0:33)

--- Math terms
- $P(\bar{x}-\ \sigma < \text{value}<\bar{x}+\ \sigma)$ = 68%
- $P(\bar{x}-2\sigma < \text{value}<\bar{x}+2\sigma)$ = 95%
- $P(\bar{x}-3\sigma < \text{value}<\bar{x}+3\sigma)$ = 99.75%
--- Simple terms
 Probability of a value between $\pm c\times \sigma$ from the mean. 
 When $c$ = ...
- $c=1$, $P=68\%$
- $c=2$, $P=95\%$
- $c=3$, $P=99.75\%$