1. Standard Deviation
It is a measure that is used to quantify the amount of variation or dispersion of a set of data values.
The "Population Standard Deviation ": [biased] σ = 1 N ∑ i = 1 N ( x i − μ ) 2 \displaystyle\sigma = \sqrt{\frac{1}{N}\sum_{i=1}^{N}(x_i - \mu)^2} σ = N 1 i = 1 ∑ N ( x i − μ ) 2 , μ \mu μ is the average of x.
The "Sample Standard Deviation ": [unbiased]σ = 1 N − 1 ∑ i = 1 N ( x i − μ ) 2 \displaystyle\sigma = \sqrt{\frac{1}{N-1}\sum_{i=1}^{N}(x_i - \mu)^2} σ = N − 1 1 i = 1 ∑ N ( x i − μ ) 2
It's an important change that to
divide by N-1 (instead of N) when calculating a Sample Variance.
Refers:
2. Exponents
Exponents (e e e ) And Logarithms(l n ( ) ln() l n ( ) )
Ask yourself this question: are we talking about inputs (cause of the change) or outputs (the actual change that happened?)
Logarithms reveal the inputs that caused the growth
Exponents find the final result of growth
e e e : is defined to be the rate of growth if we continually compound 100% return on smaller and smaller time periods:e = lim o → ∞ ( 1 + 1 n ) n ≈ 2 . 7 1 8 e = \displaystyle\lim_{o \to \infty}(1+\frac{1}{n})^{n} \approx 2.718 e = o → ∞ lim ( 1 + n 1 ) n ≈ 2 . 7 1 8 ,g r o w t h = e r t ; ( r : r a t e , t : t i m e ) growth = e^{rt}; (r: rate, t:time) g r o w t h = e r t ; ( r : r a t e , t : t i m e )
An Intuitive Guide To Exponential Functions & e
Demystifying the Natural Logarithm (ln)