Population vs Standard: Deviation and Covariance

 Population vs Standard: Deviation and Covariance

Population Deviation vs Standard Deviation

How is the population deviation related to the standard deviation?

Population Deviation (of a data set x_i):

σx = √( Σ(x_i - mean(x)) / n)

where mean(x) is the arithmetic mean of the data set over x_i

Standard Deviation:

sx = √( Σ(x_i - mean(x)) / (n - 1))

n is the size of the data set x_i.  

Suppose we can calculate the standard deviation by multiplying a factor (let's call it ß for the purpose of this example) to the population deviation.   

ß * σx = sx

ß * √( Σ(x_i - mean(x)) / n) = √( Σ(x_i - mean(x)) / (n - 1))

ß  * √( Σ(x_i - mean(x))) /  √n = √( Σ(x_i - mean(x))) / √(n - 1)

ß * √( Σ(x_i - mean(x)))  / √( Σ(x_i - mean(x))) = √n / √(n - 1)

ß  = √n / √(n - 1)

ß  = √(n/(n - 1))

Hence:

sx =  √(n/(n - 1)) * σx

and

σx = sx * √((n-1)/n)

Example:

x = {4, 7, 10, 16, 38}   

n = 5

σx = 12.16552506

sx = 12.16552506 * √(5/4) = 13.60147051

Population Covariance vs Standard Covariance

For the data sets x_i and y_i, population covariance:

cov_σ = 1/n * Σ((x_i - mean(x)) * (y_i - mean(y)))

And the sample covariance:

cov_s = 1/(n - 1) * Σ((x_i - mean(x)) * (y_i - mean(y)))

We will use the similar tactic above to find a relationship between population covariance and sample covariance:

ß * cov_σ = cov_s

ß * 1/n * Σ((x_i - mean(x)) * (y_i - mean(y))) = 

1/(n - 1) * Σ((x_i - mean(x)) * (y_i - mean(y)))

ß * Σ((x_i - mean(x)) * (y_i - mean(y))) / Σ((x_i - mean(x)) * (y_i - mean(y))) =

n/(n - 1)

ß = n/(n - 1)

Hence:

cov_ s = n/(n - 1) * cov_σ 

and

cov_σ = (n - 1)/n * cov_s

Example:

x = {4, 5, 6, 8}

y = {-2, -1, 2, 0}

n = 4

mean(x) = 5.75

mean(y) = -0.25

cov_σ = 1.1875

cov_s = 1.1875 * 4/3 = 1.5833333333

Hope you find this helpful,

Eddie 

All original content copyright, © 2011-2022.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author. 

Arcsine and Arccosine in terms of Arctangent

Arcsine and Arccosine in terms of Arctangent

Some Motivation

I recall reading about the Sinclair Scientific Programmable, a vintage scientific calculator that was introduced in 1975.  This calculator just had three trigonometric functions:  sine, cosine, and arctangent.  It is the aim of this blog to fill in the blanks.  

You can find more information here:  

http://www.vintagecalculators.com/html/scientific_prog_.html

In this blog entry, angles will have radian measure.  

For tangent, it's pretty easy use of the trig identity:

tan x = sin x ÷ cos x

Determining Arcsine 

Imagine the right triangle shown below:

Then:

sin Θ = x

Θ = arcsin x

By the Pythagorean Theorem:

t^2 + x^2 = 1

t^2 = 1 - x^2

t = √(1 - x^2)

tan Θ = x ÷ t

tan Θ = x ÷ √(1 - x^2)

Θ = arctan(x ÷ √(1 - x^2))

Then:

arcsin x = arctan(x ÷ √(1 - x^2))

Determining Arccosine

Most calculators for the arccos function have the range [0, π].  To accomplish this, we are going to use the identity

cos Θ = sin(π÷2 - Θ).

Let w = cos Θ

Then:  

w = cos Θ

w = sin(π÷2 - Θ)

arcsin w = π÷2 - Θ

Θ = π÷2 - arcsin w

Θ = π÷2 - arctan(w ÷ √(1 - w^2))

For any angle w:

arccos w = π÷2 - arctan(w ÷ √(1 - w^2))

Eddie 

All original content copyright, © 2011-2021.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.