- Statistics
- : Also See
- : Online Statistics and Probability Calculators
- : Notes and definitions on statistics.
- : Definition of a (finite) sample
- : Statistics on one sample
- : Statistics on Two samples
- Notes on MATHS Notation
- Notes on the Underlying Logic of MATHS
- Glossary

- STATISTICS::=following,

Net- I:: Finite_Sets=given.
I standards for index. Typically it is a range 1..n where n is the sample size. However
in some languages/cultures I could be 0..n-1. In fact there is no reason to limit I to a range.
Any finite set of indices will work.
- Sample::=I>->Real. Each index item has a measured value.
Example. For example the list

- (1,2,3)
has I = 1..3 and size=3.
- size::Real= |I|.
n:=size. Local shorthand.
- For p:Real, p::Sample = I +> p. A coercion that converts a single value into a sample with the same value for each index.
This turns out to be useful when we subtract the mean of
a sample (a number) from every item in the sample.
- For x, y::Sample I will use x and y as the names of samples of data.
## Statistics on one sample

- mean(x)::= +x/n. In STANDARD (+) is a serial operator that adds up all the items in its arguments.
- +(1,2,3) = (1+2+3) = 6.
- mean((1, 2, 3)) = +(1, 2, 3)/3 = 6/3 = 2.
min, max, range, mode, histogram are to be done. [click here stats1 if you can fill this hole]

- ss(x)::=+(x*x). Sum of squares.
- ss((1, 2, 3)) = +(1*1, 2*2 , 3*3) = +(1,4,9) = 14.
- ms(x)::= ss(x - mean(x) )/n. Mean squares about mean.
- (-1)|-ms(x) =( ss(x) - (+x)*mean(x))/n. Better for small hand calculations.
- ms((1,2,3)) = (14 - 6*2)/3 = 2/3.
- var(x)::= n * ms(x)/(n-1). Sample variance -- rescale to allow for estimating the mean.
- root_mean_square(x)::=sqrt(ms(x)).
- rms::= root_mean_square.
- standard_deviation(x)::=sqrt( var (x) ).
- sd(x)::=standard_deviation(x).
## Statistics on Two samples

- SP(x,y)::=+((x-mean(x))*(y-mean(y))).
- (-1)|-ss(x) = SP(x,x).
- MS(x,y)::=SP(x,y)/n.
- r(x,y)::= MS(x,y)/( sd(x)*sd(y)). Correlation coefficient -- Pearson.
More... [click here stats2 if you can fill this hole]

## Definition of a (finite) sample

(End of Net STATISTICS)

- I:: Finite_Sets=given.
I standards for index. Typically it is a range 1..n where n is the sample size. However
in some languages/cultures I could be 0..n-1. In fact there is no reason to limit I to a range.
Any finite set of indices will work.
- STANDARD::= See http://www.csci.csusb.edu/dick/maths/math_11_STANDARD.html
# Glossary

- above::reason="I'm too lazy to work out which of the above statements I need here", often the last 3 or 4 statements. The previous and previous but one statments are shown as (-1) and (-2).
- given::reason="I've been told that...", used to describe a problem.
- given::variable="I'll be given a value or object like this...", used to describe a problem.
- goal::theorem="The result I'm trying to prove right now".
- goal::variable="The value or object I'm trying to find or construct".
- let::reason="For the sake of argument let...", introduces a temporary hypothesis that survives until the end of the surrounding "Let...Close.Let" block or Case.
- hyp::reason="I assumed this in my last Let/Case/Po/...".
- QED::conclusion="Quite Easily Done" or "Quod Erat Demonstrandum", indicates that you have proved what you wanted to prove.
- QEF::conclusion="Quite Easily Faked", -- indicate that you have proved that the object you constructed fitted the goal you were given.
- RAA::conclusion="Reducto Ad Absurdum". This allows you to discard the last
assumption (let) that you introduced.

. . . . . . . . . ( end of section Statistics in MATHS) <<Contents | End>>

Proofs follow a natural deduction style that start with assumptions ("Let") and continue to a consequence ("Close Let") and then discard the assumptions and deduce a conclusion. Look here [ Block Structure in logic_25_Proofs ] for more on the structure and rules.

The notation also allows you to create a new network of variables and constraints. A "Net" has a number of variables (including none) and a number of properties (including none) that connect variables. You can give them a name and then reuse them. The schema, formal system, or an elementary piece of documentation starts with "Net" and finishes "End of Net". For more, see [ notn_13_Docn_Syntax.html ] for these ways of defining and reusing pieces of logic and algebra in your documents. A quick example: a circle might be described by Net{radius:Positive Real, center:Point, area:=π*radius^2, ...}.

For a complete listing of pages in this part of my site by topic see [ home.html ]

For a more rigorous description of the standard notations see