Original list

First, you measure something, and you make a list, adding entries at the end as you measure it.

What you then get is the Original List, for example:

y⃗:=(4,7,5,7,4,9,1,4)

Let n be the number of measurements, then:

n=8

Frequency

You order the list (for convenience), find common values and count how often each value appears:

Measured y and frequency, grouped by value.
row#ValueFrequency
H
Relative Frequency
h:=÷{H}{n}
Frequency graph
1111/8#
2433/8###
3511/8#
4722/8##
5911/8#

Average

y⃗̄=5.125

Variance

V(y⃗):=÷{1}{n}⋅\sum\limits_{i=1}^{n} (y_i-y⃗̄)²
V(y⃗)=÷{1}{n}⋅(\sum\limits_{i=1}^{n} y_i²)-y⃗̄²
V(y⃗)\approx 5.36

Standard Deviation

s:=√V
s=2.32

Average Standard Deviation of the ENTIRE set of items where you drew some items from

σ:=√{V⋅÷{n}{n-1}}
σ\approx 6.13

Probabilities

A probability [that an event occurs] is a number between 0 and 1.

Probability distribution

A probability distribution gives the probability of a random variable being (exactly|in_the_range_of) some value.