Mathematics/Statistics/Core Measurements

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Below are some of the most basic, and regularly used forms of measurements in statistics.

Mean/Average

The "mean" and "average" are effectively two different words for the same thing.

Effectively, this attempts to get the most "middle" value given a set of value.

Standard (Unweighted) Mean

Unweighted Mean is what most people think of when someone says "mean" or "average.
Effectively, take all values in a list and add them together. Then divide this sum by the total count of original values.

Scientific Notation:

Given a list of n terms, [x_0, x_1, ..., x_{n-1}, x_{n}]:
 

Direct Notation:

Given a list of n terms, [x_0, x_1, ..., x_{n-1}, x_n]:
 

Example:

Given a list of [1, 4, 7, 5, 9, 9, 2, 10].
 
Step 1) Sum: 1 + 4 + 7 + 5 + 9 + 9 + 2 + 10 = 47
Step 2) Divide sum by count:  = 5.875

Weighted Mean

Weighted Mean is similar to above, except that each value has a "weight" associated with it.

Scientific Notation:

Given a list of n terms, [x_0, x_1, ..., x_{n-1}, x_{n}], with associated weights [w_0, w_1, ..., w_{n-1}, w_{n}]:
 

Direct Notation:

Given a list of n terms, [x_0, x_1, ..., x_{n-1}, x_n], with associated weights [w_0, w_1, ..., w_{n-1}, w_{n}]:
 

Example:

Given a list of [1, 4, 7, 5, 9, 9, 2, 10], the first 4 values are twice as important as the last 4.
 
Step 1) Sum: 2*1 + 2*4 + 2*7 + 2*5 + 1*9 + 1*9 + 1*2 + 1*10 = 64
Step 2) Divide sum by weights:  =  = 5.333