Mathematics/Statistics/Core Measurements

Below are some of the most basic, and regularly used forms of measurements in statistics.
All of these measurements are used to gather information about a list of items.

Note: Most of these are easier to use when the list of items is sorted by some meaningful ordering. For some, such as #Median, they will only work on sorted lists.

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 items.

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}]:
 
${\frac {1}{n}}*\sum _{i=1}^{n}x_{i}$

Direct Notation:

Given a list of n terms, [x_0, x_1, ..., x_{n-1}, x_n]:
 
${\frac {n_{0}+n_{1}+,...,+x_{n-1}+x_{n}}{n}}$

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: ${\frac {47}{8}}$ = 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}]:
 
${\frac {\sum _{i=1}^{n}w_{i}x_{i}}{\sum _{i=1}^{n}w_{i}}}$

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}]:
 
${\frac {w_{0}n_{0}+w_{1}n_{1}+,...,+w_{n-1}x_{n-1}+w_{n}x_{n}}{w_{0}+w_{1}+,...,+w_{n-1},w_{n}}}$

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: ${\frac {64}{2+2+2+2+1+1+1+1}}$ = ${\frac {64}{12}}$ = 5.333

Median

Similarly to mean, the median attempts to get the most "middle" item given a set of items.
However, instead of doing so by literal value, it does this by count of items.

For odd numbered sets, the median is the exact middle number.

Given a list of [1, 2, 3, 5, 7], the median is 3.

For even numbered sets, the median is the middle two numbers.

Given a list of [1, 2, 3, 4, 5, 7], the medians are 3 and 4.

Mode

The mode is the value that occurs most frequently in a set of items.

Given a list of [1, 2, 2, 3, 3, 4, 4, 4], the mode is 4.

Range

Template:ToDo
Range is the difference between the lowest and highest values. Theoretically, it is yet another attempt to get the most "middle" value out of a set of items.

Given a list of [2, 4, 5, 7, 8], the lowest and highest values are 2 and 8.
Thus, the range is 8 - 2 = 6

Note that this can be less than useful when the data set has outliers.
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If we introduce a new value of 100 to our above list, we get [2, 4, 5, 7, 8, 100].
The lowest and highest values are now 2 and 100.
Thus, the range is now 100 - 2 = 98.
This isn't very descriptive of our data anymore.