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}]:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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]:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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}]:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\sum_{i=1}^n w_ix_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}]:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{w_0n_0 + w_1n_1 +, ..., + w_{n-1}x_{n-1} + w_nx_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: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{64}{2 + 2 + 2 + 2 + 1 + 1 + 1 + 1}}
= Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{64}{12}}
= 5.333