Mathematics/Statistics/Core Measurements - Revision history

Brodriguez: Brodriguez moved page Statistics/Core Measurements to Mathematics/Statistics/Core Measurements

2020-10-25T17:20:51Z

Brodriguez moved page Statistics/Core Measurements to Mathematics/Statistics/Core Measurements

← Older revision

Revision as of 17:20, 25 October 2020

Brodriguez: Add reference link

2020-05-18T12:40:56Z

Add reference link

← Older revision		Revision as of 12:40, 18 May 2020
Line 157:		Line 157:
	* <math>\bar{x}</math> is the [[#Mean\|mean]] of our x values.		* <math>\bar{x}</math> is the [[#Mean\|mean]] of our x values.
	* <math>\bar{y}</math> is the [[#Mean\|mean]] of our y values.		* <math>\bar{y}</math> is the [[#Mean\|mean]] of our y values.


			For further explanation, see [https://youtu.be/0nZT9fqr2MU this youtube video].

Brodriguez: Create covariance section

2020-05-18T06:44:14Z

Create covariance section

← Older revision		Revision as of 06:44, 18 May 2020
Line 139:		Line 139:
	At this point, all that's missing is the square root.		At this point, all that's missing is the square root.
	<math>\sqrt{10.109375}</math> = 3.179524335		<math>\sqrt{10.109375}</math> = 3.179524335


			== Covariance ==
			While [[#Variance\|variance]] indicates how values of a single variable tend to behave, '''covariance''' indicates how two variables behave in relation to each other.

			A value of 0 indicates that there is nearly no correlation between variables.

			A positive value indicates a positive correlation. That is, as x increases, so does y. And as y increases, so does x. Larger values indicate a stronger correlation.

			A negative value indicates a negative correlation. That is, as x increases, y decreases. And as y increases, x decreases. Larger values indicate a stronger correlation.

			{{ Note \| Covariance values will change depending on the scale of initial variables. }}

			Scientific Notation:
			<math>\frac{1}{n - 1}\sum_{i=1}^n (x_i-\bar{x})(y_i-\bar{y})</math>
			Where
			* <math>\bar{x}</math> is the [[#Mean\|mean]] of our x values.
			* <math>\bar{y}</math> is the [[#Mean\|mean]] of our y values.

Brodriguez: Brodriguez moved page Statistics - Core Measurements to Statistics/Core Measurements: Clean url with subpages

2020-05-15T08:51:58Z

Brodriguez moved page Statistics - Core Measurements to Statistics/Core Measurements: Clean url with subpages

← Older revision

Revision as of 08:51, 15 May 2020

Brodriguez: Formatting

2020-05-12T12:35:23Z

Formatting

← Older revision		Revision as of 12:35, 12 May 2020
Line 65:		Line 65:
	Given a list of [1, 2, 3, 4, 5, 7], the medians are 3 and 4.		Given a list of [1, 2, 3, 4, 5, 7], the medians are 3 and 4.
	We can average this together to get a single value of 3.5		We can average this together to get a single value of 3.5


	== Mode ==		== Mode ==

Brodriguez: Expand medians

2020-05-12T12:35:03Z

Expand medians

← Older revision		Revision as of 12:35, 12 May 2020
Line 61:		Line 61:
	Given a list of [1, 2, 3, 4, 5, 7], the medians are 3 and 4.		Given a list of [1, 2, 3, 4, 5, 7], the medians are 3 and 4.

			For some applications, you always want a single number as the median, including for even numbered sets.<br>
			In these instances, average the two median numbers together.
			Given a list of [1, 2, 3, 4, 5, 7], the medians are 3 and 4.
			We can average this together to get a single value of 3.5

	== Mode ==		== Mode ==

Brodriguez: Correct standard deviation representation

2020-05-11T14:30:53Z

Correct standard deviation representation

← Older revision		Revision as of 14:30, 11 May 2020
Line 117:		Line 117:

	The end result is a value that indicates, generally speaking, how far each item in our list is from the mean.		The end result is a value that indicates, generally speaking, how far each item in our list is from the mean.

			{{ Tip \| Often, this is represented as <math>\sigma</math>, which is the lowercase character for "sigma".}}


	Scientific Notation:		Scientific Notation:
	<math>\sigma^2 = \sqrt{\frac{\sum_{i=1}^n (x_i - \bar{x})^2}{n}}</math>		<math>\sigma = \sqrt{\frac{\sum_{i=1}^n (x_i - \bar{x})^2}{n}}</math>

	Direct Notation:		Direct Notation:
	Given a list of n terms, [x_0, x_1, ..., x_{n-1}, x_n] with a mean of <math>\bar{x}</math>:		Given a list of n terms, [x_0, x_1, ..., x_{n-1}, x_n] with a mean of <math>\bar{x}</math>:

	<math>\sigma^2 = \sqrt{\frac{(n_0 - \bar{x})^2 + (n_1 - \bar{x})^2 +, ..., + (x_{n-1} - \bar{x})^2 + (x_n - \bar{x})^2}{n}}</math>		<math>\sigma = \sqrt{\frac{(n_0 - \bar{x})^2 + (n_1 - \bar{x})^2 +, ..., + (x_{n-1} - \bar{x})^2 + (x_n - \bar{x})^2}{n}}</math>

	Example:		Example:

Brodriguez: Add standard deviation section

2020-05-11T14:29:13Z

Add standard deviation section

← Older revision		Revision as of 14:29, 11 May 2020
Line 108:		Line 108:
	Step 2) Sum result: 80.875		Step 2) Sum result: 80.875
	Step 3) Divide sum by count: <math>\frac{80.875}{8}</math> = 10.109375		Step 3) Divide sum by count: <math>\frac{80.875}{8}</math> = 10.109375


			== Standard Deviation ==
			The Standard Deviation is the same as the Variance, but taken a step further.

			Note the the variance is meant to determine "how far values in a set stray from our mean". But in doing so, we square our values in order to counteract inevitable negative values.<br>
			Standard Deviation is meant to correct this, by taking our final value and square rooting it.

			The end result is a value that indicates, generally speaking, how far each item in our list is from the mean.

			Scientific Notation:
			<math>\sigma^2 = \sqrt{\frac{\sum_{i=1}^n (x_i - \bar{x})^2}{n}}</math>

			Direct Notation:
			Given a list of n terms, [x_0, x_1, ..., x_{n-1}, x_n] with a mean of <math>\bar{x}</math>:

			<math>\sigma^2 = \sqrt{\frac{(n_0 - \bar{x})^2 + (n_1 - \bar{x})^2 +, ..., + (x_{n-1} - \bar{x})^2 + (x_n - \bar{x})^2}{n}}</math>

			Example:
			Using our above [[#Mean/Average]] example, we have a list of [1, 4, 7, 5, 9, 9, 2, 10] with a variation of 10.109375.

			At this point, all that's missing is the square root.
			<math>\sqrt{10.109375}</math> = 3.179524335

Brodriguez: Create variance section

2020-05-11T14:20:54Z

Create variance section

← Older revision		Revision as of 14:20, 11 May 2020
Line 18:		Line 18:
	Given a list of n terms, [x_0, x_1, ..., x_{n-1}, x_{n}]:		Given a list of n terms, [x_0, x_1, ..., x_{n-1}, x_{n}]:

	<math>\frac{1}{n} * \sum_{i=1}^n x_i</math>		<math>\bar{x} = \frac{1}{n} * \sum_{i=1}^n x_i</math>

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

	<math>\frac{n_0 + n_1 +, ..., + x_{n-1} + x_n}{n}</math>		<math>\bar{x} = \frac{n_0 + n_1 +, ..., + x_{n-1} + x_n}{n}</math>

	Example:		Example:
Line 37:		Line 37:
	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}]:		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}]:

	<math>\frac{\sum_{i=1}^n w_ix_i}{\sum_{i=1}^n w_i}</math>		<math>\bar{x} = \frac{\sum_{i=1}^n w_ix_i}{\sum_{i=1}^n w_i}</math>

	Direct Notation:		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}]:		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}]:

	<math>\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}</math>		<math>\bar{x} = \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}</math>

	Example:		Example:
Line 76:		Line 76:
	Thus, the range is 8 - 2 = 6		Thus, the range is 8 - 2 = 6

	Note that this can be less than useful when the data set has outliers.<br>		Note that this can be less than useful when the data set has outliers. This is because it only uses extreme endpoints.<br>
	{{ ToDo \| Link to outlier description. }}		{{ ToDo \| Link to outlier description. }}
	If we introduce a new value of 100 to our above list, we get [2, 4, 5, 7, 8, 100].		If we introduce a new value of 100 to our above list, we get [2, 4, 5, 7, 8, 100].
Line 82:		Line 82:
	Thus, the range is now 100 - 2 = 98.		Thus, the range is now 100 - 2 = 98.
	This isn't very descriptive of our data anymore.		This isn't very descriptive of our data anymore.


			== Variance ==
			This is an alternative to the Range, which also attempts to measure how much values tend to spread apart in a given set.<br>
			Unlike Range, this approach handles outliers much better, so the result is generally more representative of your data.

			This is one of the more regularly used values in statistics, and fairly important to understand.<br>
			For a full, detailed explanation, see [[https://www.youtube.com/watch?v=sOb9b_AtwDg this youtube video]].

			{{ Tip \| Often, this is represented as <math>\sigma^2</math>, which is the lowercase character for "sigma".}}


			Scientific Notation:
			<math>\sigma^2 = \frac{\sum_{i=1}^n (x_i - \bar{x})^2}{n}</math>

			Direct Notation:
			Given a list of n terms, [x_0, x_1, ..., x_{n-1}, x_n] with a mean of <math>\bar{x}</math>:

			<math>\sigma^2 = \frac{(n_0 - \bar{x})^2 + (n_1 - \bar{x})^2 +, ..., + (x_{n-1} - \bar{x})^2 + (x_n - \bar{x})^2}{n}</math>

			Example:
			Using our above [[#Mean/Average]] example, we have a list of [1, 4, 7, 5, 9, 9, 2, 10] with a mean of 5.875.

			Step 1) Sum with mean: <math>(1 - 5.875)^2 + (4 - 5.875)^2 + (7 - 5.875)^2 + (5 - 5.875)^2 + (9 - 5.875)^2 + (9 - 5.875)^2 + (2 - 5.875)^2 + (10 - 5.875)^2</math>
			Step 2) Sum result: 80.875
			Step 3) Divide sum by count: <math>\frac{80.875}{8}</math> = 10.109375

Brodriguez: Add mean notation

2020-05-11T13:43:05Z

Add mean notation

← Older revision		Revision as of 13:43, 11 May 2020
Line 6:		Line 6:

	== Mean/Average ==		== Mean/Average ==
	The "mean" and "average" are effectively two different words for the same thing.		The "mean" and "average" are effectively two different words for the same thing.<br>
			Effectively, this attempts to get the most "middle" value given a set of items.

	~~Effectively~~, this ~~attempts to get the most~~ "~~middle~~" ~~value given a set of items~~.~~<br>~~		{{ Tip \| Often, this is represented as <math>\bar{x}</math>, pronounced "x bar".}}

	=== Standard (Unweighted) Mean ===		=== Standard (Unweighted) Mean ===
Line 69:		Line 70:
	{{ ToDo \| Differentiate between statistical range and mathematical function range. }}<br>		{{ ToDo \| Differentiate between statistical range and mathematical function range. }}<br>
	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.		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.

			In other words, this attempts to measure how much values tend to spread apart in a given set.

	Given a list of [2, 4, 5, 7, 8], the lowest and highest values are 2 and 8.		Given a list of [2, 4, 5, 7, 8], the lowest and highest values are 2 and 8.