Normal distribution - Revision history

Dirkjan at 20:01, 27 August 2020

2020-08-27T20:01:12Z

Dirkjan: Replaced content with "Normal distribution Category:mbi Category:Decision Category:M7 Category:Methods Category:Quantitative Category:Term"

2020-08-27T19:20:27Z

Replaced content with "Normal distribution Category:mbi Category:Decision Category:M7 Category:Methods Category:Quantitative Category:Term"

Dirkjan: Created page with "The normal distribution is the most widely used statistical distribution and is defined by its mean $\mu$ and standard variance $\sigma$ $P(x)<..."$

2020-08-25T08:05:17Z

Created page with "The normal distribution is the most widely used statistical distribution and is defined by its mean <math> \mu </math> and standard variance <math> \sigma </math> <math>P(x)<..."

New page

The normal distribution is the most widely used statistical distribution and is defined by its mean <math> \mu </math> and standard variance <math> \sigma </math>

<math>P(x)</math> = <math>\frac{1}</math>{{\sigma \sqrt {2\pi } }}e^{{{ - \left( {x - \mu } \right)^2 } \mathord{\left/ {\vphantom {{ - \left( {x - \mu } \right)^2 } {2\sigma ^2 }}} \right. } {2\sigma ^2 }}}<math></math>

The ''standard normal distribution'' is scaled back to a mean of 0 and a standard deviation of 1:

<math>Z= \frac {X-\mu}{\sigma}</math>

<html>
<img src="/static/files/MBI/Module%207/stdnormaldist.jpg" width=500
</html>

Useful properties of the normal distribution are:
* The total area under the standard normal distribution curve is 1.
* At Z=0 the total area under the curve is 0.5
* To measure the area between two Z-values you can subtract the z values (Look up in the [[Standard Normal Distribution Table|Standard Normal Distribution Table]]) To find the probability of a value between 1 and 1.1 standard deviations above the mean you calculate:
{{{
Area above Z = 1 is 0.1587
Area above Z = 1.1 is 0.1357
}}}
Therefore the area between Z=1 and Z=1.1 is 0.3413 - 0.3413 = 0.0230
* It is symmetric. The probability to be -Z away from the mean is equal to being away +Z.

The normal distribution can be used to approximate the binomial distribution, <math>np</math> is the mean and <math>\sqrt{np(1-p)}</math> is the standard deviation of a binomial distribution. Therefore Z can be written as:
Z = <math>\Huge \frac{X=np}{\sqrt {np(1-p)}}</math>

And the poison distribution:
Z = <math>\Huge \frac{X-\lambda}{\sqrt \lambda}</math>

Example of the latter. If the average # of broken eggs is 50. What is the probability there will be more than 70 broken eggs?

Mean = <math>\lambda</math> = 50
Standard deviation= <math>\sqrt \lambda</math> = 7.07107

The area requires is the area above X=70.5 so Z will be:
Z = <math>(70.5 - 50)/7.07107 = 2.8988</math>
This makes the probability 1 - 0.5 - 0.4980 = 0.002

Part of: [[M7-Quantitative Decision Methods-S3 - Normal Distribution and confidence intervals|M7-Quantitative Decision Methods-S3 - Normal Distribution and confidence intervals]]
[[Category:mbi]]

[[Category:Decision]]

[[Category:M7]]

[[Category:Methods]]

[[Category:Quantitative]]

[[Category:Term]]

← Older revision		Revision as of 19:20, 27 August 2020
Line 1:		Line 1:
−	~~The normal distribution is the most widely used statistical distribution and is defined by its mean <math> \mu </math> and standard variance <math> \sigma </math>~~	+	Normal distribution
−
−	<math>P(x)</math> = <math>\frac{1}</math>{{\sigma \sqrt {2\pi } }}e^{{{ - \left( {x - \mu } \right)^2 } \mathord{\left/ {\vphantom {{ - \left( {x - \mu } \right)^2 } {2\sigma ^2 }}} \right. } {2\sigma ^2 }}}<math></math>
−
−	~~The ''standard normal distribution'' is scaled back to a mean of 0 and a standard deviation of 1:~~
−
−	~~<math>Z= \frac {X-\mu}{\sigma}</math>~~
−
−	~~<html>~~
−	~~<img src="/static/files/MBI/Module%207/stdnormaldist.jpg" width=500~~
−	~~</html>~~
−
−	~~Useful properties of the normal distribution are:~~
−	* The total area under the standard normal distribution curve is 1.
−	* At Z=0 the total area under the curve is 0.5
−	* To measure the area between two Z-values you can subtract the z values (Look up in the [[Standard Normal ~~Distribution Table\|Standard Normal Distribution Table]]) To find the probability of a value between 1 and 1.1 standard deviations above the mean you calculate:~~
−	~~{{{~~
−	~~Area above Z = 1 is 0.1587~~
−	~~Area above Z = 1.1 is 0.1357~~
−	~~}}}~~
−	~~Therefore the area between Z=1 and Z=1.1 is 0.3413 - 0.3413 = 0.0230~~
−	* It is symmetric. The probability to be -Z away from the mean is equal to being away +Z.
−
−	~~The normal~~ distribution ~~can be used to approximate the binomial distribution, <math>np</math> is the mean and <math>\sqrt{np(1-p)}</math> is the standard deviation of a binomial distribution. Therefore Z can be written as:~~
−	~~Z = <math>\Huge \frac{X=np}{\sqrt {np(1-p)}}</math>~~
−
−	~~And the poison distribution:~~
−	~~Z = <math>\Huge \frac{X-\lambda}{\sqrt \lambda}</math>~~
−
−	~~Example of the latter. If the average # of broken eggs is 50. What is the probability there will be more than 70 broken eggs?~~
−
−	~~Mean = <math>\lambda</math> = 50~~
−	~~Standard deviation= <math>\sqrt \lambda</math> = 7.07107~~
−
−	~~The area requires is the area above X=70.5 so Z will be:~~
−	~~Z = <math>(70.5 - 50)/7.07107 = 2.8988</math>~~
−	~~This makes the probability 1 - 0.5 - 0.4980 = 0.002~~
−
−	~~Part of: [[M7-Quantitative Decision Methods-S3 - Normal Distribution and confidence intervals\|M7-Quantitative Decision Methods-S3 - Normal Distribution and confidence intervals]]~~
	[[Category:mbi]]		[[Category:mbi]]

← Older revision		Revision as of 20:01, 27 August 2020
Line 1:		Line 1:
−	Normal distribution	+	The normal distribution is the most widely used statistical distribution and is defined by its mean <math> \mu </math> and standard variance <math> \sigma </math>
		+
		+	<math>P(x)</math> = <math>\frac{1}</math>{{\sigma \sqrt {2\pi } }}e^{{{ - \left( {x - \mu } \right)^2 } \mathord{\left/ {\vphantom {{ - \left( {x - \mu } \right)^2 } {2\sigma ^2 }}} \right. } {2\sigma ^2 }}}<math></math>
		+
		+	The ''standard normal distribution'' is scaled back to a mean of 0 and a standard deviation of 1:
		+
		+	<math>Z= \frac {X-\mu}{\sigma}</math>
		+
		+	<html>
		+	<img src="/static/files/MBI/Module%207/stdnormaldist.jpg" width=500
		+	</html>
		+
		+	Useful properties of the normal distribution are:
		+	* The total area under the standard normal distribution curve is 1.
		+	* At Z=0 the total area under the curve is 0.5
		+	* To measure the area between two Z-values you can subtract the z values (Look up in the [[Standard Normal Distribution Table\|Standard Normal Distribution Table]]) To find the probability of a value between 1 and 1.1 standard deviations above the mean you calculate:
		+	{{{
		+	Area above Z = 1 is 0.1587
		+	Area above Z = 1.1 is 0.1357
		+	}}}
		+	Therefore the area between Z=1 and Z=1.1 is 0.3413 - 0.3413 = 0.0230
		+	* It is symmetric. The probability to be -Z away from the mean is equal to being away +Z.
		+
		+	The normal distribution can be used to approximate the binomial distribution, <math>np</math> is the mean and <math>\sqrt{np(1-p)}</math> is the standard deviation of a binomial distribution. Therefore Z can be written as:
		+	Z = <math>\frac{X=np}{\sqrt {np(1-p)}}</math>
		+
		+	And the poison distribution:
		+	Z = <math>\frac{X-\lambda}{\sqrt \lambda}</math>
		+
		+	Example of the latter. If the average # of broken eggs is 50. What is the probability there will be more than 70 broken eggs?
		+
		+	Mean = <math>\lambda</math> = 50
		+	Standard deviation= <math>\sqrt \lambda</math> = 7.07107
		+
		+	The area requires is the area above X=70.5 so Z will be:
		+	Z = <math>(70.5 - 50)/7.07107 = 2.8988</math>
		+	This makes the probability 1 - 0.5 - 0.4980 = 0.002
		+
		+	Part of: [[M7-Quantitative Decision Methods-S3 - Normal Distribution and confidence intervals\|M7-Quantitative Decision Methods-S3 - Normal Distribution and confidence intervals]]
	[[Category:mbi]]		[[Category:mbi]]

Normal distribution - Revision history

Dirkjan at 20:01, 27 August 2020

Dirkjan: Replaced content with "Normal distribution Category:mbi Category:Decision Category:M7 Category:Methods Category:Quantitative Category:Term"

Dirkjan: Created page with "The normal distribution is the most widely used statistical distribution and is defined by its mean \mu and standard variance \sigma P(x)<..."

Dirkjan: Created page with "The normal distribution is the most widely used statistical distribution and is defined by its mean $\mu$ and standard variance $\sigma$ $P(x)<..."$