AP Statistics Curriculum 2007 EDA Center - Revision history

Tdlee: Text replacement - "{{translate|pageName=http://wiki.stat.ucla.edu/socr/" to ""{{translate|pageName=http://wiki.socr.umich.edu/"

2020-03-03T20:23:57Z

Text replacement - "{{translate|pageName=http://wiki.stat.ucla.edu/socr/" to ""{{translate|pageName=http://wiki.socr.umich.edu/"

2014-09-02T19:50:23Z

‎Mean

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‎Mean

2013-05-02T02:04:34Z

‎Medimean

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‎Medimean

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‎Medimean

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‎Medimean

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‎Medimean

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‎Medimean

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‎Medimean

← Older revision		Revision as of 20:23, 3 March 2020
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	* SOCR Home page: http://www.socr.ucla.edu		* SOCR Home page: http://www.socr.ucla.edu

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 ===Mean===
-The '''sample-mean''' is the arithmetic average of a finite sample of numbers. For instance, the mean of the sample <math>x_1, x_2, x_3, \cdots, x_{n-1}, x_n</math>. Short hand notation: <math>\{x_{i}\}_{i=1}^n,</math>, is given by:
+The '''sample-mean''' is the arithmetic average of a finite sample of numbers. For instance, the mean of the sample <math>x_1, x_2, x_3, \cdots, x_{n-1}, x_n</math>. Short hand notation: <math>\{x_{i}\}_{i=1}^n,</math> is given by:
 <math>\bar{x}={1\over n}\sum_{i=1}^{n}{x_{i}}.</math>

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 ===Mean===
-The '''sample-mean''' is the arithmetic average of a finite sample of numbers. For instance, the mean of the sample <math>x_1, x_2, x_3, \cdots, x_{n-1}, x_n</math> (short hand notation: <math>\{x_{i}\}_{i=1}^n</math>), is given by:
+The '''sample-mean''' is the arithmetic average of a finite sample of numbers. For instance, the mean of the sample <math>x_1, x_2, x_3, \cdots, x_{n-1}, x_n</math>. Short hand notation: <math>\{x_{i}\}_{i=1}^n,</math>, is given by:
-<center><math>\bar{x}={1\over n}\sum_{i=1}^{n}{x_{i}}.</math>
+<math>\bar{x}={1\over n}\sum_{i=1}^{n}{x_{i}}.</math>
 In the long-jump example, the sample-mean is calculated as follows:
 <center><math>\overline{y} = {1 \over 8} (74+78+106+80+68+64+60+76)=75.75 in.</math></center>
 ===Median===

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 : \( argmin_x \big ( \int_{s<m} {(m-s)P(s)ds}  + \int_{s>m} {(m-s)P(s)ds} \big )\)=
 : \( argmin_x \big ( m \int_{s<m} {P(s)ds} - \int_{s<m} {xP(s)ds} + m \int_{s>m} {P(s)ds} -\int_{s>m} {sP(s)ds}  \big )\)=
-: \( argmin_x \big ( m (F(m) -(1-F(m))) + \int_{s>m} {sP(s)ds} -\int_{s<m} {sP(s)ds}  \big )\), where the cumulative distribution function \( F(m)= \int_{s<m} {P(s)ds} \).
+: \( argmin_x \big ( m (F(m) -(1-F(m))) + \int_{s>m} {sP(s)ds} -\int_{s<m} {sP(s)ds}  \big )\), where the [[AP_Statistics_Curriculum_2007_Distrib_RV#Probability_density.2Fmass_and_.28cumulative.29_distribution_functions|cumulative distribution function]] \( F(m)= \int_{s<m} {P(s)ds} \).
 : And this function, \( m (2F(m) -1) + \int_{s>m} {sP(s)ds} -\int_{s<m} {sP(s)ds}\), is minimized for \(m=F^{-1}(0.5)\), which is the standard definition of the [[AP_Statistics_Curriculum_2007_EDA_Center#Median|median]] .

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 : And this function, \( m (2F(m) -1) + \int_{s>m} {sP(s)ds} -\int_{s<m} {sP(s)ds}\), is minimized for \(m=F^{-1}(0.5)\), which is the standard definition of the [[AP_Statistics_Curriculum_2007_EDA_Center#Median|median]] .
-* For the '''Mean''' = \( argmin_x \big ( \sum_{s\in S} {|s-x|^2}\big )\), observe that minimizing \( \sum_{s\in S} {|s-x|^2}\big )\) is similar to minimizing the [[AP_Statistics_Curriculum_2007_Distrib_MeanVar#Variance|variance]] of the process \( \sum_{s\in S} {(s-x)^2P(x)}\big )\), which is identical to minimizing \(E(X^2)-\mu^2\), for \(\mu\).
+* For the '''Mean''' = \( argmin_x \big ( \sum_{s\in S} {|s-x|^2}\big )\), observe that minimizing \( \sum_{s\in S} {|s-x|^2}\) is similar to minimizing the [[AP_Statistics_Curriculum_2007_Distrib_MeanVar#Variance|variance]] of the process \( \sum_{s\in S} {(s-x)^2P(x)}\), which is identical to minimizing \(E(X^2)-\mu^2\), for \(\mu\).
 ===[[EBook_Problems_EDA_Center | Problems]]===

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 : \( argmin_x \big ( m \int_{s<m} {P(s)ds} - \int_{s<m} {xP(s)ds} + m \int_{s>m} {P(s)ds} -\int_{s>m} {sP(s)ds}  \big )\)=
 : \( argmin_x \big ( m (F(m) -(1-F(m))) + \int_{s>m} {sP(s)ds} -\int_{s<m} {sP(s)ds}  \big )\), where the cumulative distribution function \( F(m)= \int_{s<m} {P(s)ds} \).
-This function is minimized for \(m=F^{-1}(0.5)\), which is the standard definition of the [[AP_Statistics_Curriculum_2007_EDA_Center#Median|median]] .
+And this function, \( argmin_x \big ( m (2F(m) -1) + \int_{s>m} {sP(s)ds} -\int_{s<m} {sP(s)ds}  \big )\), is minimized for \(m=F^{-1}(0.5)\), which is the standard definition of the [[AP_Statistics_Curriculum_2007_EDA_Center#Median|median]] .
 * For the '''Mean''' = \( argmin_x \big ( \sum_{s\in S} {|s-x|^2}\big )\), observe that minimizing \( \sum_{s\in S} {|s-x|^2}\big )\) is similar to minimizing the [[AP_Statistics_Curriculum_2007_Distrib_MeanVar#Variance|variance]] of the process \( \sum_{s\in S} {|s-x|^2P(x)}\big )\), which is identical to minimizing \(E(X^2)-\mu^2\), for \(\mu\).

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 We have already seen the standard definitions for [[AP_Statistics_Curriculum_2007_EDA_Center#Mean|mean]] and [[AP_Statistics_Curriculum_2007_EDA_Center#Median|median]]. There is an alternative ways to define these measures of centrality using optimization theory.
 : '''Median''' = \( argmin_x \big ( \sum_{s\in S} {|s-x|^1} \big )\)
-: '''Average''' = \( argmin_x \big ( \sum_{s\in S} {|s-x|^2}\big )\)
+: '''Mean''' = \( argmin_x \big ( \sum_{s\in S} {|s-x|^2}\big )\)
 : where \( argmin_x \) provides the value x which minimizes the distance function in the operator.
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 The medimean may be used as a compromise between the mean and the median. For example, if you are interested in something like a median, but which moves just a little bit in the direction of far-away elements, if there are any. In that case you may want to use a medimean(1.2) in the analysis.
 ===[[EBook_Problems_EDA_Center | Problems]]===