Page 1 Lecture 3: Measures of central tendency and dispersion Module 7 Page 2 Lecture 3: Measures of central tendency and dispersion Module 7 Measures of Central Tendency Module 7 1 1 E(x) = = ( ) , for parameter estimate Arithmatic Mean ( ) = .......for sample estimate Most frequently occuring value 1) 0 & 0 2) Value of x associated Mean: Mo de: with ma a a µ +  = = ?? = < ?? ? ? n i i n i x f x dx xx ff xx f ( ) The pdf takes the maximum value at the mode. x xi x Page 3 Lecture 3: Measures of central tendency and dispersion Module 7 Measures of Central Tendency Module 7 1 1 E(x) = = ( ) , for parameter estimate Arithmatic Mean ( ) = .......for sample estimate Most frequently occuring value 1) 0 & 0 2) Value of x associated Mean: Mo de: with ma a a µ +  = = ?? = < ?? ? ? n i i n i x f x dx xx ff xx f ( ) The pdf takes the maximum value at the mode. x xi x Measures of Central Tendency Contd… Module 7 It divides the area under the pdf curve into two halves. i.e Area is 50% = ( ) = P[X ] = 0.5 : [ It is the observation such that half the values lie on either side of it ] Median: µ µµ  = ? med med x med d n P x dx Def (b) Median Median (c) Median (a) Page 4 Lecture 3: Measures of central tendency and dispersion Module 7 Measures of Central Tendency Module 7 1 1 E(x) = = ( ) , for parameter estimate Arithmatic Mean ( ) = .......for sample estimate Most frequently occuring value 1) 0 & 0 2) Value of x associated Mean: Mo de: with ma a a µ +  = = ?? = < ?? ? ? n i i n i x f x dx xx ff xx f ( ) The pdf takes the maximum value at the mode. x xi x Measures of Central Tendency Contd… Module 7 It divides the area under the pdf curve into two halves. i.e Area is 50% = ( ) = P[X ] = 0.5 : [ It is the observation such that half the values lie on either side of it ] Median: µ µµ  = ? med med x med d n P x dx Def (b) Median Median (c) Median (a) [ ] [ ] [ ] [ ] [ ] [ ] aa aa aa aa aa aa    += + + + ? += + ?? ?? ?? L , E ( ) ( , ) = ( , ) y ( , ) =E E E E E et X Y x y f x y dxdy x f x y dxdy f x y dxdy X Y XY X Y [ ] aa aa a a   = ?? ? (X, Y) an independent RVs with a joint pdf of f(x,y) E , ( , ) For independent variable f(x,y) = g(x) ( ) ( x and y are indpendent) = ( X Y xy f x y dxdy x h y because xy g x [ ] [ ] [ ] [ ] [ ] a aa a aa   •+ ? += × ? ?? ) ( ) = ( ) y h( ) = E E E E E (If x and Y are independent) h y dxdy x g x dx y dy X Y XY X Y Module 7 Measures of Central Tendency Contd… Page 5 Lecture 3: Measures of central tendency and dispersion Module 7 Measures of Central Tendency Module 7 1 1 E(x) = = ( ) , for parameter estimate Arithmatic Mean ( ) = .......for sample estimate Most frequently occuring value 1) 0 & 0 2) Value of x associated Mean: Mo de: with ma a a µ +  = = ?? = < ?? ? ? n i i n i x f x dx xx ff xx f ( ) The pdf takes the maximum value at the mode. x xi x Measures of Central Tendency Contd… Module 7 It divides the area under the pdf curve into two halves. i.e Area is 50% = ( ) = P[X ] = 0.5 : [ It is the observation such that half the values lie on either side of it ] Median: µ µµ  = ? med med x med d n P x dx Def (b) Median Median (c) Median (a) [ ] [ ] [ ] [ ] [ ] [ ] aa aa aa aa aa aa    += + + + ? += + ?? ?? ?? L , E ( ) ( , ) = ( , ) y ( , ) =E E E E E et X Y x y f x y dxdy x f x y dxdy f x y dxdy X Y XY X Y [ ] aa aa a a   = ?? ? (X, Y) an independent RVs with a joint pdf of f(x,y) E , ( , ) For independent variable f(x,y) = g(x) ( ) ( x and y are indpendent) = ( X Y xy f x y dxdy x h y because xy g x [ ] [ ] [ ] [ ] [ ] a aa a aa   •+ ? += × ? ?? ) ( ) = ( ) y h( ) = E E E E E (If x and Y are independent) h y dxdy x g x dx y dy X Y XY X Y Module 7 Measures of Central Tendency Contd… Measure of “Spread” or “Dispersion” Range : (Max  min) value a a µ µ s   ? 2 2 2 Most important measure of dispersion = Second moment about Variance: the mean = ( ) ( ) = x f x dx = E(X) = Expected value of x Module 7Read More
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