Unit E: Statistics and Probability

Chapter 1: Statistics

Summary


Mean

  • The mean is the sum of a set of values divided by the number of values.

    • The mean formula is

      «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd»«mover»«mi»x«/mi»«mo»§#175;«/mo»«/mover»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mi»sum«/mi»«mo»§#160;«/mo»«mi»of«/mi»«mo»§#160;«/mo»«mi»the«/mi»«mo»§#160;«/mo»«mi»values«/mi»«mo»§#160;«/mo»«mi»in«/mi»«mo»§#160;«/mo»«mi»the«/mi»«mo»§#160;«/mo»«mi»data«/mi»«mo»§#160;«/mo»«mi»set«/mi»«/mrow»«mrow»«mi»total«/mi»«mo»§#160;«/mo»«mi»number«/mi»«mo»§#160;«/mo»«mi»of«/mi»«mo»§#160;«/mo»«mi»values«/mi»«mo»§#160;«/mo»«mi»in«/mi»«mo»§#160;«/mo»«mi»the«/mi»«mo»§#160;«/mo»«mi»data«/mi»«mo»§#160;«/mo»«mi»set«/mi»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«msub»«mi»x«/mi»«mn»1«/mn»«/msub»«mo»+«/mo»«msub»«mi»x«/mi»«mn»2«/mn»«/msub»«mo»+«/mo»«msub»«mi»x«/mi»«mn»3«/mn»«/msub»«mo»+«/mo»«mo»§#8230;«/mo»«mo»+«/mo»«msub»«mi»x«/mi»«mi»n«/mi»«/msub»«/mrow»«mi»n«/mi»«/mfrac»«/mtd»«/mtr»«/mtable»«/math»
      where
      x1 is the first value in the data set
      x2 is the second value in the data set
      x3 is the third value in the data set
      xn is the last value in the data set
      n is the number of values in the data set


    Weighted Mean

    • The weighted mean is an average in which each value is assigned a weight depending on its frequency.

    • The weighted mean formula is

      «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd»«mover»«mi»x«/mi»«mo»§#175;«/mo»«/mover»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mi»sum«/mi»«mo»§#160;«/mo»«mi»of«/mi»«mo»§#160;«/mo»«mi»the«/mi»«mo»§#160;«/mo»«mi»values«/mi»«mo»§#160;«/mo»«mi»in«/mi»«mo»§#160;«/mo»«mi»the«/mi»«mo»§#160;«/mo»«mi»data«/mi»«mo»§#160;«/mo»«mi»set«/mi»«/mrow»«mrow»«mi»total«/mi»«mo»§#160;«/mo»«mi»number«/mi»«mo»§#160;«/mo»«mi»of«/mi»«mo»§#160;«/mo»«mi»values«/mi»«mo»§#160;«/mo»«mi»in«/mi»«mo»§#160;«/mo»«mi»the«/mi»«mo»§#160;«/mo»«mi»data«/mi»«mo»§#160;«/mo»«mi»set«/mi»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«msub»«mi»w«/mi»«mn»1«/mn»«/msub»«mfenced»«msub»«mi»x«/mi»«mn»1«/mn»«/msub»«/mfenced»«mo»+«/mo»«msub»«mi»w«/mi»«mn»2«/mn»«/msub»«mfenced»«msub»«mi»x«/mi»«mn»2«/mn»«/msub»«/mfenced»«mo»+«/mo»«msub»«mi»w«/mi»«mn»3«/mn»«/msub»«mfenced»«msub»«mi»x«/mi»«mn»3«/mn»«/msub»«/mfenced»«mo»+«/mo»«mo»§#8230;«/mo»«mo»+«/mo»«msub»«mi»w«/mi»«mi»n«/mi»«/msub»«/mrow»«mrow»«msub»«mi»w«/mi»«mn»1«/mn»«/msub»«mo»+«/mo»«msub»«mi»w«/mi»«mn»2«/mn»«/msub»«mo»+«/mo»«msub»«mi»w«/mi»«mn»3«/mn»«/msub»«mo»+«/mo»«mo»§#8230;«/mo»«mo»+«/mo»«msub»«mi»w«/mi»«mi»n«/mi»«/msub»«/mrow»«/mfrac»«/mtd»«/mtr»«/mtable»«/math»
      where
      w1 is the weight of the first value in the data set
      w2 is the weight of the second value in the data set
      w3 is the weight of the third value in the data set
      wn is the weight of the last value in the data set
      n is the number of values in the data set

    When Should Weighted Mean be Used to Find the Mean?

    • Both the mean equation and weighted mean formula an be used to find the mean.

    • The weighted mean formula should be used when
      • the data values are repeated and it is easier to calculate using the weighted mean formula
      • the data is grouped
      • the values in the set of data are weighted differently