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					Describing Data:
Numerical Measures




                     Chapter 3
    GOALS

    1. Calculate the arithmetic mean, weighted mean,
       median, mode, and geometric mean.
    2. Explain the characteristics, uses, advantages, and
       disadvantages of each measure of location.
    3. Identify the position of the mean, median, and mode
       for both symmetric and skewed distributions.
    4. Compute and interpret the range, mean deviation,
       variance, and standard deviation.
    5. Understand the characteristics, uses, advantages,
       and disadvantages of each measure of dispersion.
    6. Understand Chebyshev’s theorem and the Empirical
       Rule as they relate to a set of observations.
2
    Characteristics of the Mean

     The arithmetic mean is the most widely used
       measure of location. It requires the interval
       scale. Its major characteristics are:
        – All values are used.
        – It is unique.
        – The sum of the deviations from the mean
          is 0.
        – It is calculated by summing the values
          and dividing by the number of values.


3
    Population Mean

     For ungrouped data, the population mean is the
     sum of all the population values divided by the
     total number of population values:




4
                Population Mean
 For ungrouped data (data not in a frequency distribution)

                      Sum of all the values in the population
Population Mean   =
                       Number of values in the population

                      X
                   
                       N
                Define Variables & Symbols
  µ    =   The population mean = “mu”
  N    =   Total number of observations
  X    =   A particular value
  Σ    =   Indicates the operation of adding = “sigma”
  ΣX   =   Sum of the X values                                  5
    A Parameter is a measurable characteristic of a
                   population.

The Kiers                         56,000
family owns                       42,000
four cars. The
                                  23,000
following is the
current mileage                   73,000
on each of the
four cars. The
mean mileage for
the cars is 40,333
1/3 miles.

                                                      6
    EXAMPLE – Population Mean




7
    Sample Mean

       For ungrouped data, the sample mean
        is the sum of all the sample values
        divided by the number of sample
        values:




8
                  Sample Mean
For ungrouped data (data not in a frequency distribution)

                        Sum of all the values in the sample
Sample Mean        =
                         Number of values in the sample

                          X
                       X
                           n
                 Define Variables & Symbols
   X
   X    =   Sample Mean = "X bar"
   n    =   Total number of observations
   X    =   A particular value
   Σ    =   Indicates the operation of adding = “sigma”
   ΣX   =   Sum of the X values                               9
     EXAMPLE – Sample Mean




10
A statistic is a measurable characteristic of a sample.

 A sample of
 five                       14.0,
 executives                 15.0,
 received the               17.0,
 following
                            16.0,
 bonus last
 year ($000):               15.0


 Find The
 Sample
 Mean:                                             11
     Properties of the Arithmetic Mean

        Every set of interval-level and ratio-level data has a mean.
        All the values are included in computing the mean.
        A set of data has a unique mean.
        The mean is affected by unusually large or small data values.
        The arithmetic mean is the only measure of central tendency
         where the sum of the deviations of each value from the mean is
         zero.




12
Sum Of The Deviations Of Each Value From
            The Mean Is Zero
       Deviation  ( X  X )
       X  A particular value, X  Mean
Consider the set of values: 3, 8, and 4. The mean is 5.




( X  X )  (3  5)  (8  5)  (4  5)  0

Later, this will become relevant information when we calculate the variation and
                                                                               13
standard deviation (This is the reason we will have to square the deviations).
     Weighted Mean


        The weighted mean of a set of numbers X1,
         X2, ..., Xn, with corresponding weights w1,
         w2, ...,wn, is computed from the following
         formula:




14
                  Weighted Mean
  • Instead of adding all the observations, we multiply each
    observation by the number of times it happens.
  • The weighted mean of a set of numbers X1, X2, ..., Xn,
    with corresponding weights w1, w2, ...,wn, is computed
    from the following formula:


Xw 
     (w 1X1  w 2 X 2  ...  w n X n )
                                                            Xw      
                                                                       (wX)
                                                                       w
                                                       or
          (w 1  w 2  ...w n )
                          Define Variables & Symbols
              X w = Weighted Mean = "X bar sub w"
             X1   =   A particular value
             X2   =   A particular value
             w1   =   A particular weight
             w2   =   A particular weight
                                                                         15
             Σ    =   Indicates the operation of adding = “sigma”
     EXAMPLE – Weighted Mean

     The Carter Construction Company pays its hourly
       employees $16.50, $19.00, or $25.00 per hour.
       There are 26 hourly employees, 14 of which are paid
       at the $16.50 rate, 10 at the $19.00 rate, and 2 at the
       $25.00 rate. What is the mean hourly rate paid the
       26 employees?




16
     The Median

        The Median is the midpoint of the values after
         they have been ordered from the smallest to
         the largest.
         –   There are as many values above the median as
             below it in the data array.
         –   For an odd set of values, the median will be the
             middle number.
         –   For an even set of values, the median will be the
             arithmetic average of the two middle numbers.
         –   Median is the measure of central tendency usually used by
             real estate agents. Why?...

17
                  Median Example
       $80,000.00 $70,000.00 $275,000.00 $65,000.00 $60,000.00

           Prices Ordered                  Prices Ordered
           from Low to High                from High to Low
                   $60,000                 $275,000
                     65,000                80,000
                     70,000 ← Median →     70,000
2nd                  80,000                65,000
example:           275,000                 60,000
   Prices Ordered                                 Prices Ordered
   from Low to High                               from High to Low
           $60,000                                $275,000
             65,000                               80,000
             70,000   (70000 + 75000)             75,000
                                        = $72,500
             75,000          2                    70,000
             80,000                               65,000
           275,000                                60,000
                                                                     18
Median Example in Excel




                          19
     Properties of the Median


          There is a unique median for each data set.
          It is not affected by extremely large or small
           values and is therefore a valuable measure
           of central tendency when such values occur.
          It can be computed for ratio-level, interval-
           level, and ordinal-level data.
          It can be computed for an open-ended
           frequency distribution if the median does not
           lie in an open-ended class.



20
     EXAMPLES - Median

     The ages for a sample of   The heights of four
       five college students    basketball players, in
       are:                     inches, are:
       21, 25, 19, 20, 22             76, 73, 80, 75

                                Arranging the data in
     Arranging the data in
                                ascending order gives:
       ascending order gives:
                                       73, 75, 76, 80.
       19, 20, 21, 22, 25.
                                Thus the median is 75.5
     Thus the median is 21.
21
                   Mode
•   The Mode is the value of the observation
    that appears most frequently.
•   The mode is especially useful in
    describing nominal and ordinal levels of
    measurement.
•   There can be more than one mode.




                                           22
                        Mode Example – Nominal Level
                                   Data
                                         Company has developed five bath oils
                        Company conducts marketing survey to determine which oil customers prefer
                            Number of Respondents Favoring Various Bath Oils

                            400
                                                 358
Number of Respondents




                            300


                            200                              193

                                     110
                            100                                            92
                                                                                      43
                              0
                                    Amor       Lamoure     Soothing    Smell Nice   Far Out

                                                                                                23
                                               Mode       Bath Oil
  Mode Example – Nominal Level
             Data
• With Nominal Data      Respondents
                         Smell Nice     Frequency Table
  you would count to     Lamoure       Smell Nice     92   =COUNTIF($A$2:$A$797,C3)
  see which occurs       Lamoure       Lamoure      358    =COUNTIF($A$2:$A$797,C4)
                         Soothing      Soothing     193    =COUNTIF($A$2:$A$797,C5)
  most frequently        Smell Nice    Amor         110    =COUNTIF($A$2:$A$797,C6)

• You can build a
                         Smell Nice    Far Out        43   =COUNTIF($A$2:$A$797,C7)
                         Soothing
  Frequency Table        Amor
                         Lamoure
  using the              Soothing
  COUNTIF function       Soothing
                         Smell Nice
  in Excel.              Smell Nice
                         Lamoure
• The one that           Amor
                         Lamoure
  occurs the most is
  the Mode
                       More data…

                                                                                24
     Example – Mode – Ratio Level Data

      Salaries
      $   35,000
      $   49,100
      $   60,000
      $   60,000
      $   40,000
      $   58,000
      $   60,000
      $   60,000
      $   40,000
      $   65,000
      $   55,000
      $   60,000
      $   71,400
      $   60,000
      $   55,000 In Excel:
      $   60,000 =MODE(G2:G16)

25
     Mean, Median, Mode Using Excel
     Table 2–4 in Chapter 2 shows the prices of the 80 vehicles sold last month at Whitner Autoplex in
     Raytown, Missouri. Determine the mean and the median selling price. The mean and the median
     selling prices are reported in the following Excel output. There are 80 vehicles in the study. So the
     calculations with a calculator would be tedious and prone to error.




26
     Mean, Median, Mode Using “Descriptive
     Statistics” in Excel (From Textbook)




27
     The Relative Positions of the Mean,
     Median and the Mode




28
     The Geometric Mean
        Useful in finding the average change of percentages, ratios, indexes, or growth
         rates over time.
        It has a wide application in business and economics because we are often
         interested in finding the percentage changes in sales, salaries, or economic
         figures, such as the GDP, which compound or build on each other.
        The geometric mean will always be less than or equal to the arithmetic mean.
        The GM gives a more conservative figure that is not drawn up by large values
         in the set.
        The geometric mean of a set of n positive numbers is defined as the nth root of
         the product of n values.
        The formula for the geometric mean is written:




29
             Geometric Mean
• The GM of a set of n positive numbers is defined
  as the nth root of the product of n values. The
  formula is either (both are true):

   GM %  1  n ( X 1)( X 2)( X 3)...(Xn)
    GM %  n ( X 1)( X 2)( X 3)...(Xn) 1
              Define Variables & Symbols
      GM       = Geometric Mean
      X1       = A particular number (1 + %)
      X2       = A particular number (1 + %)
      n        = Number of postive numbers in set

                                                    30
Geometric Mean Example 1:
   Percentage Increase

   Starting Salary           $41,000.00
   Increase in salary Year 1     5%
   Increase in salary Year 2    15%


GM  2 (1.05)(1.15)  1.09886

              In Excel:
   1.05 * 1.15 = 1.2075
   GM = 1.2075 ^ (1/2) - 1 = 9.886%

                                          31
     Verify Geometric Mean Example
                             Verify 1:
           Raise 1 = $41,000.00 *       5% =    $2,050.00
           Raise 2 = 43,050.00 *       15% =     6,457.50
           Total                                $8,507.50

                             Verify 2:
           Raise 1 = $41,000.00 * 0.09886 =     $4,053.39
           Raise 2 = 45,053.39 * 0.09886 =       4,454.11
           Total                                $8,507.50

The GM gives a more
                       If We used Arithmetic Mean (5%+15%)/2 = 10%
conservative figure
that is not drawn     Raise 1 = $41,000.00 *      10% = $4,100.00
up by large values    Raise 2 = 45,100.00 *       10% =    4,510.00
in the set.           Total                              $8,610.00
                                                               32
     EXAMPLE – Geometric Mean (2)

     The return on investment earned by Atkins
       construction Company for four successive
       years was: 30 percent, 20 percent, -40
       percent, and 200 percent. What is the
       geometric mean rate of return on investment?


       GM  4 (1.3 )(1.2 )( 0.6 )( 3.0 )  4 2.808  1.294



33
         Another Use Of GM:
      Ave. % Increase Over Time
 • Another use of the geometric mean is to
   determine the percent increase in sales,
   production or other business or economic series
   from one time period to another
 • Where n = number of periods

            (Value at end of all the periods)
GM  n                                           1
         (Value at beginning of all the periods)


                                                 34
 Example for GM: Ave. % Increase Over
                 Time
• The total number of females enrolled in
  American colleges increased from 755,000 in
  1992 to 835,000 in 2000. That is, the geometric
  mean rate of increase is 1.27%.

                  835,000
      GM  8               1  .0127
                  755,000
•The annual rate of increase is 1.27%
•For the years 1992 through 2000, the rate of
female enrollment growth at American colleges
was 1.27% per year                                  35
     Dispersion

      Why Study Dispersion?
        –   A measure of location, such as the mean or the median,
            only describes the center of the data. It is valuable from
            that standpoint, but it does not tell us anything about the
            spread of the data.
        –   For example, if your nature guide told you that the river
            ahead averaged 3 feet in depth, would you want to wade
            across on foot without additional information? Probably not.
            You would want to know something about the variation in
            the depth.
        –   A second reason for studying the dispersion in a set of data
            is to compare the spread in two or more distributions.


36
     Dictionary Definitions:
 Dispersion, Variation, Deviation
• Dispersion
  • The spatial property of being scattered about over
    an area or volume
  • The degree of scatter of data, usually about an
    average value, such as the median or mean
• Variation
  • The act of changing or altering something slightly but
    noticeably from the norm or standard
• Deviation
  – A variation that deviates from the standard or norm

                Deviation  ( X  X )
                                                             37
                  Dispersion
1. How spread out is the data?
  •   What is the average of all the deviations?
2. A small value for a measure of dispersion
   indicates that the data are clustered around
   the typical value (mean)
  •   Mean can fairly represent the data
3. A large value for a measure of dispersion
   indicates that the data are not clustered
   around the typical value (mean)
  •   Mean may not fairly represent the data

                                                   38
     Samples of Dispersions




39
     Measures of Dispersion

         Range



         Mean Deviation



         Variance and Standard
          Deviation




40
                      Range
• Range = Highest Value – Lowest Value
• Excel = MAX – MIN functions
  – The difference between the largest and the smallest
    value

• Advantage:
  – It is easy to compute and understand
• Disadvantage:
  – Only two values are used in its calculation
  – It is influenced by an extreme value
                                                          41
                 Range Example
             Boomerangs made per day over ten day period
             Boomerangs made per day over ten day period
                              ?
                              ?
                           ? ?       ?
                        ? ? ?        ? ?
                  52 54 56 58 60     62 64 66 68

                                 X
           Boomerangs made per day at Colorado Boomerangs


                              ?
                              ?
            ?                 ?                 ?
?           ?                 ?                 ?           ?
40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70 72 74 76 78 80

                                 X
             Boomerangs made per day at Gel Boomerangs      42
    Mean Deviation (MA or MAD)

• Mean Deviation measures the mean amount by
  which the values in a population, or sample, vary
  from their mean
                                X X
                 MD 
                                        n
                      Define Variables & Symbols
            X
            X    =   Sample Mean = "X bar"
            n    =   Total number of observations
            X    =   A particular value
            Σ    =   Adding = “sigma”
            MD   =   Mean Deviation
            ?    =   Absolute Value (distance from zero)   43
                                Variance
              ( X   )             2
                                             ( X  X )                    2
       2
                                        s   2
                  N                              n 1
        Define Variables & Symbols                    Define Variables & Symbols
 2 = Pop. Variance = "sigma squared"    s2       = Sample Variance
                                                      Total number of observations
                                          n       =   (n as denominator tends to
N   = Total number of observations
                                                      underestimate variance)
                                                      Provides the appropriate correction
                                         n-1 =
X   = A particular value                              for underestimation
                                         X        =   A particular value
μ   = Pop. Mean                          X        =   Sample Mean



In Excel: 1) Population Variance  VARP function
          2) Sample Variance  VAR function
                                                                                     44
                      Standard Deviation
        ( X   )                    2              ( X  X )2
           2                            s  s2 
            N                                           n 1
         Define Variables & Symbols                  Define Variables & Symbols

  2     = Pop. Standard Deviation        s  s2   = Sample Standard Deviation
                                                       Total number of observations
                                              n      = (n as denominator tends to
   N       = Total number of observations              underestimate variance)
                                                       Provides the appropriate correction
                                             n-1     =
                                                       for underestimation
  X        = A particular value               X      = A particular value
   μ       = Pop. Mean                       X       = Sample Mean


  In Excel: 1) Population Variance  STDEVP function
            2) Sample Variance  STDEV function
 The primary use of the statistic s2 is to estimate σ2, therefore
 (n-1) is necessary to get a better representation of the
                                                                45
 deviation or dispersion in the data (n tends to underestimate)
     EXAMPLE – Range

     The number of cappuccinos sold at the Starbucks location in the
       Orange Country Airport between 4 and 7 p.m. for a sample of 5
       days last year were 20, 40, 50, 60, and 80. Determine the mean
       deviation for the number of cappuccinos sold.



                 Range = Largest – Smallest value
                       = 80 – 20 = 60




46
     EXAMPLE – Mean Deviation

     The number of cappuccinos sold at the Starbucks location in the
       Orange Country Airport between 4 and 7 p.m. for a sample of 5
       days last year were 20, 40, 50, 60, and 80. Determine the mean
       deviation for the number of cappuccinos sold.




47
     EXAMPLE – Variance and Standard
     Deviation

      The number of traffic citations issued during the last five months in
        Beaufort County, South Carolina, is 38, 26, 13, 41, and 22. What
        is the population variance?




                                               2 ^(1/2) =  = 10.22441
48
     EXAMPLE – Sample Variance and
     Sample Standard Deviation

     The hourly wages for
       a sample of part-
       time employees at
       Home Depot are:
       $12, $20, $16, $18,
       and $19. What is
       the sample
       variance and
       sample standard
       deviation?




                             s2 ^(1/2) = s = 3.162278
49
     EXAMPLE – Sample Standard
          Deviation (p1)




50
  EXAMPLE – Sample Standard Deviation (p2)
              Conclusions
The Variance is 14.48 square grams
The sample standard deviation is 3.8 grams
On average the weight of each box deviated from the sample mean by 3.8
grams. If the manufacturer of the cereal has 254 grams printed on the box, is
the production efficient? Is their claim of 254 grams accurate or fair?

If the historical sample standard deviation (s) is 2.7 grams, then an s = 3.8
indicates that it is reasonable to assume that the production is not efficient.
Also, s = 2.7 grams would indicate that the claim on the box does not seem
reasonable.




                                                                                  51
     Chebyshev’s Theorem

     The arithmetic mean biweekly amount contributed by the Dupree
       Paint employees to the company’s profit-sharing plan is $51.54,
       and the standard deviation is $7.51. At least what percent of the
       contributions lie within plus 3.5 standard deviations and minus
       3.5 standard deviations of the mean?




52
              Chebyshev’s Theorem
• For any set of observations (sample or
  population & does not have to be normal
  distributed), the proportion of the values
  that lie within (+/-) k standard deviations
  of the mean is at least:

                        1
                  1
                                   2) The smaller this
  3) The
bigger this                2
                       k           1) The bigger this


• where k is any constant greater than 1          53
     The Empirical Rule (also called Normal Rule)




54
                           34.13                       34.13
                           % area                      % area




                  13.58                                         13.58
                  % area                                        % area



        2.15%                                                            2.15%
         area                                                             area
0.13%
                                                                                 0.13%
 area
                                                                                  area




                Xbar-3   Xbar-2 Xbar-1      Xbar   Xbar +1 Xbar +2 Xbar +3

                                         68.26% area


                                         95.44% area


                                         99.74% area
                                                                                    55
A sample of the rental rates at for 2 bedroom apartments in Seattle approximates
   a symmetrical, bell-shaped distribution. The sample mean is $925.00; The
standard deviation is $110.00. Using the Empirical Rule, answer these questions:

Sample Mean = Xbar =                                                  $925.00
Sample Standard Deviation = s =                                       $110.00

       Q1: About 68% of the rental rates are between what two amounts?
 Xbar +/- 1*s = $925.00 +/- 1*$110.00 ==> About 68% of the values lie between
                            $815.00 and $1,035.00.
       Q2: About 95% of the rental rates are between what two amounts?
 Xbar +/- 2*s = $925.00 +/- 2*$110.00 ==> About 95% of the values lie between
                            $705.00 and $1,145.00.
       Q3: Almost all of the rental rates are between what two amounts?
  Xbar +/- 3*s = $925.00 +/- 3*$110.00 ==> About Almost all of the values lie
                       between $595.00 and $1,255.00.

                                                                         56
     The Arithmetic Mean of Grouped Data




57
     The Arithmetic Mean of Grouped Data -
     Example

      Recall in Chapter 2, we
        constructed a frequency
        distribution for the vehicle
        selling prices. The
        information is repeated
        below. Determine the
        arithmetic mean vehicle
        selling price.




58
     The Arithmetic Mean of Grouped Data -
     Example




59
     Standard Deviation of Grouped Data




60
     Standard Deviation of Grouped Data -
     Example
      Refer to the frequency distribution for the Whitner Autoplex data
        used earlier. Compute the standard deviation of the vehicle
        selling prices




61
   Approximate Median from Frequency Distribution
1. Locate the class in which the median lies
    • Divide the total number of data values by 2.
    • Determine which class will contain this value. For
      example, if n=50, 50/2 = 25, then determine which
      class will contain the 25th value
                                                 n
2. Use formula to arrive                            CF
   at estimate of Median:           Median  L  2      (i )
                                                    f
                         Define Variables & Symbols
    L    =   Lower limit of the class containing the median
    n    =   Total number of frequencies
    f    =   Frequency in the median class
             Cumulative number of frequencies in all the classes
    CF   =
             preceeding the class containing the medain
    i    =   width of the class in which the median lies           62
  Approximate Median from Frequency Distribution Example 1
 Selling
  Price               Cumulative         Total Number Frequencies
                                   1st                              = 40
   ($       Frequency Frequency                     2
 thousands)     (f)      CF
12 up to 15      8        8                40th value must be here
15 up to 18     23        31
18 up to 21     17        48       2nd         L            =         18
21 up to 24     18        66                   n            =         80
24 up to 27      8        74                   f            =         17
27 up to 30      4        78                  CF            =         31
30 up to 33      2        80
                                               i            =          3
    Total       80

             n                  80
                CF                 31
Median  L  2      (i )  18  2       (3)  19.588235294
                f                 17
The estimated median vehicle selling price is $19,588.00
                                                                      63
                   Selling
                    Price               Cumulative
                     ($       Frequency Frequency
                  thousands )     (f)       CF
                 12 up to 15       8         8
                 15 up to 18      23        31
                 18 up to 21      17        48
                 21 up to 24      18        66
                 24 up to 27       8        74
                 27 up to 30       4        78
                 30 up to 33       2        80
                     Total        80
• Let’s think about this:
   – n/2 – CF = 80/2-31 = 9 vehicles to get from the 31st to
     the 40th
   – (n/2 – CF)/f = 9/17 = 9/17th of the distance through the
     class width of $3000
   – ((n/2 – CF)/f)(i) = (9/17)($3000) = $1,588
   – ((n/2 – CF)/f)(i) + L = $1,588 + $18,000 = $19,588,
     Median estimate
                                                            64
Estimate Range from
    Grouped Data
        Employee $ contributed to profit   Frequency
                sharing plan                    f
          $50      Up To        $55             4
           55      Up To        $60             7
           60      Up To         65             9
           65      Up To         70            22
           70      Up To         75            40
           75      Up To         80            24
           80      Up To         85            15
           85      Up To         90             9
                    Total                     130


              Upper Limit of        Lower Limit of
Estimated      the Highest           the Lowest
 Range    =       Class           -     Class

Estimated
 Range    =         $90           -         $50        = $40

                                                               65
     End of Chapter 3




66

				
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