Patent Publication Number: US-2012029873-A1

Title: Machine-implemented method and an electronic device for graphically illustrating a statistical display based on a set of numerical data, and a computer program product

Description:
CROSS-REFERENCE TO RELATED APPLICATION 
     This application claims priority of Taiwanese Application No. 099125344, filed on Jul. 30, 2010. 
     BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The invention relates to a machine-implemented method and an electronic device for graphically illustrating a statistical display and a computer program product for implementing the method, more particularly to a machine-implemented method and an electronic device for graphically illustrating a statistical display based on a set of numerical data, the display being easy to read and requiring little display resources, and a computer program product for implementing the method. 
     2. Description of the Related Art 
     Statistical analysis tools are used to collect and organize data, and to present an objective interpretation of the collection of data through a statistical plot. Currently known types of statistical plots include dot plot, histogram, probability plot, residual plot, box plot, block plot, etc. An appropriate statistical plot can capture important information about the collection of data. Thus, statistics is widely used in the fields of medicine, finance and social studies and by governments. 
     When reading a statistical plot, an observation emphasis is the central tendency and statistical dispersion of the distribution of the data. Central tendency is generally measured by mean, median, geometric mean and mode values. Statistical dispersion indicates variability in a set of data (i.e., the degree to which values are scattered around a central point, e.g., mean or median), and is measured by range, variance, standard deviation, etc. An observation that is numerically distant from the mean by more than twice the standard deviation is generally referred to as an unusual observation, and an observation that is numerically distant from the mean by more than three times the standard deviation is generally referred to as an outlier if the distribution of the data is Gaussian distribution. Outliers represent the most extreme observations, which rarely occur, but may not be naively ignored. 
     Shown in  FIG. 1  is a probability density function for the “Gaussian distribution” (also known as the “normal distribution”) with the mean denoted by (μ) and the standard deviation denoted by (σ). When a probability distribution is close to the Gaussian distribution, approximately 68.26% (34.13%×2) of the values of the distribution would fall within the range of one standard deviation (σ) away from the mean (μ), approximately 95.44% (68.26%+13.59%×2) of the values of the distribution fall within the range of twice the standard deviation (σ) away from the mean (μ), and approximately 99.72% (95.44%+2.14%×2) of the values of the distribution fall within the range of three times the standard deviation (σ) away from the mean (μ). 
     Shown in  FIG. 2  is a box plot (also known as box-and-whisker plot), which depicts a collection of data through what is known as the “five-number summary”, which includes five important sample percentiles: the smallest observation (sample minimum), lower quartile (Q 1 ) (i.e., the 25 th  percentile), medium (Q 2 ) (i.e., the 50 th  percentile), upper quartile (Q 3 ) (i.e., the 75th percentile), and largest observation (sample maximum). The two sides of the box of a box plot respectively represent the lower and upper quartiles (Q 1 , Q 3 ), the band at the middle of the box represents the median (Q 2 ), and the ends of the whiskers normally represent the sample minimum and the sample maximum, respectively. The box plot is commonly used in open-high-low-close chart (OHLC) for illustrating price fluctuations of a financial instrument in a unit time. Since the box plot uses percentiles, the information that can be obtained out of the box plot shown in  FIG. 2  only includes whether the distribution is symmetrical (not symmetrical in this case). 
     Shown in  FIG. 3  is another example of the box plot where two fences are illustrated by dashed lines to the right of the box. The two fences respectively indicate an inner fence percentile equal to Q 3 +1.5×(Q 3 −Q 1 ), and an outer fence percentile equal to Q 3 +3×(Q 3 −Q 1 ). The whisker beyond the inner fence percentile is not displayed, and symbols, such as a hollow dot (∘) and a star ( ), are used for indicating observations beyond the inner fence percentile and observations beyond the outer fence percentile (collectively referred to as “extreme observations” herein). In cases where there are data with negative values, there would be two fences to the left of the box, respectively indicating an inner fence percentile equal to Q 1 −1.5×(Q 3 −Q 1 ), and an outer fence percentile equal to Q 1 −3×(Q 3 −Q 1 ). As compared to the box plot of  FIG. 2 , the box plot of  FIG. 3  presents additional information regarding the locations of extreme observations. However, the extreme observations are different in definition from the unusual observations and outliers of the Gaussian distribution, which are the commonly used reference in statistics. In other words, meanings of extreme observations with reference to Gaussian distribution cannot be known from the box plot. 
     Shown in  FIG. 4  is a box plot for illustrating multiple sets of data. It is evident from  FIG. 4  that to simultaneously present boxes and extreme data, the plot becomes difficult to read and interpret by a user, especially when the extreme observations are very distant from the boxes, where the boxes are compressed significantly. 
     Moreover, a shortcoming common to both dot plot and histogram is that, significant plot-generating and displaying resources are required when the data is large in quantity. 
     SUMMARY OF THE INVENTION 
     Therefore, the object of the present invention is to provide a machine-implemented method and an electronic device for graphically illustrating a statistical display based on a set of numerical data, where the display is easy to read, requires little display resources, and is capable of providing objective meanings of extreme observations, and a computer program product for implementing the method. 
     According to one aspect of the present invention, there is provided a machine-implemented method for graphically illustrating a statistical display based on a set of numerical data. The machine-implemented method includes the steps of: (a) finding, with a processor, a median of the set of numerical data, and finding, with the processor, a subset of the numerical data, each corresponding to a member of a predetermined set of cumulative distribution probabilities of the Gaussian distribution; (b) computing, with the processor, a mean of the set of numerical data and a standard deviation of the set of numerical data; (c) computing, with the processor, a plurality of reference values, each differing from the mean of the set of numerical data by a corresponding predetermined number multiplied by the standard deviation of the set of numerical data; (d) generating, with the processor, a plot that includes a first line, a second line and a plurality of connecting lines, the first line extending in an axis direction and having the median and the subset of the numerical data found in step (a) marked thereon, the second line extending in the axis direction, being spaced apart from the first line, and having the mean and the reference values computed in step (c) marked thereon, the connecting lines respectively connecting the median and the mean, and corresponding pairs of the subset of the numerical data and the reference values; and (e) outputting the plot for viewing by a user. 
     According to another aspect of the present invention, there is provided a computer program product, including a computer readable storage medium that includes program instructions, which when executed by an electronic device, cause the electronic device to perform the above described method. 
     According to still another aspect of the present invention, there is provided an electronic device for graphically illustrating a statistical display based on a set of numerical data. The electronic device includes a data selecting unit, a computing unit, a plot generating unit, and an output unit. 
     The data selecting unit is for finding a median of the set of numerical data, and for finding a subset of the numerical data, each corresponding to a member of a predetermined set of cumulative distribution probabilities of the Gaussian distribution. 
     The computing unit is for computing a mean of the set of numerical data and a standard deviation of the set of numerical data, and for computing a plurality of reference values, each differing from the mean by a corresponding predetermined number multiplied by the standard deviation of the set of numerical data. 
     The plot generating unit is coupled to the data selecting unit and the computing unit for generating a plot that includes a first line, a second line and a plurality of connecting lines. The first line extends in an axis direction and has the median and the subset of the numerical data found by the data selecting unit marked thereon. The second line extends in the axis direction, is spaced apart from the first line, and has the mean and the reference values computed by the computing unit marked thereon. The connecting lines respectively connect the median and the mean, and corresponding pairs of the subset of the numerical data and the reference values. 
     The output unit is coupled to the plot generating unit for outputting the plot for viewing by a user. 
     The advantages and effects of the present invention lie in that it requires less plot-generating and displaying resources as compared to the conventional statistical graphs, such as dot plots and histograms, and that it presents more information regarding the distribution of the numerical data as compared to the conventional statistical graphs, such as the box plot. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       Other features and advantages of the present invention will become apparent in the following detailed description of the preferred embodiments with reference to the accompanying drawings, of which: 
         FIG. 1  is a plot for illustrating a probability density function for a Gaussian distribution known in the prior art; 
         FIG. 2  is an example of a conventional box plot; 
         FIG. 3  is another example of the conventional box plot; 
         FIG. 4  is an example of the conventional box plot for illustrating multiple sets of data; 
         FIG. 5  is a block diagram, illustrating the first preferred embodiment of an electronic device for graphically illustrating a statistical display based on a set of numerical data according to the present invention; 
         FIG. 6  is a block diagram, illustrating the second preferred embodiment of an electronic device for graphically illustrating a statistical display based on a set of numerical data according to the present invention; 
         FIG. 7  is a flow chart, illustrating the first preferred embodiment of the machine-implemented method for graphically illustrating a statistical display based on a set of numerical data according to the present invention; 
         FIG. 8  is a plot, illustrating the statistical display generated using the machine-implemented method of the present invention; 
         FIG. 9  illustrates a table containing first and second sets of numerical data used for a first exemplary embodiment of the present invention; 
         FIG. 10  illustrates a table containing a median and a subset of numerical data for each of the first and second sets of numerical data of  FIG. 9 ; 
         FIG. 11  illustrates a table containing a mean and a plurality of reference values for each of the first and second sets of numerical data of  FIG. 9 ; 
         FIG. 12  is a plot, illustrating the statistical display generated for the first exemplary embodiment; 
         FIG. 13  illustrates a table containing first and second sets of numerical data used for a second exemplary embodiment of the present invention, which are natural logarithmic equivalents of the first and second sets of numerical data of  FIG. 9 ; 
         FIG. 14  illustrates a table containing the median and the subset of numerical data for each of the first and second sets of numerical data of  FIG. 13 ; 
         FIG. 15  illustrates a table containing the mean and the reference values for each of the first and second sets of numerical data of  FIG. 13 ; 
         FIG. 16  is a plot, illustrating the statistical display generated for the second exemplary embodiment; and 
         FIG. 17  is a plot, illustrating the statistical display generated for a third exemplary embodiment of the present invention. 
     
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     Before the present invention is described in greater detail, it should be noted that like elements are denoted by the same reference numerals throughout the disclosure. 
     With reference to  FIG. 5 , the first preferred embodiment of an electronic device  100  for graphically illustrating a statistical display based on a set of numerical data according to the present invention includes a processor  10 , and a storage unit  11 , an input unit  12  and an output unit  13  all coupled electrically to the processor  10 . The processor  10  is for performing steps of the machine-implemented method for graphically illustrating a statistical display based on a set of numerical data according to the present invention. In particular, the processor  10  is capable of executing program instructions of a computer readable storage medium of a computer program product  110  that causes the processor  10  to perform steps of the machine-implemented method of the present invention. In addition, the storage unit  11  has a numerical database  111  and a parameter setting table  112  established therein. 
     The electronic device  100  can be, but is not limited to, a personal computer, a workstation, a notebook computer, a palmtop computer, data processing equipment, audiovisual equipment, personal digital assistant (PDA), etc. 
     The computer program product  110  may be written in programming languages such as C, Visual C++, Visual Basic, JAVA, etc. The processor  10  is a central processor in this embodiment. The input unit  12  permits input of the set of numerical data from an external data source  2 , such as a host containing financial data, to the processor  10 , which then stores the set of numerical data in the numerical database  111 . The parameter setting table  112  may contain parameters that are pre-established or that are inputted by a user via the input unit  12 . To associate operably with the external data source  2 , the input unit  12  may be an Internet interface, or other transmission interfaces capable of communicating with the external data source  2 , or may be a keyboard, a mouse, a remote controller, a voice recognition system, a touch panel of a mobile phone, etc. in cases where the set of numerical data is inputted by a user. The output unit  13  may include a display device (not shown) for displaying the plot  300 . The output unit  13  can be a computer monitor, a TV screen, a display screen of a mobile phone, or a printer, as long as the statistical display may be viewed by the user in some way. 
     With reference to  FIG. 7  and  FIG. 8 , the first preferred embodiment of the machine-implemented method for graphically illustrating a statistical display based on a set of numerical data according to the present invention includes the following steps. 
     In step  401 , with the processor  10 , a median (M) of the set of numerical data and a subset of the numerical data are found. Each member of the subset corresponds to a member of a predetermined set of cumulative distribution probabilities of the Gaussian distribution. 
     In step  402 , with the processor  10 , a mean (μ) of the set of numerical data and a standard deviation (σ) of the set of numerical data are computed. 
     In step  403 , with the processor  10 , a plurality of reference values are computed. Each of the reference values differs from the mean (μ) of the set of numerical data by a corresponding predetermined number multiplied by the standard deviation (σ) of the set of numerical data. 
     In step  404 , with the processor  10 , a plot  300  including a first line  501 , a second line  502  and a plurality of connecting lines  503  is generated. The first line  501  extends in an axis direction and has the median (M) and the subset of the numerical data found in step  401  marked thereon. The second line  502  extends in the axis direction, is spaced apart from the first line  501 , and has the mean (μ) and the reference values computed in step  403  marked thereon. The connecting lines  503  respectively connect the median (M) and the mean (μ), and corresponding pairs of the subset of the numerical data and the reference values. 
     In step  405 , the plot  300  is outputted for viewing by a user. To facilitate viewing, the plot  300  may be outputted together with an X-axis and a Y-axis. In this embodiment, the Y-axis defines the axis direction. However, to satisfy the requirements and needs of particular applications, the X-axis, instead of the Y-axis, may also define the axis direction in other embodiments. 
     According to this embodiment, in step  401 , the predetermined set of cumulative distribution probabilities includes first, second, third, fourth, fifth and sixth cumulative distribution probabilities. The first cumulative distribution probability (d 1 ) corresponds to a range of within one standard deviation smaller than the mean of the Gaussian distribution. The second cumulative distribution probability (d 2 ) corresponds to a range of within one standard deviation greater than the mean of the Gaussian distribution. The third cumulative distribution probability (d 3 ) corresponds to a range of within two standard deviations smaller than the mean of the Gaussian distribution. The fourth cumulative distribution probability (d 4 ) corresponds to a range of within two standard deviations greater than the mean of the Gaussian distribution. The fifth cumulative distribution probability (d 5 ) corresponds to a range of within to three standard deviations smaller than the mean of the Gaussian distribution. The sixth cumulative distribution probability (d 6 ) corresponds to a range of within three standard deviations greater than the mean of the Gaussian distribution. 
     In particular, with reference to  FIG. 1 , the first cumulative distribution probability (d 1 ) can be computed as 
     
       
         
           
             
               
                 
                   1 
                   - 
                   
                     68.26 
                      
                     % 
                   
                 
                 2 
               
               = 
               
                 15.87 
                  
                 % 
               
             
             , 
           
         
       
     
     and the second cumulative distribution probability (d 2 ) can be computed as 
     
       
         
           
             
               
                 
                   1 
                   + 
                   
                     68.26 
                      
                     % 
                   
                 
                 2 
               
               = 
               
                 84.13 
                  
                 % 
               
             
             , 
           
         
       
     
     where 68.26% is the distribution probability within one standard deviation from the mean of the Gaussian distribution. The third cumulative distribution probability (d 3 ) can be computed as 
     
       
         
           
             
               
                 
                   1 
                   - 
                   
                     95.44 
                      
                     % 
                   
                 
                 2 
               
               = 
               
                 2.28 
                  
                 % 
               
             
             , 
           
         
       
     
     and the fourth cumulative distribution probability (d 4 ) can be computed as 
     
       
         
           
             
               
                 
                   1 
                   + 
                   
                     95.44 
                      
                     % 
                   
                 
                 2 
               
               = 
               
                 97.72 
                  
                 % 
               
             
             , 
           
         
       
     
     where 95.44% is the distribution probability within two standard deviations from the mean of the Gaussian distribution. The fifth cumulative distribution probability (d 5 ) can be computed as 
     
       
         
           
             
               
                 
                   1 
                   - 
                   
                     99.73 
                      
                     % 
                   
                 
                 2 
               
               = 
               
                 0.135 
                  
                 % 
               
             
             , 
           
         
       
     
     and the sixth cumulative distribution probability (d 6 ) can be computed as 
     
       
         
           
             
               
                 
                   1 
                   + 
                   
                     99.73 
                      
                     % 
                   
                 
                 2 
               
               = 
               
                 99.87 
                  
                 % 
               
             
             , 
           
         
       
     
     where 99.87% is the distribution probability within three standard deviations from the mean of the Gaussian distribution. The first, second, third, fourth, fifth and sixth cumulative distribution probabilities (d 1 , d 2 , d 3 , d 4 , d 5 , d 6 ) may be pre-established in the parameter setting table  112 . 
     Accordingly, the subset of the numerical data includes first, second, third, fourth, fifth and sixth members (n 1 , n 2 , n 3 , n 4 , n 5 , n 6 ). The numerical order of the first member (n 1 ) of the subset among the set of the numerical data corresponds to the total number of the numerical data multiplied by the first cumulative distribution probability (d 1 ). The numerical order of the second member (n 2 ) of the subset among the set of the numerical data corresponds to the total number of the numerical data multiplied by the second cumulative distribution probability (d 2 ). The numerical order of the third member (n 3 ) of the subset among the set of the numerical data corresponds to the total number of the numerical data multiplied by the third cumulative distribution probability (d 3 ). The numerical order of the fourth member (n 4 ) of the subset among the set of the numerical data corresponds to the total number of the numerical data multiplied by the fourth cumulative distribution probability (d 4 ). The numerical order of the fifth member (n 5 ) of the subset among the set of the numerical data corresponds to the total number of the numerical data multiplied by the fifth cumulative distribution probability (d 5 ). The numerical order of the sixth member (n 6 ) of the subset among the set of the numerical data corresponds to the total number of the numerical data multiplied by the sixth cumulative distribution probability (d 6 ). 
     In particular, the subset of the numerical data is found in the following way. Assuming the total number of the numerical data in the set is (k), with six cumulative distribution probabilities (d 1 ˜d 6 ), six intermediate values (i 1 ˜i 6 ) can be obtained through the following equation k·d x =i x , where (x) is an integer between 1 and 6. Each of the first to sixth members (n 1 ˜n 6 ) of the subset of numerical data is found by selecting, among the set of numerical data, a member whose numerical order is the closest integer to a corresponding one of the intermediate values (i 1 ˜i 6 ). 
     Moreover, the reference values computed in step  403  include first, second, third, fourth, fifth and sixth reference values (v 1 ˜v 6 ). The first reference value (v 1 ) is smaller than the mean (μ) of the set of numerical data by one standard deviation (σ) of the set of numerical data. The second reference value (v 2 ) is greater than the mean (μ) of the set of numerical data by one standard deviation (σ) of the set of numerical data. The third reference value (v 3 ) is smaller than the mean (μ) of the set of numerical data by two standard deviations (σ) of the set of numerical data. The fourth reference value (v 4 ) is greater than the mean (μ) of the set of numerical data by two standard deviations (σ) of the set of numerical data. The fifth reference value (v 5 ) is smaller than the mean (μ) of the set of numerical data by three standard deviations (σ) of the set of numerical data. The sixth reference value (v 6 ) is greater than the mean (μ) of the set of numerical data by three standard deviations (σ) of the set of numerical data. 
     It should be noted herein that, since the predetermined set of cumulative distribution probabilities is defined to include cumulative distribution probabilities that correspond to ranges of within integer multiples of the standard deviation smaller/greater than the mean of the Gaussian distribution, the reference values are also defined to differ from the mean (μ) by integer multiples of the standard deviation (σ) of the set of numerical data. However, the present invention also encompasses those applications where the cumulative distribution probabilities correspond to ranges whose upper limits are smaller/greater than the mean of the Gaussian distribution by values computed by multiplying non-integers by the standard deviation of the Gaussian distribution. In such cases, the reference values are also defined to differ from the mean (μ) of the set of numerical data by non-integers multiplied by the standard deviation (σ) of the set of numerical data. 
     Further, the plot  300  generated in step  404  includes seven of the connecting lines  503 , respectively connecting the median (M) to the mean (μ), and the first, second, third, fourth, fifth and sixth members (n 1 ˜n 6 ) of the subset of numerical data respectively to the first, second, third, fourth, fifth and sixth reference values (v 1 ˜v 6 ). Preferably, the connecting lines  503  that connect the median (M) to the mean (μ), the first member (n 1 ) of the subset of numerical data to the first reference value (v 1 ), and the second member (n 2 ) of the subset of numerical data to the second reference value (v 2 ) are shown in solid lines, while the connecting lines  503  that connect the third, fourth, fifth and sixth members (n 3 ˜n 6 ) of the subset of numerical data respectively to the third, fourth, fifth and sixth reference values (v 3 ˜v 6 ) are shown in dashed lines when outputted for viewing by the user. 
     Preferably, among the points marked on the first and second lines  501 ,  502 , if there is one which is numerically distant from the mean (μ) by more than twice the standard deviation (σ), it will be marked using a (⊙) symbol, and is referred to as an unusual observation when the set of numerical data has a Gaussian distribution, and if there is one which is numerically distant from the mean (μ) by more than three times the standard deviation (σ), it will be marked using a (*) symbol, and is referred to as an outlier when the set of numerical data has a Gaussian distribution. Otherwise, the points are marked using a () symbol. 
     When reading the plot  300  generated according to the present invention, the more the connecting lines  503  approach a perpendicular relationship relative to the axis direction (Y), the more likely that the set of numerical data has a Gaussian (normal) distribution. In particular, if a group of the connecting lines  503  are approximately perpendicular to the axis direction while the rest of the connecting lines  503  are not, statistical analysis and estimations based on the Gaussian distribution may be applied to values close to the points to which the connecting lines  503  of the group are connected, while statistical analysis and estimations based on the Gaussian distribution are not applicable to the values close to the points to which the rest of the connecting lines  503  are connected. 
     Alternatively, with reference to  FIG. 6 , the second preferred embodiment of an electronic device  100  for graphically illustrating a statistical display based on a set of numerical data according to the present invention differs from the first preferred embodiment in that the processor  10  includes a data selecting unit  101 , a computing unit  102  and a plot generating unit  103 . The data selecting unit  101  finds the median (M) of the set of numerical data, and further finds the subset of the numerical data, each corresponding to a member of the predetermined set of cumulative distribution probabilities of the Gaussian distribution. The computing unit  102  computes the mean (μ) and the standard deviation (σ) of the set of numerical data, and computes the plurality of reference values. Each of the reference values differs from the mean (μ) by a corresponding predetermined number multiplied by the standard deviation (σ) of the set of numerical data. The plot generating unit  103  is coupled to the data selecting unit  101  and the computing unit  102  for generating the plot  300  (shown in  FIG. 8 ). The output unit  13  is coupled to the plot generating unit  103  for outputting the plot  300  for viewing by the user. 
     The present invention will be better understood with reference to the following exemplary embodiments. 
     With reference to  FIG. 9 , the first exemplary embodiment includes two sets of numerical data, namely a first set having a total number of numerical data of (k 1 =30), and a second set having a total number of numerical data of (k 2 =68). The first and second sets of numerical data respectively include profit/earnings (P/E) ratios of stocks of 30 and 68 companies. 
     According to step  401 , the median (M) and the subset of the numerical data are found for each of the first and second sets of numerical data. As shown in  FIG. 10 , the median (M) for the first set of numerical data is denoted by n 10 (M, 0σ) and is equal to 19, and the median (M) for the second set of numerical data is denoted by n 20 (M, 0σ) and is equal to 19. With the first, second, third, fourth, fifth and sixth cumulative distribution probabilities (d 1 , d 2 , d 3 , d 4 , d 5 , d 6 ) defined as previously described, i.e., 15.87%, 84.13%, 2.28%, 97.72%, 0.135% and 99.87%, the subset of numerical data for the first set includes first, second, third, fourth, fifth and sixth members n 11 (M, −1σ), n 12 (M, 1σ), n 13 (M, −2σ), n 14 (M, 2σ), n 15 (M, −3σ), n 16 (M, 3σ) that respectively equal to 13, 29, 8, 68, 8 and 68, and the subset of numerical data for the second set includes first, second, third, fourth, fifth and sixth members n 21 (M, −1σ), n 22 (M, 1σ), n 23 (M, −2σ), n 24 (M, 2σ) n 25 (M, −3σ), n 26 (M, 3σ) that are respectively equal to 13, 31, 8, 68, 7 and 91. 
     Taking the first set of numerical data for illustration, since the total number (k 1 ) of the numerical data in the first set is 30, the median n 10 (M, 0σ) is the numerical data whose numerical order corresponds to approximately half of the total number (k 1 ), i.e., 15 or 16. In this embodiment, the median n 10 (M, 0σ) is taken to be the 15 th  numerical data in ascending order, and has the value of 19. It should be noted herein that the numerical data shown in  FIG. 9  are arranged in numerical order for easy reference. The first member n 11 (M, −1σ) of the subset is found by first obtaining the corresponding intermediate value i 11 =k 1 ·d 1 =30·15.87%=4.761, and with 5 being the closest integer to i 11 , locating the 5 th  numerical data in ascending order to be the first member n 11 (M, −1σ) of the subset. The second member n 12 (M, 1σ) of the subset is found by obtaining the corresponding intermediate value i 12 =k 1 ·d 2 =30·84.130=25.239, and with 25 being the closest integer to i 12 , locating the 25 th  numerical data in ascending order to be the second member n 12 (M, 1σ) of the subset. The rest of the members of the subset for the first set of numerical data, and members of the subset for the second set of numerical data are found in a similar fashion, and further details of the same are omitted herein for the sake of brevity. 
     According to step  402 , the mean (μ) and the standard deviation (σ) of each of the first and second sets of numerical data are computed. The mean (μ) is computed by dividing the sum of all numerical data in the set with the total number (k) of the numerical data, and the standard deviation (σ) is computed using a standard formula of 
     
       
         
           
             σ 
             = 
             
               
                 
                   
                     1 
                     
                       k 
                       - 
                       1 
                     
                   
                    
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         1 
                       
                       k 
                     
                      
                     
                       [ 
                       
                         
                           x 
                           i 
                         
                         - 
                         μ 
                       
                       ] 
                     
                   
                 
               
               . 
             
           
         
       
     
     Accordingly, for the first set of numerical data, the mean (μ1), which is denoted by v 10 (μ, 0σ) in  FIG. 11 , is computed to be 22.5, and the standard deviation (σ1) is computed as 
     
       
         
           
             
               σ 
                
               
                   
               
                
               1 
             
             = 
             
               
                 
                   
                     1 
                     
                       
                         k 
                         1 
                       
                       - 
                       1 
                     
                   
                    
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         1 
                       
                       
                         k 
                         1 
                       
                     
                      
                     
                       [ 
                       
                         
                           x 
                           i 
                         
                         - 
                         22.5 
                       
                       ] 
                     
                   
                 
               
               = 
               
                 
                   
                     
                       1 
                       29 
                     
                      
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         30 
                       
                        
                       
                         [ 
                         
                           
                             x 
                             i 
                           
                           - 
                           22.5 
                         
                         ] 
                       
                     
                   
                 
                 = 
                 
                   13.5564 
                   . 
                 
               
             
           
         
       
     
     In addition, for the second set of numerical data, the mean (μ2), which is denoted by v 20 (μ, 0σ) in  FIG. 11 , is computed to be 22.72, while the standard deviation (σ2) is computed as 
     
       
         
           
             
               σ 
                
               
                   
               
                
               2 
             
             = 
             
               
                 
                   
                     1 
                     
                       
                         k 
                         2 
                       
                       - 
                       1 
                     
                   
                    
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         1 
                       
                       
                         k 
                         2 
                       
                     
                      
                     
                       [ 
                       
                         
                           x 
                           i 
                         
                         - 
                         22.72 
                       
                       ] 
                     
                   
                 
               
               = 
               
                 
                   
                     
                       1 
                       67 
                     
                      
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         68 
                       
                        
                       
                         [ 
                         
                           
                             x 
                             i 
                           
                           - 
                           22.72 
                         
                         ] 
                       
                     
                   
                 
                 = 
                 
                   14.0827 
                   . 
                 
               
             
           
         
       
     
     According to step  403 , the plurality of reference values are computed for each of the first and second sets of numerical data. As described earlier, the plurality of reference values for the first set of numerical data include the first, second, third, fourth, fifth and sixth reference values v 11 (μ, −1σ), v 12 (μ, 1σ), v 13 (μ, −2σ), v 14 (μ, 2σ), v 15 (μ, −3σ), v 16 (μ, 3σ) that are respectively equal to 8.9, 36.1, −4.6, 49.6, −18.2 and 63.2, and the plurality of reference values for the second set of numerical data include the first, second, third, fourth, fifth and sixth reference values v 21 (μ, −1σ), v 22 (μ, 1σ), v 23 (μ, −2σ), v 24 (μ, 2σ), v 25 (μ, −3σ), v 26 (μ, 3σ) that are respectively equal to 8.64, 36.8, −5.44, 50.88, −19.52 and 64.96. 
     Taking the first set of numerical data for illustration, the first reference value v 11 (μ, −1σ) is computed as μ−1σ=22.5-13.5564=8.9, the second reference value v 12 (μ, 1σ) is computed as μ+1σ=22.5+13.5564=36.1, the third reference value v 13 (μ, −2σ) is computed as μ−2σ=22.5−2×13.5564=−4.6, the fourth reference value v 14 (μ, 2σ) is computed as μ+2σ=22.5+2×13.5564=49.6, the fifth reference value v 15 (μ, −3σ) is computed as μ−3σ=22.5−3×13.5564=−18.2, and the sixth reference value v 16 (μ, 3σ) is computed as μ+3σ=22.5+3×13.5564=63.2. The reference values for the second set of numerical data are found in a similar fashion, and further details of the same are omitted herein for the sake of brevity. 
     According to step  404 , the plot  300   a  shown in  FIG. 12  is generated. For this particular exemplary embodiment, since there are two sets of numerical data, the plot  300   a  includes two sets of first, second and connecting lines  501   a   1 ,  501   a   2 ,  502   a   1 ,  502   a   2 ,  503   a   1 ,  503   a   2 . On the first line  501   a   1  corresponding to the first set of numerical data, there are marked the median n 10 , and the members of the subset of numerical data n 11 , n 12 , n 13 , n 14 , n 15 , n 16 . On the second line  502   a   1  corresponding to the first set of numerical data, there are marked the mean v 10 , and the reference values v 11 , v 12 , v 13 , v 14 , v 15 , v 16 . Three solid connecting lines  503   a   1  respectively connect the median n 10  to the mean v 10 , the first member n 11  of the subset to the first reference value v 11 , and the second member of the subset n 12  to the second reference value v 12 . Four dashed connecting lines  503   a   1  respectively connect the third member of the subset n 13  to the third reference value v 13 , the fourth member of the subset n 14  to the fourth reference value v 14 , the fifth member of the subset n 15  to the fifth reference value v 15 , and the sixth member of the subset n 16  to the sixth reference value v 16 . Similarly, on the first line  501   a   2  corresponding to the second set of numerical data, there are marked the median n 20 , and the members of the subset of numerical data n 21 , n 22 , n 23 , n 24 , n 25 , n 26 . On the second line  502   a   2  corresponding to the second set of numerical data, there are marked the mean v 20 , and the reference values v 21 , v 22 , v 23 , v 24 , v 25 , v 26 . Three solid connecting lines  503   a   2  respectively connect the median n 20  to the mean v 20 , the first member n 21  of the subset to the first reference value v 21 , and the second member of the subset n 22  to the second reference value v 22 . Four dashed connecting lines  503   a   2  respectively connect the third member of the subset n 23  to the third reference value v 23 , the fourth member of the subset n 24  to the fourth reference value v 24 , the fifth member of the subset n 25  to the fifth reference value v 25 , and the sixth member of the subset n 26  to the sixth reference value v 26 . 
     It is noted herein that since n 14  and n 16  are greater than v 16 , i.e., that n 14  and n 16  are numerically distant from the mean (μ1) by more than three times the standard deviation (σ1), these two points are considered outliers if the first set of numerical data has a Gaussian distribution and are marked using the ( ) symbol, and since n 24  and n 26  are greater than v 26 , i.e., that n 24  and n 26  are numerically distant from the mean (μ2) by more than three times the standard deviation (σ2), these two points are considered outliers if the second set of numerical data has a Gaussian distribution and are marked using the ( ) symbol. 
     As is evident from  FIG. 12 , the means v 10 , v 20  and the standard deviations (σ1, σ2) of the first and second sets of numerical data are approximately the same, and the distributions of the first and second sets of numerical data are also approximately the same. In particular, none of the first and second sets of numerical data is normally (Gaussian) distributed. Instead, the first and second sets of numerical data are positively-skewed with the long tail on the positive side. 
     In the alternative, the second preferred embodiment of a machine-implemented method for graphically illustrating a statistical display based on a set of numerical data according to the present invention differs from the first preferred embodiment in that the plot  300  is presented in a logarithmic scale. The machine-implemented method of the second preferred embodiment further includes, prior to step  401 , step  400 , where, with the processor  10 , natural logarithms ( 1   n ) of a set of source numerical data are taken so as to generate the set of numerical data used in the subsequent steps. Alternatively, in the absence of step  400 , the set of numerical data may be a natural logarithmic equivalent of a set of source numerical data. This is especially useful when the set of source numerical data involve financial stats, such as P/E ratios, or for applications in the analysis of operational risks (e.g., key risk indicator (KRI)) and investments. 
     Accordingly, with reference to  FIG. 13 , the second exemplary embodiment includes two sets of numerical data that are respectively natural logarithmic equivalents of the first and second sets of numerical data used in the previous exemplary embodiment. In other words, the first and second sets of numerical data used in the first exemplary embodiment are respectively first and second sets of source numerical data for the second exemplary embodiment. Shown in  FIG. 14  are the medians n 10 ′, n 20 ′ and the members of the subsets of numerical data n 11 ′, n 12 ′, n 13 ′, n 14 ′, n 15 ′, n 16 ′, n 21 ′, n 22 ′, n 23 ′, n 24 ′, n 25 ′, n 26 ′ for the first and second sets of numerical data shown in  FIG. 13  as found in the manner described with reference to the first exemplary embodiment. Shown in  FIG. 15  are the means v 10 ′, v 20 ′ and the reference values v 11 ′, v 12 ′, v 13 ′, v 14 ′, v 15 ′, v 16 ′, v 21 ′, v 22 ′v 23 ′, v 24 ′, v 25 ′, v 26 ′ for the first and second sets of numerical data shown in  FIG. 13  as computed in the manner described with reference to the first exemplary embodiment. 
     It is noted herein that since n 14 ′ and n 16 ′ are greater than v 14 ′, i.e., that n 14 ′ and n 16 ′ are numerically distant from the mean v 10 ′ by more than twice the standard deviation, these two points are considered unusual observations if the first set of numerical data has a Gaussian distribution and are marked using the (⊙) symbol. For a similar reason, n 24 ′ is considered an unusual observation if the second set of numerical data has a Gaussian distribution, and is marked using the (⊙) symbol. Moreover, since n 26 ′ is greater than v 26 ′, i.e., n 26 ′ is numerically distant from the mean v 20 ′ by more than three times the standard deviation, this point is considered an outlier if the second set of numerical data has a Gaussian distribution and is marked using the ( ) symbol. 
     The plot  300   b  shown in  FIG. 16  is generated for the second exemplary embodiment, where there are two sets of first, second and connecting lines  501   b   1 ,  501   b   2 ,  502   b   1 ,  502   b   2 ,  503   b   1 ,  503   b   2 . On the first line  501   b   1  corresponding to the first set of numerical data, there are marked the median n 10 ′, and the members of the subset of numerical data n 11 ′, n 12 ′, n n ′, n 14 ′, n 15 ′, n 16 ′. On the second line  502   b   1  corresponding to the first set of numerical data, there are marked the mean v 10 ′, and the reference values v 11 ′, v 12 ′, v 13 ′, v 14 ′, v 15 ′, v 16 ′. Three solid connecting lines  503   b   1  respectively connect the median n 10 ′ to the mean v 10 ′, the first member n 11 ′ of the subset to the first reference value v 11 ′, and the second member of the subset n 12 ′ to the second reference value v 12 ′. Four dashed connecting lines  503   b   1  respectively connect the third member of the subset n 13 ′ to the third reference value v 13 ′, the fourth member of the subset n 14 ′ to the fourth reference value v 14 ′, the fifth member of the subset n 15 ′ to the fifth reference value v 15 ′, and the sixth member of the subset n 16 ′ to the sixth reference value v 16 ′. Similarly, on the first line  501   b   2  corresponding to the second set of numerical data, there are marked the median n 20 ′, and the members of the subset of numerical data n 21 ′, n 22 ′, n 23 ′, n 24 ′, n 25 ′, n 26 ′. On the second line  502   b   2  corresponding to the second set of numerical data, there are marked the mean v 20 ′, and the reference values v 21 ′, v 22 ′, v 23 ′, v 24 ′, v 25 ′, v 26 ′. Three solid connecting lines  503   b   2  respectively connect the median n 20 ′ to the mean v 20 ′, the first member n 21 ′ of the subset to the first reference value v 21 ′, and the second member of the subset n 22 ′ to the second reference value v 22 ′. Four dashed connecting lines  503   b   2  respectively connect the third member of the subset n 23 ′ to the third reference value v 23 ′, the fourth member of the subset n 24 ′ to the fourth reference value v 24 ′, the fifth member of the subset n 25 ′ to the fifth reference value v 25 ′, and the sixth member of the subset n 26 ′ to the sixth reference value v 26 ′. 
     As is evident in  FIG. 16 , the solid connecting lines  503   b   1  corresponding to the first set of numerical data and respectively connecting the median n 10 ′ to the mean v 10 ′, the first member of the subset of numerical data n 11 ′ to the first reference value v 11 ′, and the second member of the subset of numerical data n 12 ′ to the second reference value v 12   1 , as well as the dashed connecting line  503   b   1  corresponding to the first set of numerical data and connecting the third member of the subset of numerical data n 13 ′ to the third reference value v 13 ′ direction (Y). Therefore, in the range between one standard deviation greater than the mean v 10 ′ and two standard deviations smaller than the mean v 10 ′, the first set of numerical data exhibits a distribution proximate to the Gaussian distribution. Under the Gaussian distribution, this range encompasses roughly 82% of the distribution probability. Since the first set of numerical data is the natural logarithmic equivalent of the first set of source numerical data (i.e., the first set of numerical data for the first exemplary embodiment), the first set of source numerical data may be said to have a lognormal distribution in the range. A similar observation may be found for the second set of numerical data with reference to  FIG. 16 . 
     With reference to  FIG. 17 , in the third exemplary embodiment of the present invention, the plot  300   c  is obtained by directly plotting exponentials of the medians n 10 ′, n 20 ′, the members of the subsets of numerical data n 11 ′, n 12 ′, n 13 ′, n 14 ′, n 15 ′, n 16 ′, n 21 ′, n 22 ′, n 23 ′, n 24 ′, n 25 ′, n 26 ′, the means v 10 ′, v 20 ′, and the reference values v 11 ′, v 12 ′, v 13 ′, v 14 ′, v 15 ′, v 16 ′, v 21 ′, v 22 ′, v 23 ′, v 24 ′, v 25 ′, v 26 ′ for the first and second sets of numerical data of the second exemplary embodiment. Therefore, points e n     10   ′, e n     11   ′, e n     12   ′, e n     13   ′, e n     14   ′, e n     15   ′, and e n     16   ′ are marked on the first line  501   c   1 , and points e v     10   ′, e v     11   ′, e v     12   ′, e v     13   ′, e v     14   ′, e v     15   ′ and e v     16   ′ are marked on the second line  502   c   1  corresponding to the first set of numerical data, and points e n     20   ′, e n     21   ′, e n     22   ′, e n     23   ′, e n     24   ′, e n     25   ′ and e n     26   ′ are marked on the first line  501   c   2 , endpoints e v     20   ′, e v     21   ′, e v     22   ′, e v     23   ′, e v     24   ′, e v     25   ′ and e v     26   ′ are marked on the second line  502   c   2  corresponding to the second set of numerical data. 
     Since and e n     14   ′ and e n     16   ′ are greater than e v ′, these two points are marked using the (⊙) symbol. For a similar reason, e n     24   ′ is also marked using the (⊙) symbol. Moreover, since e n     26   ′ is greater than e v     26   ′, this point is marked using the ( ) symbol. 
     As is evident in  FIG. 17 , the second to fifth connecting lines  503   c   1  counting from the bottom of  FIG. 17  and corresponding to the first set of numerical data are all approximately perpendicular to the axis direction (Y). Therefore, in the range between e n     13   ′ and e n     12   ′, all critical points of the first set of numerical data exhibits an exponential distribution, which encompasses roughly 82% of the lognormal distribution probability. A similar observation may be found for the second set of numerical data with reference to  FIG. 17 . 
     With reference to  FIGS. 12-17 , the third exemplary embodiment in essence brings the critical values (i.e., median, members of the subset, mean, reference values) of each of the first and second sets of numerical data from the natural logarithmic scale (as shown in  FIG. 16 ) back into the scale shown in  FIG. 12 . However,  FIG. 17  differs from  FIG. 12  in that the critical values marked on the plot  300   a  of  FIG. 12  are determined using the data shown in  FIG. 9 , while the critical values marked on the plot  300   c  of  FIG. 17  are determined using the data shown in  FIG. 13  and converted back into the scale of  FIG. 12 . This technique is useful when, as in  FIG. 12 , the distributions of the data are clearly not Gaussian distributions and cannot be analyzed using statistical tools, but a range of values might be found to correspond to log normal distribution and can be analyzed using statistical tools in the logarithmic scale. 
     It should be noted herein that although the above exemplary embodiments are presented as applications in investment and finance, the present invention is not limited to such applications, and can be used for analyzing numerical data of any nature. It should also be noted herein that the present invention is not limited to the degree of approximations taken for the determinations of the median, the subset of numerical data, the mean, the standard deviations, and the reference values. 
     In summary, the present invention provides a machine-implemented method and an electronic device for graphically illustrating a statistical display based on a set of numerical data, and a computer program product for implementing the method, where the statistical display is easy to read, requires little display resources (as compared to dot plots and histograms, especially when the data is large in quantity), and is capable of providing objective meanings of extreme observations (as compared to bar plots). 
     While the present invention has been described in connection with what are considered the most practical and preferred embodiments, it is understood that this invention is not limited to the disclosed embodiments but is intended to cover various arrangements included within the spirit and scope of the broadest interpretation so as to encompass all such modifications and equivalent arrangements.