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Data   collection Data   collection   is   defined   as   the   procedure   of   collecting,   measuring   and   analyzing   accurate   insights   for    research   using   standard   validated   techniques. A   researcher   can   evaluate   their   hypothesis   on   the   basis   of   collected   data.   Data   collection   is   the   primary   and    most important step  for   research,   irrespective  of  the   field  of  research. The   most   critical   objective   of   data   collection   is   ensuring   that   information-rich   and   reliable   data   is   collected   so    that   statistical analysis data-driven   decisions  can be  made   for  research. Data   Collection   Methods:   Phone   vs.   Online   vs.   In-Person   Interviews Essentially   there   are   four   choices   for   data   collection   –   in-person   interviews,   mail,   phone   and   online.   There   are    pros   and cons  to  each of  these   modes. 1.	 In-Person   Interviews Pros: In-depth   and a   high  degree   of  confidence   on  the   data Cons:   Time   consuming,   expensive   and   can   be   dismissed  as   anecdotal

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Mail   Surveys Pros:   Can  reach   anyone   and   everyone   –   no  barrier    Cons: Expensive,   data   collection   errors,   lag   time Phone   Surveys Pros: High   degree   of   confidence   in   the   data   collected,   reach   almost  anyone    Cons: Expensive,  cannot   self-administer,  need  to   hire  an agency Web/Online   Surveys Pros: Cheap,   can   self-administer,   very   low   probability   of   data   errors Cons:  Not  all   your   customers   might  have   an   email   address/be   on   the   internet,   customers   may   be   wary   of   divulging    information online. An  example  for   data   collection   is shown below Let’s   consider   an   example   of   a   mobile   manufacturer,   company   X,   which   is   launching   a   new   product  variant.   To    conduct   research   about   features,   price   range,   target   market,   competitor   analysis   etc,   data   has   to   be   collected   from    appropriate   sources.   The   marketing   team  can   conduct  various   data   collection   activities   such   as   online surveys   or   focus   groups. Data   collection   methods   are   chosen   depending   on   the   available   resources.   For   example,   conducting   questionnaires    and   surveys   would   require   the   least   resources   while   focus groups  require   moderately   high   resources.

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Organization   of   data   means   classification,   tabulation,   graphical   presentation   and   diagrammatic   presentation    of  data. The   methods   that   we   use   to   organize   data   include   classification,   tabulation,   graphical   presentation   and    diagrammatic presentation. Classification  -   Classification  of  data  refers  to categorization  of  data.  It  includes the summary  of  the   frequency of  individual scores  or ranges of  scores  for a  variable. Data is  grouped on  the basis  of  their    similarities. Tabulation  -  Tabulation  of  data  refers  to  a way  to systematically  arrange  the data in  rows and  columns.  The    objective is to simplify the presentation  and  to facilitate comparisons keeping in view the objectives  of  the    study. 	 Graphical   presentation   -   Data   is   plotted   on   a   pictorial   platform   formed   of   horizontal   and   vertical   lines.   The    purpose   is   to   provide  a   systematic   way   of  “looking   at”   and   understanding  of   the   data.   Graphs   can   be    polygon,  chart   or  diagram Diagrammatic   presentation   -   It   is   used   to   present   statistical   data   in   simple,   readily   comprehensible   form.    Diagrammatic   presentation   is   a   different   form   from   graphica   used   only   for   presentation   of   the   data   in   visual    form.   There   are  different   forms  of  diagrams  e.g.,   Bar   diagram,   Sub-divided   bar  diagram,   Multiple   bar    diagram,   Pie diagram  and  Pictogram.

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Data   representations   are   graphics   that   display   and   summarize   data   and   help   us   to   understand   the    data's   meaning. Data   representations   can   help   us   answer   the   following   questions: How  much   of   the   data   falls   within  a   specified   category   or   range   of  values? What   is  a  typical value  of  the data? How  much spread   is   in   the data? Is   there  a  trend   in the   data  over  time? Is  there   a   relationship   between   two   variables? What   skills  are   tested? Matching   a   data   set   to   its   graphical   representation Matching   a   graphical   representation   to   a   description Using   data   representations   to   solve   problems

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Mean   :   The   arithmetic   mean   of   a   given   data   is   the   sum  of   all   observations   divided   by   the   number   of    observations.   For   example:   A   cricketer's   scores   in   five   ODI   matches   are   as   follows:  12,   34,   45,   50,   24. To   find   his  average   score   in   a   match,   we   calculate   the   arithmetic   mean   of   data   using   the   mean   formula:

Mean  =  (12+34+45+50+24)/5   =165/5   =33 Mean is   denoted  by   ¯x   (pronounced   as   x   bar) Data  can   be  present   in  raw form   or   tabular   form . Let's find   the   mean   in   both   cases.

Mean   =   Sum   of   all   observations   /   Number   of    observations

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1)   Raw   form Let   x1,   x2,   x3   ……xn   be   n   observations. We  can   find   the   arithmetic   mean   using   the   mean   formula Mean,   ¯x   =   (x1+x2+x3…….xn)/n Example   1 If  the   heights  of 5  people  are 142  cm,  150  cm,  149  cm,  156  cm,  and 153 cm.    Find   the   mean height. Mean   height   ,   ¯x=142+150+149+156+153=750/5=150 2)  Frequency   Distribution   (Tabular)  Form When   the   data   is   present   in   tabular   form,   we   use   the   following   formula: Mean,   ¯x=  (x1f1+x2f2+....xnfn)   /  (f1+f2+.....fn  )

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Consider the   following   example. Example   1 Find the   mean  of   the   following   distribution:

Solution Calculation table  for  arithmetic   mean:

x i f i x i f i 4 5 20 6 10 60 9 10 90 10 7 70 15 8 120 ∑fi=40∑fi=40 ∑xifi=360∑xifi=360

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∴   Mean=¯x=∑xifi/∑fi=360/40=9    Thus, Mean  =   9

Number  of   patients Number   of   days   visiting  hospital 0-10 2 10-20 6 20-30 9 30-40 7 40-50 4 50-60 2

Example   2 Here  is  an   example  where  the   data   is in   the  form of   class   intervals. The   following   table   indicates   the   data   on   the   number   of   patients   visiting   a   hospital   in  a   month.    Find the  average  number  of  patients   visiting   the   hospital in  a day .

Solution In   this case,   we   find   the   class mark   (also   called   as   mid-point  of   a   class)   for   each   class. Note:   Class mark  =lower  limit  + upper  limit2 Let   x1,   x2,   x3   ……xn   be   the   class   marks  of   the   respective   classes. Hence,   we   get  the   following  table:

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Class   mark   (x i ) frequency   (f i ) x i f i 5 2 10 15 6 90 25 9 225 35 7 245 45 4 180 55 2 110 Total ∑fi=30 ∑fixi=860

∴ Mean=¯x=∑xifi/∑fi=860/30=28.67 Median The   value   of   the   middlemost   observation,   obtained   after   arranging   the   data   in   ascending   order,   is   called    the median  of  the   data. For   example,   consider   the   data:  4, 4, 6, 3, 2. Let's arrange  this data in ascending  order: 2, 3, 4, 4, 6.    There   are 5  observations. Thus,   median  =  middle value   i.e.  4 We   can  see  here:   2,   3,  4, 4   , 6   (Thus,  4  is the   median)

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Step   1:   Arrange   the   data   in   ascending   or  descending   order.    Step   2:   Let  the  total   number   of   observations  be   n. To  find   the   median,   we   need   to   consider if   n   is   even   or odd. If  n  is   odd ,  then  use   the   formula: Median  = [ ( n+1)/2] th  observation Example   1 Let's   consider   the  data:   56, 67,   54, 34, 78, 43, 23.    What is  the   median?

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Solution Arranging in   ascending   order,  we get:   23, 34, 43, 54, 56, 67, 78.    Here,   n (no of  observations)  =   7 So,   ( 7+1 )/ 2=4	 ∴   Median  =   4th observation Median   =   54 If   n  is even , then   use   the   formula: Median   =   n2th   obs   +(n2+1)th   obs Example   2 Let's  consider the data:  50, 67, 24, 34, 78, 43.    What  is the   median?

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Solution Arranging  in ascending  order,  we  get: 24, 34, 43, 50, 67, 78.    Here,  n (no.of observations)   =   6 Using   the   median   formula, Median   =   (3 rd   obs.+4 th   obs)/2   =   (43+50)/2

Case   2:   Grouped   Data When the data is continuous   and in the form   of   a frequency distribution, the median   is found as  shown   below:    Step   1:   Find the median class. Let  n = total number of observations i.e.  ∑f    Note:  Median   Class   is   the   class  where  n2   lies. Step   2:  Use  the   following   formula   to   find   the   median.

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Median   =   l+[n2−cf]×h where, l   =   lower limit   of  median class c =   cumulative  frequency   of   the   class   preceding   the   median   class    f   =   frequency of  the median   class h   =   class   size Let's   consider   the   following   example   to   understand   this   better. Example   1 Find   the   median   marks  for   the   following   distribution:

Classes 0-10 10-20 20-30 30-40 40-50 Frequency 2 12 22 8 6

Solution We   need   to   calculate   the   cumulative   frequencies   to   find   the    median. Calculation   table:

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Classes Number   of   students Cumulative   frequency 0-10 2 2 10-20 12 2   +   12   =   14 20-30 22 14   +   22   =   36 30-40 8 36   +   8   =   44 40-50 6 44   +   6   =   50

N=50    N2=50/2=25 Median  Class  =20−30    l =20,f=22,c . f=14, h =10 Using   Median  formula: Median  =l+[n2−cf]×h =20+(25−14)/22 × 10=20+11/22 × 10=20+5=25 ∴∴   Median  =   25

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Mode The   value   which   appears  most   often   in   the   given   data   i.e.   the   observation   with   the   highest   frequency   is   called    a   mode   of  data. Case   1:  Ungrouped   Data For  ungrouped   data,   we   just  need   to   identify   the   observation   which   occurs  maximum   times. Mode  =  Observation   with   maximum   frequency For   example   in   the   data:  6, 8,   9, 3, 4,   6, 7,   6, 3   the   value  6   appears  the   most number   of  times. Thus,   mode   =   6. An  easy   way   to   remember   mode   is:   M ost   O ften   D ata   E ntered. Note:   A   data   may  have   no  mode,   1  mode,   or  more   than  1   mode. Depending   upon   the   number   of   modes   the   data   has,   it   can   be   called   unimodal,   bimodal,   trimodal,   or   multimodal. The  example   discussed   above has  only  1   mode,   so   it is   unimodal.

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Bimodal List
List
Mode IAI =
List A has 2 modes.
Therefore, it is a bimodal list.
Trimodal List
List B =
Mode [Bl =
List B has 3 modes.
Therefore, it is a trimodal list.

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Use   of   Graphs Uses   of   graph Graphs and   charts  are   very   useful   because   they   are   used   for   communicating   information   in   visual   form. These   are   used   in   all   fields   like   in   newspapers,   businesses,   magazines,   to   show   the   economic   data,   annual    reports, profits  and  loss These   are   also   used   in   maths  and   all   other   branches of   science   like   chemistry   and   Physics   to   show   the    variations  and  relation   between   two   variables. The graph   is   easily   understood   by   everyone   without  any  prior   knowledge. Graph   saves   time The graph   enables  us  to   relate   and  compare   the   data  for   different   time   periods The graph   is   used   in   statistics   to   determine   the   mean,   median   and   mode   for   different data,   as  well  as  in   the    interpolation  and  the   extrapolation  of  data. When   some   complicated   information   is  appeared   to   problematic   and   requires  a   design   then   a   graph   is   the   most    suitable   idea   to   represent   them.  It  shows   all features quickly   and  visually .

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Type   of   Graphs: Bar   Graph A bar chart is a graph represented by spaced rectangular bars that describe the data points in a set of data. It is usually used to    plot discrete and categorical   data. The  horizontal axis of the chart represents categorical data while the vertical axis of the chart defines discrete data. Although the    rectangular   bars in a bar chart   are mostly placed  vertically,  they can also be horizontal. Types   of   Bar   Graph Grouped   Bar   Chart Grouped  bar charts are used  when  the datasets   have subgroups that   need to be visualized   on the graph.   Each  subgroup is usually    differentiated  from   the other by shading them   with distinct   colors. Stacked   Bar   Chart The  stacked bar graphs are also used to show subgroups in a dataset.  But  in this case, the rectangular bars defining each group    are   stacked on top of each  other. Segmented   Bar   Chart This is the type of stacked   bar chart   where  each stacked bar   shows  the percentage of its discrete   value from   the total   value.   The    total percentage is 100%

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Advantages   of   a  Bar   Chart Summarizes  a   large   amount  of   data   in   an   understandable   form. Easily   accessible   to  a  wide   audience. Disadvantages   of   a  Bar   Chart It   does not reveal key  assumptions   like   causes,   effects,   patterns,   etc. May   require   further   explanation. Pie   Chart A  pie  chart  is  a  circular  graph  used to illustrate numerical proportions in  a  dataset. This  graph  is usually    divided   into   various   sectors,   where   each   sector   represents   the   proportion   of   a   particular   numerical   element   in    the set. Just   like   a   pizza   is   divided   into   different   slices,   each   sector   in   a   pie   chart  represents   the   proportion   of   an    element   in   the   dataset.   The   proportion   is   defined   by   the   degree   of   the   sector   and   the   percentage   area   with    respect   to   the  area of  the   circle.

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Simple   Pie   Chart This is   the   most   basic   type   of   pie  chart and can   also   be  simply   called   a  pie   chart. Exploded   Pie   Chart In   an   exploded   pie   chart,   one   of   the   sectors   of   the   circle   is   separated   (or   exploded)   from  the   chart.   It   is   used   to    lay   emphasis  on a  particular   element in   the   data   set. Pie   of   Pie As   the   name   suggests,   a   pie   of   pie   is  a   chart  that   generates  an   entirely   new   (usually   small)   pie   chart from  the    existing  one.   It   can   be  used   to  reduce   clutteredness  and   lay   emphasis  on a   particular  group   of  elements. Bar   of   Pie This   is   similar   to   the   pie   of   pie,   with   the   main   difference   being   that  a   bar   chart  is  what  is   generated   in   this   case    rather than  a  pie   chart. 3D   Pie   Chart This   is  a   type   of   pie   chart  that   is   represented   in   a   3-dimensional   space.

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Uses   of   Pie   Chart It summarizes   data   into   a   visually   appealing   form. It   is   quite   simple   compared   to   many   graph   types. Disadvantages   of   Pie   Chart It   is   inapplicable   for   large   datasets. It   cannot   visualize   groups   of   data. Line   Graph Line graphs are represented by a group of data points joined together by a straight   line.  Each  of these data points describes the    relationship  between  the horizontal   and the vertical   axis on the graph. The  graph may ascend, descend, or do both depending on  what  kind of data is being visualized. When studying the    relationship  between  price and  supply,  it   goes  down  and for peace and demand, it   goes up. Types   of   Line   Graph Simple   Line   Graph In a simple line graph, only one line is plotted on the   graph.  One  of the axes defines the independent   variables while the   other    axis   contains dependent   variables.

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Multiple   Line   Graph Multiple line  graphs   contain   two   or  more   lines   representing   more   than   one  variable   in   a   dataset.   This   type   of  graph    can  be  used  to study  two  or  more variables   over  the  same period  of  time. Compound   Line   Graph A   compound   line   graph   is   an   extension   of the   simple line   graph,   which   is   used   when   dealing   with   different   groups   of    data   from   a  larger   dataset.   Each   line in   a  compound   line  graph   is   shaded   downwards   to   the  x-axis.   In   a  compound   line    graph,   each   group  of  data represented  by a   simple   line  graph   is  stacked  upon one   another. Uses   of   a   Line   Graph It   helps  in studying   data trends   over  a  period  of  time. They   are   easy  to   read   and  plot. Disadvantages   of   a   Line   Graph It   can   only   be  used   to   visualize data   over  a  short   period   of  time. It  is   not   convenient   to   plot  when   dealing   with   fractions   and   decimals

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Histogram Histogram   chart   visualizes   the   frequency   of   discrete   and   continuous   data   in   a   dataset   using   joined   rectangular   bars.   Each    rectangular bar defines   the  number  of  elements   that   fall  into   a  predefined   class   interval. Types   of   Histogram   Chart The   histogram   chart   is   classified   into   different   parts   depending   on   their   distribution Normal   Distribution A   normally   distributed   histogram   chart   is   usually   bell-shaped.   As   the   name   suggests,   this   distribution   is   normal   and   is    the  standard   for  how   a  normal   histogram   chart  should look  like. Bimodal   Distribution In   a   bimodally   distributed   histogram   chart,   we   have   two   groups   of   histogram   charts   that   are   of   normal   distribution.   It    is  formed   because  of  combining   two   processes  in a  dataset. Skewed   Distribution This   is   an   asymmetric   graph   with   an   off-centre   pick   usually   tending   towards   the   end   of   the   graph.   A   histogram   chart    can  be  said  to   be  right  or  left-skewed   depending  on   the  direction   where  the  peak   tends   towards. Random   Distribution This   type of  histogram   chart   does   not   have  a   regular   pattern.   It   produces   multiple  peaks   and   can   also   be  called   a    multimodal   distribution.

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Uses   of   Histogram   Chart It   helps   in  visualizing   large   amounts  of  data. It   reveals   the  variation,   centering,   and   distribution   of   the  data. Cons It   does  not  visualize  the  exact   values  in a  dataset. It  only  visualizes   continuous   data. Uses   of   Graphs   &   Chart In   Mathematics   and   Statistics Graphs   and   charts   are   used   in   analytical   geometry   to   map   out   the  functions   of   two   or  more   variables   in   a  cartesian    coordinate.   They   are   also   used   to   determine  the  correlation   and   regression   of a  statistical   dataset. In   Data   Interpretation Charts   and   graphs   are   mainly   used   in   data   interpretation   to   make   sense   of   a   dataset.   The   charts   summarize   the    information  in a  dataset,   therefore,   easing  the  data   interpretation   process.

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Chances Chance   is   the   occurrence   of   events   in   the   absence   of   any   obvious   intention   or   cause. Probability Probability   means   possibility.   It  is  a   branch   of   mathematics   that   deals   with   the   occurrence   of   a   random  event.    The  value   is   expressed   from zero  to   one.   Probability   has been  introduced   in   Maths   to   predict  how   likely    events   are  to  happen.

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1.   Trials  -   A   trial   is  a   single   performance   of   well-defined   experiment,   such   as   the   flipping   of   a   coin,   the   generation    of a   random   number,   the   dropping  of a  ball  down  the  apex of a   triangular   lattice   and   having   it   fall   into   a   single   bin   at  the   bottom,   etc.

Experiment   -   A   fixed   number   of   repetitions   of   the   same   experiment   can   be   thought  of   as   a   composed    experiment,   in   which   case   the   individual repetitions  are   called   trials. Random   experiment   -   An   experiment   is   random  if   it   is   repeated   in   the   same   manner   every   time   is   called    random   experiments.

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Outcomes  -   An   outcome   is   a   possible   result  of   an   experiment   or   trial.   Each   possible   outcome   of   a   particular    experiment is unique,  and  different outcomes.

Events Elementary   events   -   elementary   events   are   what   one   calls   the   results   of   random   experiments.   When   throwing   a    six-sided dice. Compound   events   -   A   compound   event   involves   finding   the   probability   of   more   than   one   event   occurring   at   the    same time.

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Occurrence  of   an  event   -   Probability   of occurrence   explores   the   likelihood   that  an  identified   risk   could    occur.   Probability   of   occurrence   uses  a   rating   and   value   scale   ranging   from   Not  Present  (0)   to   Almost   Certain    to   Certain.   Example:   The  probability   of an occurrence  happening   would  be   low. Negation  of   an  event   -   Calculates   the   probability   of   the   occurrence   of   an   event  given   the   probability   of   its    non-occurrence   (or  the   reverse). Types   of   probability Experimental probability  -  Experimental   probability,   also  known   as Empirical  probability,   is based  on    actual  experiments   and   adequate recordings  of   the   happening   of  events.   To   determine the   occurrence  of    any   event,  a   series  of   actual  experiments are   conducted.   Experiments which  do not  have   a   fixed   result    are  known  as random experiments.  The  outcome  of  such experiments is uncertain. Random experiments    are  repeated   multiple   times to   determine   their   likelihood. An  experiment is  repeated   a   fixed   number   of  times and   each   repetition   is  known   as  a   trial.   Mathematically,    the   formula   for  the   experimental probability   is defined  by; Probability   of   an  Event   P(E)   =  Number   of   times  an   event   occurs  /   Total   number   of   trials.

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Theoretical   probability   -   Theoretical   probability   is   the   theory   behind   probability.   To   find   the   probability   of   an    event  using   theoretical   probability,   it   is  not   required   to   conduct   an   experiment.   Instead   of   that,   we   should   know    about  the   situation   to   find   the   probability   of   an   event   occurring.   The   theoretical   probability   is   defined   as  the    ratio  of  the   number  of  favourable   outcomes to   the   number  of   possible   outcomes. Probability   of   Event   P(E)   =   No.   of.   Favourable   outcomes/   No.   of.   Possible   outcomes Basic   Probability   Formulas Probability   Range	 :	 0   ≤   P (A)   ≤   1 Rule   of   Complementary   Events .   :	 P (A C )   +   P (A)   =   1    Rule   of   Addition .   :	 P (A ? B)   =   P (A)   +   P (B)   -   P (A∩B)

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Disjoint   Events   -   Events   A   and   B   are   disjoint   if	 :	 P (A∩B)   =   0 Conditional   Probability	 :	 P (A   |   B)   =   P (A∩B)   /   P (B) Bayes   Formula	 :	 P (A   |   B)  =   P (B   |   A)   ⋅   P (A)   /   P (B) Independent   Events   -   Events   A   and   B  are   independent   if.   :	P(A∩B)   =   P(A)   ⋅   P(B) Cumulative   Distribution   Function	 :	FX(x)   =   P(X   ≤   x) Probability   Mass   Function	 : Probability  Density   Function   :

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Use  of  cards,   dice   and   coins  to  find  probability. In   the   probability   part of   the   course,   I   have   based   many   of   my   examples  on   various  real  world   objects.   It    struck   me,   as a   result  of   a   question   from   a   student,   that   it   might  be   helpful   to   some  of   you   if   I   were   to   explain    these,  and  that is the   purpose  of  this  page. Coins When   you   toss   a   coin,   you   flip   it   up   into   the   air   in   such   a   way   that   it   is   spinning   rapidly   while   in   flight.   You    then   catch   it  and   place   it  on   the  back   of your   hand.   Being   disc-shaped,   it  can   have   one of   two  faces  uppermost.    Conventionally,   the   two   sides  are   known   as  "heads"   (the   portrait  of   the   Queen)   and  "tails"   (the   other   side).   When    we   use   coin   tossing   in   probability   examples,   we   make   the   assumption   that  heads   and   tails  are   equally   likely,   that   is, P(heads)   =   P(tails)   =   1/2.

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Dice N.B.   DIE   is   singular   ("one   die")   and   DICE   is   plural   ("twelve   dice"). A  die is  a  solid  cube, and  so  has  six faces.  The faces are  marked with the numbers  1  to  6 (1  being opposite  6, 2    being   opposite   5   and   3   being   opposite   4).   When   the   die   is   thrown   onto   a   flat   surface,   it   rolls   and   eventually    comes to   a   stop.   The   number   on   the   uppermost  face   is   taken   to   be   the   result      a   random  number   between   1    and 6  inclusive.   We   generally   assume   that  : P(1)  =  P(2)  =  P(3)  =  P(4)  =  P(5)  =  P(5)  =  1/6.

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Cards Sometimes  a   more   complicated   example   is  needed,   and   a   standard   pack   of  playing   cards can   be   useful  here. There   are   52   cards  in  a   pack,   and  they look like this:

You   will   notice   that   there   are   four   different   "flavours"   of   cards,   called   suits ,   one   in   each   row   in   the   picture. There   are  52/4   =   13   cards   in  each   suit.

Scene 37

The   suits   are   named   clubs   (black,   top   row);  spades,   (black,   second   row),   hearts   (red,   third   row)   and   diamonds    (red,  bottom  row).   Taking   the   first   row,   the  cards are   named   logically   (third   from  left is   the   "3   of  clubs"   for    instance)  except  that the   first is  known as  the  "ace   of  clubs"  (as  opposed   to   the  "one   of  clubs"); The  last   three   in  each   row   are   known  collectively   as face   cards   and   are   more   picturesque   than   the   other   cards  in    the  pack. The  three   face   cards are   known  respectively   as  the   Jack,   Queen   and  King   of   their   respective   suits; so,   for    example,   that rightmost  card of  the   second  row  is the   king   of  spades. That's  about  all   there   is   to   it.   Check   that   you   have   taken   all   the   above   in   by   finding   the   probabilities  of   picking   a    card   at   random   from   a  standard  pack and  finding   that it is a   red   queen  [answer:   2/52  =  1/26] an   ace  [answer:   4/52  =   1/13] a  spade   [answer: 13/52  =  1/4] a   face   card  [answer: 12/52  =   3/13]

MODULE 7 : DATA HANDLING

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