STATISTICS CHAPTER 1

STATISTICS CHAPTER 1

1. DECISION MAKING IN UNCERTAIN ENVIRONMENT

   • Decisions made on uncertain info.
     • We must carefully define the problem, determine needed data, collect it and use statistics to summarize it
   • Accounts need to select a portion of recordings for auditing purposes
   • RANDOM AND SYSTEMATIC SAMPLING
     • Before a new product is launched, manufacturer must assess the likely level of demand, undertaking a market survery.
       • Manufacturer interested in all n potential buyers
         • PROBLEM: populations too large to analyze
           • POPULATION
             • Complete set of all items that interest an investigator.
             • Size N can be very large/infinite
           • SAMPLE
             • An observed subset (portion) of population with sample size given by n
             • One important principle to be followed in sample selection process is sampling
               • SAMPLING
                 • Simple random sampling
                   • Procedure used to select a sample of n objects from a population
                     • This to make sure each member chosen by chance
                   • Adjective "simple" is generally dropped
                 • Systematic Sampling
                   • Involves the selection of every jth item in the population, where j is the ratio of population size N
                     • j=N/n
                   • Provides a good representation of population if there isnt no cyclical variation
   • SAMPLING AND NONSAMPLING ERRORS
     • PARAMETER
       • Numerical measure that describes a specific characteristic of population
     • STATISTIC
       • Numerical measure describing specific characteristic of a sample
     • The value of any population parameter cannot be known precisely
       • SAMPLING ERROR
         • Information is available on only a subset of population members
       • NONSAMPLING ERROR
         • Error unconnected with sampling procedure used
         • 3 examples
           • 1. Population sampled is not the relevant one
           • 2. Survey subjects may give inaccurate or dishonest answers
             • May happen with questions phrased in a difficult manner or difficult to understand
           • 3. There may be no response to survey questions
         • There is no general procedure fo/analyzing these errors
           • However, thsse errors may be important!
   • After we identify/define a problem we collect the data produced by various process
     • these include design, and data analyzzed using one or more statistical procedures
       • DESCRIPTIVE STATISTICS
         • Focus on graphical and numerical procedures used to summarize and process data
       • INFERENTIAL STATISTICS
         • Focus on data used to summarize and process data to make predictions, forecasts and estimates for better decisions

2. CLASSIFICATION OF VARIABLES

   • A variable is a specific characteristic of an individual/object
     • CATEGORICAL VARIABLE
       • produces responses belonging to groups of people
         • eg. yes/no
     • NUMERICAL VARIABLE
       • Include both discrete and continuous variables
         • DISCRETE NUM. VARIABLE
           • May (but not necessarily) have a finite n° of values
           • Most common type of num. var. comes from counting process
             • eg. n° of students in a class
         • CONTINUOUS NUM. VARIABLE
           • May take on any value within a given range of real numbers and usually arises from a measurement (NOT COUNTING) process
   • MEASUREMENT LEVELS
     • Data can be quantitative or qualitative
       • QUALITATIVE
         • No measurable meaning in the difference between numbers
           • eg. soccer team jersey numbers
         • Nominal and ordinal measurements
           • NOMINAL
             • Considered lowest type of data
             • Numbers assigned to categories
               • eg. 1= male, 2=female
           • ORDINAL
             • indicate rank ordering of items and, similar to nominal data, the values are words describing responses
               • eg. productivity rating
                 • 1=poor, 2=average, 3=excellent
       • QUANTITATIVE
         • There IS a measurable meaning
           • eg. score on an exam
         • Interval and ratio measurements
           • Refer to data obtained from numerical variables and meaning is given to the difference between measurements.
           • INTERVAL SCALE
             • Measures rank and distance from arbitrary zero measured in unit intervals
               • eg. temperature
           • RATIO DATA
             • Indicate both rank and distance from a natural zero with ratios of two measures having a meaning
               • eg. a person who is 40 yrs old is twice as old as a person 20 yrs. old

3. GRAPHS TO DESCRIBE CATEGORICAL VARIABLES

   • FREQUENCY DISTRIBUTION
     • Is a table used to organize data
       • Left column (classes groups) includes possible responses on a variable studied
         • Right column is a list of frequencies
     • Reative frequency distribution
       • Obtained by dividing each freq. by the number of observations and mult. result by 100%
   • TABLES AND CHARTS
     • The classes that we use to construct frequency distribution tables of categorical variable are simply the possible responses to categorical variable.
     • If our objective is to draw attention to the frequency, then we will use bar charts
       • Each bar height represents each frequency
       • No need for the bars to touch
   • CROSS TABLES
     • There are situations in which we need to describe relationships beteen categorical or ordinal variables
     • Cross tables lists the numberof observations for every combination of values of two categorical/ordinal variables.
       • A cross table with r rows and c columns is refferred to as an rXc cross table
   • PIE CHARTS
     • Used if we want to draw attention on the proportion of frequencies in each category
     • Depicts division of a whole into its constituent parts
       • Circle = Total
       • Segments = Shares of total
   • PARETO DIAGRAMS
     • Managers who need to identify the cause of problems and attempt to correct them quickly with min. cost freq. use a special bar chart called pareto diagram
     • Bar chart displaying the frequency of defect causes
     • Bar on the left indicates most frequent cause.
     • Bar on the right indicates causes with decreasing frequencies
     • Used to separate vital few to trivial many

4. GRAPHS TO DESCRIBE TIME-SERIES DATA

   • TIME SERIES DATA
     • Data measured at successive points in time
     • Graphs of this type of data called "line chart" or "time series plot"
   • LINE CHART (TIME SERIES PLOT)
     • Time series is a set of measurements ordered over time
     • A line chart is a series of data plotted at various time intervals
     • Time measured on hor. axis and numerical quantity on vertical axis yields a point on the graph
       • Joining points adjacent in time by straignt line produces time series plot
         • eg. annual IR

5. GRAPHS TO DESCRIBE NUMERICAL VARIABLES

   • FREQUENCY DISTRIBUTION
     • CONSTRUCTION
       • 1. Determine k, the n° of classes
       • 2. Classes should be same width, w
         • Determined by:
       • 3. Classes must be inclusive and non overlapping
       • Boundaries/endpoints of each class clearly defined
     • CUMULATIVE AND RELATIVE CUMULATIVE FREQ. DISTR
       • Cumulative freq. distr. contains total number of observations whose values are less than the upper limit for each class
         • We construct a cumulative frequency distribution by adding up all the frequency distribution classes up to present class
       • In a relative cum. freq. distr. cumulative frequencies are expressed as proportions
   • HISTOGRAMS AND OGIVES
     • HISTOGRAM
       • Graph consisting of vertical bars constructed on a horizontal line that is marked off with intervals for the variable being displayed
       • Intervals correspond to the classes in a frequency distr. table.
       • Height of each bar proportional to the number of observations
     • OGIVE
       • Sometimes called "Cumulative Frequency Line"
       • Is a line connecting points that are cumulative percent of observations below the upper limit of each interval in a cum. freq. distr.
     • SYMMETRY AND SKEWNESS
       • Shape of distribution is symmetric id the observations are balanced about its center
       • Distribution skewed/asymmetric if observations are not symmetrically distributed
         • Skewed-right/skewed left depending on whwere the tail lies
   • STEM AND LEAF DISPLAY
     • Alternate graph to histogram
     • Data grouped according to its leading digits (stem) and final digits (leaves)
       • Leaves displayed in ascending order individually after each stem
   • SCATTER PLOTS
     • Prepared by locating one point for each pair of two variables

6. DATA PRESENTATION ERRORS

   • Poorly designed graphs can easily distort the truth
   • Accurate design is a fundamental feature in today's global markets
   • Misleading Histograms
     • Like unproportional widths
   • Misleading time-plot
     • Inadequate graph for time picture of the data

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