Probability and Statistics

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• Probability and Statistics
• Lecture notes 03

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Lesson Overview

• Types of Data
• Qualitative (Categorical)
• Quantitative (Numerical)
• Discrete vs. Continuous
• Levels of Measurement
• Nominal, Ordinal, Interval, Ratio
• Data Summary and Presentation
• The Stem-and-leaf Diagram
• The Frequency Distribution Tables
• Histogram
• The Box Plot
• Time Sequence Plots

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Types of Data

• Data can be classified as either numeric or
  nonnumeric.
• Specific terms are used as follows
• Qualitative data are nonnumeric.
• Poor, Fair, Good, Better, Best, colors
  (ignoring any physical causes), and types of
  material straw, sticks, bricks are examples of
  qualitative data.
• Qualitative data is often termed catagorical
  data.
• Some books use the terms individual and variable
  to reference the objects and characteristics
  described by a set of data.
• They also stress the importance of exact
  definitions of these variables, including what
  units they are recorded in.
• The reason the data were collected is also
  important.

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Types of Data

• Quantitative data are numeric.
• Quantitative data are further classified as
  either discrete or continuous.
• Discrete data are numeric data that have a finite
  number of possible values.A classic example of
  discrete data is a finite subset of the counting
  numbers, 1,2,3,4,5 perhaps corresponding to
  Strongly Disagree... Strongly Agree.
• When data represent counts, they are discrete. An
  example might be how many students were absent on
  a given day. Counts are usually considered exact
  and integer. Consider, however, if three tradies
  make an absence, then aren't two tardies equal to
  0.67 absences?

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Quantitative data / Types of Data

• Continuous data have infinite possibilities 1.4,
  1.41, 1.414, 1.4142, 1.141421...The real numbers
  are continuous with no gaps or interruptions.
• Physically measureable quantities of length,
  volume, time, mass, etc. are generally considered
  continuous. At the physical level
  (microscopically), especially for mass, this may
  not be true, but for normal life situations is a
  valid assumption.
• The structure and nature of data will greatly
  affect our choice of analysis method. By
  structure we are referring to the fact that, for
  example, the data might be pairs of measurements.

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Levels of Measurement

• The experimental (scientific) method depends on
  physically measuring things.
• The concept of measurement has been developed in
  conjunction with the concepts of numbers and
  units of measurement.
• Statisticians categorize measurements according
  to levels.
• Each level corresponds to how this measurement
  can be treated mathematically.

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Levels of Measurement (Measurement Scales) Four
common types

• Nominal Nominal data have no order and thus only
  gives names or labels to various categories.
• Ordinal Ordinal data have order, but the
  interval between measurements is not meaningful.
• Interval Interval data have meaningful intervals
  between measurements, but there is no true
  starting point (zero).
• Ratio Ratio data have the highest level of
  measurement. Ratios between measurements as well
  as intervals are meaningful because there is a
  starting point (zero).
• (Gender is something you are born with, whereas
  sex is something you should get a license for.)

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Levels of Measurement (measurement Scales) Four
common types

• Nominal scales are for things that are mutually
  exclusive/non-overlapping, but there is no order
  or ranking.  For example, professors are divided
  into departments by subject, but no subject is
  ranked as better than another.
• Ordinal Levels of Rank are categories that can be
  ordered, but not precisely.  For example, letter
  grades, movie quality (excellent, good, adequate,
  bad, terrible).
• Interval Level ranks the data in precise scales,
  but there is no meaningful zero.  For example IQ
  tests and temperature. Neither have a meaningful
  zero.
• Ratio Level Data can be ranked and there are
  precise differences between the ranks, as well as
  having a meaningful zero.  For example Height,
  weight, Salary, and Age.

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Types of Data / Levels of Measurement

• Example 1 ColorsTo most people, the colors
  black, brown, red, orange, yellow, green, blue,
  violet, gray, and white are just names of colors.
  
• To an electronics student familiar with
  color-coded resistors, this data is in ascending
  order and thus represents at least ordinal data.
• To a physicist, the colors red, orange, yellow,
  green, blue, and violet correspond to specific
  wavelengths of light and would be an example of
  ratio data.

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Types of Data / Levels of Measurement

• Example 2 TemperaturesWhat level of measurement
  a temperature is depends on which temperature
  scale is used.Specific values 0C 32F
  273.15 K 491.69R     100C 212F 373.15 K
  671.67R     -17.8C 0F 255.4 K
  459.67Rwhere C refers to Celsius F refers to
  Fahrenheit K refers to Kelvin R refers to
  Rankine.
• Only Kelvin and Rankine have zeroes (starting
  point) and ratios can be found. Celsius and
  Fahrenheit are interval data certainly order is
  important and intervals are meaningful. However,
  a 180 dashboard is not twice as hot as the 90
  outside temperature (Fahrenheit assumed)!
  Although ordinal data should not be used for
  calculations, it is not uncommon to find averages
  formed from data collected which represented
  Strongly Disagree, ..., Strongly Agree! Also,
  averages of nominal data (zip codes, social
  security numbers) is rather meaningless!

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Data Sources

• Published source
• Designed experiment
• Survey
• Observational study

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DATA SUMMARY
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DATA SUMMARY AND PRESENTATION

• The Stem-and-leaf Diagram
• The Frequency Tables
• Standard, Relative, and Cumulative
• Histograms
• The Box Plot
• Time Sequence Plots

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Graphical Displays

• The distribution of a variable describes what
  values the variable takes and how often each
  value occurs.
• The frequency of any value of a variable is the
  number of times that value occurs in the data.
• The relative frequency of any value is the
  proportion (fraction or percent) of all
  observations that have that value.

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DATA SUMMARY AND PRESENTATION

• Frequency Tables Standard, Relative, and
  Cumulative
• Histograms, Ogive, Pareto Diagrams,
• Pie Charts
• Exploratory Data Analysis
• Stem-and-Leaf Diagram
• Boxplots

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Graphical Displays

• The distribution of a variable describes what
  values the variable takes and how often each
  value occurs.
• The frequency of any value of a variable is the
  number of times that value occurs in the data.
• The relative frequency of any value is the
  proportion (fraction or percent) of all
  observations that have that value.

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Types of Variables

• Categorical variable Places an individual into
  one of several categories.
• Examples Gender, race, political party, zip code
• Quantitative variable Takes numerical values for
  which arithmetic operations make sense.
• Examples OYS score, number of vote, cost of
  textbooks

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Graphs for categorical variables

• Pie charts require relative frequencies since
  they display percentages and not raw data. The
  relative frequency of each category corresponds
  to the percent of the pie that is occupied by
  that category.
• Bar graphs display data where the categories are
  on the horizontal axis and the frequencies (or
  relative frequencies) are on the vertical axis.

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Graphs for quantitative variables

• Histograms
• The data are divided into classes of equal width
  and the number (or percentage) of observations in
  each class is counted.
• Data scale is on the horizontal axis.
• Frequency (or relative frequency) scale is on the
  vertical axis.
• Bars are draw where base of each bar covers the
  class, height of each bar covers the frequency
  (or relative frequency).

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• Stem-plots or Stem and Leaf Displays
• Separate each observation in a stem unit (all but
  the final rightmost digit of (rounded) data) and
  a leaf unit (the final digit of (rounded) data).
• Write the stems in a vertical column, smallest to
  largest from top to bottom.
• Write each leaf in the row to the right of its
  stem, in increasing order.

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Histograms vs. Stem plots

• Both are used to describe the distribution of
  data.
• Stemplots display actual data values.
• Stemplots are used for small data sets (less than
  100 values).
• Histogram can be constructed for larger data sets.

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Common Distributional Shapes

• A symmetric distribution is one where both sides
  about the center line are approximately mirror
  images of each other.
• A skewed distribution is one where one side of
  the center line contains more data than the
  other.
• Skewed to the right The right side of the
  histogram extends much farther than the left
  side.
• Skewed to the left The left side of the
  histogram extends much farther than the right
  side.

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Common Distributional Shapes

• A bimodal distribution has two humps where much
  of the data lies.  
• All classes occur with approximately the same
  frequency in a uniform distribution.
• An outlier in any graph of data is an individual
  observation that falls outside the overall
  pattern of the graph.

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DATA SUMMARY AND PRESENTATION

• THE STEM-AND-LEAF DIAGRAM
• A stem-and-leaf diagram is a good way to obtain
  an informative visual display of a data set
• x1, x2, ..., xn,
• where each number xi consists of at least two
  digits.
• To construct a stem-and-leaf diagram, we divide
  each number xi into two parts
• a stem, consisting of one or more of the leading
  digits, and
• a leaf, consisting of the remaining digits.

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• Write the stems in a vertical column, smallest to
  largest from top to bottom.
• Write each leaf in the row to the right of its
  stem, in increasing order.

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THE STEM-AND-LEAF DIAGRAM

• EXAMPLE
• Construct a stem-and-leaf display for the
  following data

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THE STEM-AND-LEAF DIAGRAM

• SOLUTION
• We will select as stem values the numbers 7, 8,
  9, 10, 11, , 24.
• The resulting stem-and-leaf diagram is presented
  in the following figure.

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THE STEM-AND-LEAF DIAGRAM
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THE STEM-AND-LEAF DIAGRAMStem is sorted in
decreasing order, leaf ordered in increasing order
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THE STEM-AND-LEAF DIAGRAM

• Inspection of this display immediately reveals
  that most of the data lie between 110 and 200 and
  that a central value is somewhere between 150 and
  160. Furthermore, the data are distributed
  approximately symmetrically about the central
  value.
• The stem-and-leaf diagram enables us to determine
  quickly some important feature of the data that
  were not immediately obvious in the original
  display in original table.

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THE FREQUENCY DISTRIBUTION TABLES

• Frequency Tables
• Frequency refers to the number of times each
  category occurs in the original data
• A frequency table lists in one column the data
  categories or classes and in another column the
  corresponding frequencies.
• A common way to summarize or present data is
  with a standard frequency table.

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Frequency Tables

• Often, the category column will have continuous
  data and hence be presented via a range of
  values. In such a case, terms used to identify
  the class limits, class boundaries, class widths,
  and class marks must be well understood.
• Class limits are the largest or smallest numbers
  which can actually belong to each class. Each
  class has a lower class limit and an upper class
  limit.
• Class boundaries are the numbers which separate
  classes. They are equally spaced halfway between
  neighboring class limits.

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Frequency Tables

• Class marks are the midpoints of the classes. It
  may be necessary to utilize class marks to find
  the mean and standard deviation, etc. of data
  summarized in a frequency table.
• Class width is the difference between two class
  boundaries (or corresponding class limits).

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Frequency Tables

• Following are guidelines for constructing
  frequency tables.
• The classes must be "mutually exclusive"no
  element can belong to more than one class.
• Even if the frequency is zero, include each and
  every class.
• Make all classes the same width. (However, open
  ended classes may be inevitable.)
• Target between 5 and 20 classes, depending on the
  range and number of data points.
• Keep the limits as simple and as convenient as
  possible.

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Frequency Tables

• Relative freqency tables contain the relative
  frequency instead of absolute frequency. Relative
  frequencies can be expressed either as
  percentages or their decimal fraction
  equivalents.
• Cumulative frequency tables contain frequencies
  which are cumulative for subsequent classes. In a
  cumulative frequency table, the words less than
  usually also appear in the left column.

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Frequency Tables

• The frequency distribution
• A frequency distribution is a more compact
  summary of data than a stem-and-leaf diagram.
• To construct a frequency distribution, we must
  divide the range of the data into intervals,
  which are usually called class intervals, cells,
  or bins.

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Frequency Distrubion Tables

• EXAMPLE
• Construct the frequency distribution table for
  the following data

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THE FREQUENCY DISTRIBUTION TABLES

• SOLUTION
• Class relative
• frequency
• Cumulative
• frequency

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Frequency Distrubion Tables
Another example containing student distributions
as follows