Data Analysis (Quantitative Methods)

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Data Analysis (Quantitative Methods)

• Lecture 1
• Fundamental Statistics

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What is statistics

• Statistics is the science of data. This involves
  collecting, classifying, summarizing, organizing,
  analyzing, and interpreting numerical information.

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Dealing with Data

• Measurement Scales
• Descriptive Statistics
• Inferential Statistics

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Dealing with Data

• Quantitative data are measurements that are
  recorded on a naturally occurring numerical
  scale.
• Qualitative data are measurements that cannot be
  measured on a natural numerical scale they can
  only be classified into one of a group of
  categories.

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Levels of Measurement.

• Nominal Scale (Qualitative category membership
  e.g. gender, eye colour, nationality).
• Ordinal Scale (Ranks or assignments, positions in
  a group e.g. 1st 2nd 3rd).
• Interval and Ratio Scales (measured on an
  independent scale with units, e.g. I.Q scale.
  Ratio scale has an absolute zero point e.g.
  distance, Kelvin scale).

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Variables

• A variable is a characteristic or property of an
  individual population unit (a set of unit we are
  interested in studying).
• Discrete Variables There are no possible values
  between adjacent units on the scale. For
  Example, number of children in a family. X1, X2,
  , Xn
• Continuous Variables Is a variable that
  theoretically can have an infinite number of
  values between adjacent units on the scale. For
  Example, Time, height, weight. X e 0,100,
  0,30), (12, 80, (1,2)

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Descriptive Statistics
Descriptive statistics utilizes numerical and
graphical methods to look for patterns in a data
set, to summarize the information revealed in a
data set, and to represent that information in a
convenient form.

• Graphical Representation of Data
• Measures of Central Tendency
• Measures of Dispersion

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Representing Data Graphically

• Bar Charts
• Histograms
• Pie Charts
• Scattergrams

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The Bar Chart

• Used for Discrete variables
• Bars are separated

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Histogram

• Columns can only represent frequencies.
• All categories represented.
• Columns are not spaced apart.

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Pie Chart

• Used to illustrate percentages

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Scattergrams - Positive Relationships
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Negative Correlation
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No Relationship
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Measures of Central Tendency

• The Mean
• The Median
• The Mode

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The Mean
Mean Sum of all values in a group divided by
the number of values in that group. So if 5
people took 135, 109, 95, 121, 140 seconds to
solve an anagram, the mean time taken is
135 109 95 121 140
600 --------------------------------------------
----------- 120 5
5
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The Mean Pros Cons

• Advantages
• Very Sensitive Measure.
• Forms the basis of most tests used in inferential
  statistics.

• Disadvantages
• Can be effected by outlying scores E.g.
• 135, 109, 95, 121,140 480. Mean 1080/6 180
  seconds.

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The Median
The median is the central value of a set of
numbers that are placed in numerical order.
For an odd set of numbers 95, 109, 121, 135, 140
The Median is 121
For an even set of numbers 95, 109, 121, 135,
140, 480 The Median is the two central scores
divided by 2. (121 135)/2 128
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The Median Pros Cons

• Advantages
• Easier and quicker to calculate than the mean.
• Unaffected by extreme values.

• Disadvantages
• Doesnt take into account the exact values of
  each item
• If values are few it can be unrepresentative. E.G
• 2,3,5,98,112 the median is 5

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The Mode
The Mode The most frequently occurring value.
1, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6,
7, 7, 7, 8
The Mode 5
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The Mode Pros cons

• Disadvantages
• Doesnt take into account the value of each item.
• Not useful for small sets of data

• Advantages
• shows the most important value of a set.
• Unaffected by extreme values

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Data Types and Central Tendency Measures.
The Mode may also be used on Ordinal and Interval
Data. The median may also be used on Interval
Data.
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Why look at dispersion?

• 17, 32, 34, 58, 69, 70, 98, 142
• Mean 65
• 61, 62, 64, 65, 65, 66, 68, 69
• Mean 65

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Measures of Dispersion

• The Range
• Variance
• The Standard Deviation

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The Range
The Range is the difference between the highest
and the lowest scores.
Range Highest score - lowest score
4, 10, 5, 12, 6, 14 Range 14 - 4 10
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Population

• Population is a set of units (usually, people,
  objects, transactions, or events) that we are
  interested in studying.

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Sample

• A sample is a subset of the units of a
  population.
• A statistical inference is an estimate,
  prediction, or some other generalization about a
  population based on information contained in a
  sample

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Variance
Population Variance
Sample Variance
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Calculating the Standard Deviation
Population standard deviation
Sample standard deviation
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Inferential Statistics

• Inferential statistics utilizes sample data to
  make estimates, decisions, predictions, or other
  generations about a larger set of data.
• Inferential statistics allows us to draw
  conclusions about populations, and to test
  research hypotheses.
• Inferential Statistics Involves
• Probability
• Statistical Tests e.g., t test.

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Summary

• All data is measured on either Nominal, Ordinal,
  Interval or Ratio Scales
• Variables can be discrete and continuos
• Descriptive Statistics such as measures of
  central tendency and dispersion are used to
  describe or characters data
• Inferential Statistics is used to make inferences
  from sample data about the population at large.

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Data Analysis(Quantitative Methods)

• Lecture 5
• Probability

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Probability

• An experiment is an act or process of observation
  that leads to a single outcome that cannot be
  predicted with certainty.
• A sample point is the most basic outcome of an
  experiment. Ob1, Ob2, , Obn.

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Sample Space Venn Diagram

• The sample space of an experiment is the
  collection of all its sample points.
• S Ob1, Ob2, , Obn
• Venn diagrams.
• Graphical representations.

Ob1 Ob2 Obn
S
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Examples

• Experiment Observe the up face on a coin
• Sample Space 1. Observe a head H
• 2. Observe a tail
  T
• S H, T

H T
S
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Examples

• Experiment Observe the up face on a die.
• Sample Space 1. Observe a 1.
• 2. Observe a 2.
• 3. Observe a 3.
• 4. Observe a 4.
• 5. Observe a 5.
• 6. Observe a 6.
• S 1,2,3,4,5,6

1 2 3 4 5 6
s
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Examples

• Experiment Observe the up face on two coins
• Sample Space 1. Observe HH
• 2. Observe HT
• 3. Observe TH
• 4. Observe TT
• S HH,HT,TH,TT

HH HT TH TT
S
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Probability Rules for Sample Points

• All sample point probabilities must lie between
  0 and 1.
• The probabilities of all the sample points within
  a sample space must sum to 1.

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Probability

• An Event is a specific collection of Sample
  points.
• Example Consider the experiment of tossing two
  balanced coins.
• Events
• A Observe exactly one head
• B Observe at least one head

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Probability

• Sample point Probability
• HH 1/4
• HT 1/4
• TH 1/4
• TT 1/4
• P(A)P(HT)P(TH)1/2
• P(B)P(HH)P(TH)P(HT)3/4

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Probability of an Event

• The probability of an event A is calculated by
  summing the probabilities of the sample points in
  the sample space for A.

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Steps for Calculating Probabilities of Events

• Define the experiment.
• List the sample points. Ob1,Ob2,,Obn
• Assign probabilities to the sample points.
• P(Ob1), , P(Obn).
• Determine the collection of sample points
  contained in the event of interest.
• Sum the sample point probabilities to get the
  event probability.

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Unions and intersections

• The Union of two events A and B is the event that
  occurs if either A or B or both occur on a single
  performance of the experiment, denoted as the
  symbol

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Unions and intersections

• The intersection of two events A and B is the
  event that occurs if both A and B occur on a
  single performance of the experiment, denoted as
  the symbol
• P(A ? B)

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Unions and intersections
A
A
B
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Unions and intersections

• Example 1.
• Consider the die-toss experiment. Define the
  following events
• A Toss an even number
• B Toss a number less than or equal to 3
• Find

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Unions and intersections
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Complementary Events

• The complement of an event A is the event that A
  does not occur -- that is, the event consisting
  of all sample points that are not in event A and
  denoted as symbol Ac
• P(A)P(Ac)1

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Probability

• Additive Rule of Probability

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Conditional Probability

• To find the conditional probability that event A
  occurs given that event B occurs, divide the
  probability that both A and B occur by the
  probability that B occurs, that is,

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Probability

• Tree diagram

HH
H
T
H
HT
TH
H
T
TT
T
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Independent Events

• Events A and B are independent events if the
  occurrence of B does not alter the probability
  that A has occurred that is, events A and B are
  independent if
• P(AB)P(A)
• When events A and B are independent, it is also
  true that P(BA)P(B)
• Events that are not independent are said to be
  dependent.

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Probability

• Probability of Intersection of Two independent
  Events
• If events A and B are independent, the
  probability of the intersection of A and B equals
  the product of the probabilities of A and B that
  is P(A ? B)P(A) P(B)
• The converse is also true
• if P(A ? B)P(A) P(B), then A and B are
  independent.

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Exercise

• Calculate the mode, mean, and median of the
  following data
• (1) 12, 13, 15, 18, 12, 56, 13, 17, 19, 20, 35,
  36
• (2) 35, 23, 18, 26, 35, 23, 39, 45, 47, 37, 23,
  35, 19

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Excercise

• Calculate the range, variance and standard
  deviation of the following data
• (1) 2, 3, 1, 6, 8, 5, 9, 4, 5
• (2) 2, 0, 8, 4, 7, 5, 3, 2, 100

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Excercise

• Two fair coins are tossed and the following
  events are defined
• AObserved at least one head
• BObserved exactly one head
• CObserved exactly one tail
• DObserved at most one head
• Find P(A), P(B ? D), P(AD)

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Exercise

• Use tree-diagram to obtain the Sample space of an
  experiment that consists of a fair coin being
  tossed four times. Consider the following
  events
• AAll four results are the same.
• Bexactly one Head occurs.
• Cat least two Heads occur.
• Find P(A),P(B),P(C), P(A)P(B)P(C), P(A? C),
  P(A? B)
• Hence, explain why all the events A,B and C are
  not Mutually Exclusive.

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Exercise

• Let P(A)0.7, P(B)0.5 and P(A? B) 0.8.
• Find (1) P(A? B) (2) P(BA)

(3) Is event A independent of event B?
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References

• Statistics, 8th Edition
• MaClave and Sincich
• Prentice Hall, 2000.