Post on 15-Oct-2014
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Analytical Tools
Factors: Things, Circumstances or conditions that cause something to happen; factors beget issues
Issues: points or questions to be disputed or decided
* Study subtleties: lightweight factors and issues
Thinking Patterns:
Convergent: bringing together and moving to a single focal point
� View a problem more narrowly; winnows the weak alternatives
Divergent: Branching out, go in different directions from a single point
� Open to creative alternatives, considers all alternatives
Four Types of Problems:
� Simplistic: There is only one answer
� Deterministic: Only one answer, but it is determined by a formula
� Random: Different answers are possible
� Indeterminate: Different answers are possible, but some answers are conjecture, not all
be identified
Sanity Checking: Does it make sense? Is it practical? Or logical?
Tool 1: Problem Restatement
Shift to divergent thinking: Redefine the problem in as
many different ways we can think of; let the ideas flow
freely.
1. Use the active voice when creating the list.
2. Paraphrase: Restate using different wo
changing the meaning
3. 180 Degrees: Reverse the situation into a negative
4. Broaden the focus: restate in larger context
5. Redirect the focus: Consciously change
6. Ask Why? Of the initial statement, then why again and
restate the problem
Analytical Tools
: Things, Circumstances or conditions that cause something to happen; factors beget issues
: points or questions to be disputed or decided
: lightweight factors and issues
: bringing together and moving to a single focal point
View a problem more narrowly; winnows the weak alternatives
: Branching out, go in different directions from a single point
Open to creative alternatives, considers all alternatives
: There is only one answer
: Only one answer, but it is determined by a formula
: Different answers are possible
: Different answers are possible, but some answers are conjecture, not all
: Does it make sense? Is it practical? Or logical?
Shift to divergent thinking: Redefine the problem in as
many different ways we can think of; let the ideas flow
when creating the list.
: Restate using different words without
: Reverse the situation into a negative
: restate in larger context
: Consciously change the focus
Of the initial statement, then why again and
Pitfalls:
• Too vague or too broad
• Focus is misdirected – definition too narrow
• Statement is assumption driven
• Statement is solution driven
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: Things, Circumstances or conditions that cause something to happen; factors beget issues
: Different answers are possible, but some answers are conjecture, not all possible answers can
definition too narrow
Statement is assumption driven
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Tool 2: Pros and Cons
Steps for Creating Pros and Cons:
1. List all Pros
2. List all Cons
3. Review and consolidate the Cons, merging and
eliminating
4. Neutralize as many Cons as possible
5. Compare the Pros and unalterable Cons for all options
6. Pick one option
Pro and Cons allows distilling what be the consequence if
one was selected.
It will be easier to do Cons humans are compulsively
critical, but sometime negatives can overwhelm the
positive.
Tool 3: Divergent / Convergent Thinking
Step 1. Brainstorm (Divergent)
Step 2. Winnow and Cluster (Convergent)
Step 3. Select practical, promising ideas (Convergent)
Rules of Divergent Thinking:
• The more ideas, the better
• Build upon one idea to another
• Wacky ideas are acceptable; break conventional
wisdom
• Don’t evaluate the ideas
Tool 4: Sorting, Chronologies and Timelines
Step 1. As you are researching a decision or problem, make
a list of relevant events and dates, Always list the dates
first.
Step 2. Construct a chronology, crossing off events as they
are included.
Benefits of Chronologies:
• Chronologies shows timing and relevant events
• Call attention to key events and to significant gaps
• Identify patterns and correlations
Timelines can be designed horizontally or vertically.
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Tool 5: Casual Flow Diagram
Casual Flow Diagram Construction Steps:
1. Identify major factors
2. Identify Cause-and-Effect Relationships
3. Characterize the relationship (Direct or Inverse)
4. Diagram the relationship
5. Analyze the behavior of the relationship when it is
integrated
CFD benefits:
� Identifies the major factors – the engines – that drive
the system; how they interact, and whether these
interactions are direct or inverse related.
� Enables us to view the cause and effect relationships
as an integrated system and discover linkages that are
formerly dimmed and obscured
� Facilitates our determining the main source of the
problem
� Enables us to conceive alternative corrective
measures and to estimate respective effects.
Casual Affect Factor Diagram
Create Causal/Affected Factor Matrix First; then diagram
Casual Flow Diagram (Visual Framework)
Tool 6: Matrix
A matrix enables us to:
� Separate elements of a problem
� Categorize information by type
� Compare one type of information with another
� Compare pieces of information of the same type
� See correlations (patterns) among the information
Setup the grid:
Change the X and Y axis according to subject matter:
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Tool 7: Decision Event Tree (DET)
Steps to Building a Decision Event Tree:
1. Identify the problem
2. Identify the major factors/issues (the decision &
events) to be addressed in the analysis
3. Identify alternatives for each of these factors/issues
4. Construct the tree portraying all important alternative
scenarios
5. Ensure that the decision/events at each branches of
the tree are mutually exclusive
6. Ensure that the decisions/events at each brach are
collectively exhaustive
Decision Event Tree Offers:
� It dissects a scenario into its sequential events
� It shows clearly the cause and effect linkages,
indicating which decisions and events precede and
follow others
� Shows which decisions and events are dependent on
others
� It shows where the linkages are the strongest &
weakest
� Enables us to visually compare how one scenario
differs from another
� Reveals alternatives we might otherwise not perceive
& enables us to analyze them – separately,
systematically and sufficiently.
Decide if the matrix is a good as using a decision tree.
It easier to decompose a matrix into a tree, but not vice-
versa.
Mutually Exclusive & Collectively Exhaustive
• Mutual exclusivity implies that at most one of the
events may occur.
One outcome precludes another one from happening, so if
we decide this, then we exclude that..
• Compare this to the concept of being collectively
exhaustive, which means that at least one of the
events must occur.
Conditionally Dependent Events:
• Occurrence of one event depends on another
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Tool 8: Weighted Ranking
Steps of Weighted Ranking:
1. List all of the major criteria for ranking
2. Pair-rank the criteria
3. Select the top several criteria and weight them in
percentiles (their sum must = 1.0)
4. Construct a Weighted Ranking Matrix and enter
items to be ranked, selected criteria, and the criteria
weights.
5. Pair-rank all the items by each criterion, recording in
the appropriate space the number of votes each
item receives.
6. Multiply the number of votes/tallies/observations by
the respective criterions weight.
7. Add the weighted values for each item and enter the
sums in the column labeled Total votes
8. Determine the final rankings and enter them in the
last column labeled ‘Final Ranking’ – item with the
most points is ranked highest.
9. Perform a sanity check
Weighted Ranking Matrix
Use sort feature to rank items
Another way to setup a weighted matrix
Weighted Arithmetic Mean can be expressed as:
The weights wi represent the bounds of the partial sample. In
other applications they represent a measure for the
reliability of the influence upon the mean by respective
values.
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Tool 9: Hypothesis Testing
Steps:
1. Generate a hypothesis. Eliminate any
implausible hypothesis and combine any
similarities.
2. Construct a matrix. Label the first column
Evidence. Lable the other columns to the
right Hypothesis and enter the descriptions
atop the columns. Hypothesis must be
mutually exclusive.
3. List significant evidence down the left hand
margin, including absent evidence (list only
significant evidence)
4. Working across the matrix, test the evidence
for consistency with each hypothesis, one
item of evidence at a time.
• Consistent – C
• Inconsistent - I
5. Refine the matrix by adding or rewording the
hypothesis
6. Working downward, evaluate each
hypothesis
7. Rank the remaining hypothesis by the
weakness of inconsistent evidence. (The
hypothesis with the weakest inconsistencies
is the most likely)
• Delete any hypothesis that are inconsistent with the
evidence.
Consistent – meaning that it could be responsible
Inconsistent – unlikely to be responsible or causation
• Confirm the validity of inconsistent evidence
• Is there an underlying assumption?
Tool 10: Devils Advocacy
Steps to Devils Advocacy List:
1. Make a list of points that support the view
2. Make a list of points countering or opposing by
providing support of Cons in the Prime Position that
Favors it.
• Favor particular outcomes or solutions in the
beginning
• Analyze it.
• Focus on the contrary or opposite viewpoint
(diametrically opposed to it)
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Tool 11a: Probability Tree
Steps for Developing a Probability Tree:
1. Identify the problem
2. Identify the major decisions and events to be analyzed
3. Construct a decision and events to be analyzed.
4. Construct a decision/event tree portraying all
important alternative scenarios
5. Ensure each branch is mutually exclusive and are
collectively exhaustive.
6. Assign a probability to each decision/event. Each
Branch must equal 1.0 *
7. Calculate the conditional probability of each individual
scenario.
8. Calculate the answers to probability questions relating
to the decision/event
* In numerical possibilities, an event cannot occur more
times than its possibility of occurring. Therefore,
probability can NEVER be greater than 1 or 100 percent.
Probability Rules (Add or Multiply)
� OR Type (Add): when the combined probability that
two or more events will occur as a result of a single
decision (“or”) the event. 0.3 + 0.5 = 0.8 (P1 + P2)
� AND Type (Multiply): when the probability two or
more events will occur in succession (the “and”
situation), we multiply their individual probabilities 0.5
x 0.5 = 0.25 (P1 x P2)
(Consider two coin tosses)
Use the IS-over-OF Rule to figure out percent.
For a series of connected (conditionally dependent)
events you must multiply.
Example of Conditionally Dependent Probability:
• History of succeeding 9 out of 10 times in the past opening
locks
• 10 Consecutive Doors (open one door to get to the next)
• Unlock 10 Doors
Probability (P
1) of opening x door?
Probability of opening the 1st
door? Is 9 out of 10 or .90
Probability of opening the 2nd
door? 9 x .91 = .82
Why? 10/11 is .91, you’ve already have 10 down and 11 to
go.
3rd door = .82 x 11/12 = .75
Probability
(Pn)
Calculation Calculated P
P1 9 /10 0.90
P2 0.9 x 10 / 11 0.82
P3 0.82 x 11 / 12 0.75
P4 0.75 x 12 / 13 0.69
P5 0.69 x 13 / 14 0.64
Multiply the probability of opening the P( 1
st door) times the
P(2nd
Door) / number of outcomes
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Tool 11b: Probability Refresher
Terms:
� An experiment is a well-defined process with observable outcomes.
� The set or collection of all outcomes of an experiment is called the sample space, S
� An event E is any collection or subset of outcomes from the sample sample.
� An event could have just one outcome and hence it is often called a simple event.
� An event with more than one outcome is called a compound event.
Classical definition of probability says that the probability of an event, P(E) is:
The assumption here is that any outcome is just as likely to occur as any other outcome. In the case where E=S, then the
numerator and denominator are equal and so the P(S)=1.
Look at the following events:
� C: The card drawn is a club.
� A: The card drawn is an ace.
� T: The card drawn is a "one-eyed" three.
The number of outcomes in S is 52.
The complement of E is the event consisting of all outcomes in S which are not in E and is often
denoted by P(E '). For example:
so both the probability of Not event is equal to the event occurring – 1 therefore:
How do you determine probability?
• Computation (deterministic)
• Frequency/Experience
Frequency is how often an event has occurred in the past; experience is what happened during each event.
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Tool 12: Utility Tree
Steps to Create A Utility Tree:
1. Identify the options and outcomes to be analyzed
2. Identify the perspectives of the analysis
3. Construct a decision/event tree for each option
4. Assign a utility value to each option-outcome
combination – each branch (scenario) of the tree by
asking the Utility Question: If we select this option,
and this outcome occurs, what is the utility from the
perspective of x
5. Assign a probability to each outcome. Determine the
estimate this probability by Asking the Probability
Question: If this option is selected, what the
probability this outcome will occur? (Must add up to
1.0)
6. Determine the expected values by multiplying each
utility by its probability and then adding the expected
value for each option.
7. Determine the ranking of the alternative solutions.
8. Perform a sanity check
Answers the question, if we select this option, and this
outcome occurs, what is the utility from the perspective of
____?
Expected Value is the product (multiplication) of the utility
and probability values for a given outcome.
Elements of a Tree:
• Option
• Outcome
• Utility
• Probability
• Expected Value
• Total Expected Value
• Ranking
Utility Tree:
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Tool 13: Utility Matrix
Steps to Construct a Utility Matrix:
1. Identify the options and outcomes to be analyzed.
2. Identify the perspective of the analysis
3. Construct a Utility Matrix
4. Assign a utility value of 0 to 100 (unless its currency) to
each option-outcome combination – each cell in the
matrix by
5. Assign a utility value to each option-outcome
combination – each branch (scenario) of the tree by
asking the Utility Question: If we select this option,
and this outcome occurs, what is the utility from the
perspective of x
6. Assign a probability to each outcome. Determine the
estimate this probability by Asking the Probability
Question: If this option is selected, what the
probability this outcome will occur? (Must add up to
1.0)
7. Determine the expected values by multiplying each
utility by its probability and then adding the expected
value for each option.
8. Determine the ranking of the alternative solutions.
9. Perform a sanity check
Easier to visualize a utility matrix vs. a utility tree
Elements of a matrix:
� Option
� Outcome
� Utility
� Probability
� Expected Value (EV)
� Total EV
� Rank
Perspective Outcomes Total EV
A B C
Option 1
Option 2
Option 3
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Tool 14: Advanced Utility Analysis
Steps for Creating an Advanced Utility Analysis:
1. Identify the options and outcomes to be analyzed
2. Identify and weight the perspectives to be analyzed
3. Construct an identical utility matrix for each
perspective – same options, same outcomes;
• Perform steps 4 -7 with each matrix
4. From each of the matrix’s particular perspective,
assign utilities from 0-100 to the outcome of each
option-outcome combination (each cell in the matrix).
There must be at least one.
5. Assign a probability to the outcome of each option-
outcome combination (each cell)
6. Compute the expected values for each option and
enter the totals in the Total EV column
7. Add expected values for each option and enter the
totals in the “Total EV” column
8. Construct a single ‘merged’ matrix with the same
options as in the perspective matrices.
9. Enter opposite each option the total expected for that
option from the perspective matrices
10. Multiply the total expected values under each
perspective by perspective’s weight.
11. Add the resulting products (weighted expected values)
for each option and enter the sums in the “Total
Weighted EV” column
12. Rank the options. The one with the greatest total
weighted expected value is the preferred option
13. Perform a sanity check
This has to be created for EACH of the perspective:
Each of the Outcomes are weighted before calculated
Example:
All the results will be ranked in the Total Perspective
Perspective Outcomes Total EV
A B C
Option 1
Option 2
Option 3
Perspective Outcomes Total EV
A
WT 0.8
B
WT 0.1
C
WT 0.1
Singapore
86 x .8
3 x 0.1 3 x 0.1
69
Tokyo
34 x 0.8
34 x 0.1 35 x 0.1
37
Kuala Lumpur
10 x 0.8
90 x 0.1 55 x 0.1
22.5
Perspective Perspectives Total
Weighted
EV
Rank
A B C
Option 1
Option 2
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Appendix A: Statistics - Probability
Determining Relative Frequency (Mathematical Probability)
Empirical Method
If the process under study can be repeated or simulated many times, we can determine the empirical probability by
keeping track of the outcomes in our (large number of) trials. The probability assigned is:
P(A happens) = (# times A happened) / (# trials)
If the number of trials is very large, then it is quite likely that this will give us a reliable estimate.
Theoretical Method
Sometimes we can make mathemitical assumptions about a situation and use Four Basic Properties of Probability to
determine the theoretical probability of an event. The accuracy of a theoretical probability depends on the validity of
the mathematical assumptions made.
The four useful rules of probability are:
1. It happens or else it doesn't. The probability of an event happening added the probability of it not happening is
always 1.
P(A happens) + P(A doesn't happen) = 1
2. Exclusivity. If A and B can't both happen at the same time (in which case we say that A and B are mutually exclusive),
then
P(either A or B happens) = P(A happens) + P(B happens)
3. Independence. If B is no more or less likely to happen when A happens than when A doesn't (in which case we say
that A and B are independent), then
P(A and B both happen) = P(A happens) * P(B happens)
4. Sub-Events. If whenever A happens B must also happen, then B must be at least as likely as A, so
P(A happens) <= P(B happens)
Empirical probabilities will also follow these rules (for a given set of trials). Because people often have a poor sense of
the likelihood of an event, personal probabilities often do not follow these rules.
A collection of personal probabilities is called coherent if it does not violate the rules for mathematical probability.
Equally likely outcomes
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One especially important use of these probability rules is the conclusions that can be drawn if we assume that a number
of events are equally likely. If there are only n such events that are possible in a given situation, and all are equally likely
and pair-wise mutually exclusive (no two can happen at once), then each must have probability 1/n.
More complicated situations can be handled by dividing a situation into a number of equally likely outcomes and
counting how many of them are "of interest" (in the event). The probability then is given by (number of interest)/(total
number.
Example 1:
For example, say you're playing with a deck of cards with a friend. Say he challenges you that if you draw an ace on your
first pick, he will have to buy you lunch the next day. How good are your chances of winning? Well, this is a probability
question. Since there are four aces in 52 cards, your chances of winning are 4/52, or 1/13. So, the probability of you're
picking an ace is 1/13, whereas the probability of you not picking an ace is 12/13. The odds are against you, but it isn't
completely unlikely.
Example 2:
Let's take a more complicated example. Let's say you and your friend are playing with two dice. He gives you the same
challenge, except this time, he challenges you to roll two sixes. The probability of rolling a six on each die is 1/6, but
how about both at the same time? To figure the probability out here, we have to multiply the probability of getting a six
on each die. So, we have 1/6 x 1/6, and our probability for rolling two sixes is 1/36. Your chances are even worse here!
Example 3:
Another interesting thing probability can show you that is somewhat practical is your chances of winning the lottery. If
you play one of those 3-digit lottery tickets where you have to guess the 3-digit number exactly, your probability of
winning is 1/1000. You can figure this out by noting that the probability of you're getting the first number right is 1/10.
The probability of getting all three right is then 1/10 x 1/10 x 1/10. Not too good of a chance!
The 4-digit number is even harder! Can you guess what the probability of winning that lottery might be? And even
more amazing is the tiny probability of winning those multi-million dollar lotteries. That would be a little trickier to
figure out, but it is a slightly different problem. Say there are 70 numbers they pick from. Your chances of getting the
first number right is 1/70. But since one ball has been removed, your chances of getting the second number right is
1/69. If there are six numbers picked, can you figure out what the probability of winning this lottery is? (Remember to
multiply all of the individual probabilities together... 1/70 x 1/69 x ...etc.)
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An event with a probability of 0 is impossible.
An event with a probability of 1 is certain.
0 P(A) 1 for any event A.
Probability can be approximated by frequency:
P(A) = number of times A occurred divided by number of times experiment is repeated.
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Appendix B: Evolution of a Matrix
Utility Matrix
Perspective 1:
My Monetary Profit
Outcome
Total
EV Rank
War
Utility x Probability
Prosperity
Utility x Probability
Recession
Utility x Probability
op
tio
ns
Speculative Stocks
100 x .1 = 10
20 x .36 = 7.2 0 x .54 = 0 17.2 3
Blue-Chip Stocks
80 x .1 = 8 70 x .36 = 25.2 10 x .54 38.6 2
Government Bonds
40 x .1 = 4 40 x .36 = 14.4 40 x 21.6 40 1
If you have n perspectives, then you must create a utility matrix for each perspective.
Advanced Utility Analysis Matrix
Perspective:
Merged
Classes of Outcome
Total
Weighted
EV
Rank War
Total EV x Weight
(0.2)
Prosperity
Total EV x Weight
(0.5)
Recession
Total EV x Weight
(0.3)
op
tio
ns
Speculative Stocks
17.2 x .2 = 10 2
nd Person’s
Monetary Profit
3rd
Person’s
Monetary Profit ∑ EV(W) n
Blue-Chip Stocks
Government Bonds
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Statistics Refresher:
Normal Distribution:
Normal curve is a mathematical model of randomness
In order to manipulate measures mathematically, we treat them was though they are located at points on a scale. Those
points are midpoints of intervals.
One reason is that variables are continuous rather than discrete.
Median and Mean (Measures of Central Tendency)
• Think of finding the fulcrum to get balance
• The point in from which all deviations sum to zero; meaning the negative and positive deviations will always
cancel each other out
The Mean ( µ) of the ENTIRE population:
µ = ∑X / N
µ is the mean of the population.
X refers to the raw scores
N is the size of the population
The Mean (X) of the sample:
x = ∑X / N
Just change symbol for the result to x “hat”; no further calculation is necessary.
Can be expressed in formal notation like this:
Geometric Mean:
Instead of sum, you use the product instead; used to calculating rates of growth.
The geometric mean is useful to determine "average factors".
For example, if a stock rose 10% in the first year, 20% in the second year and fell 15% in the third year, then we compute
the geometric mean of the factors 1.10, 1.20 and 0.85 as (1.10 × 1.20
... and we conclude that the stock rose 3.91 percent per year, on average.
Put another way...
The question about finding the average rate of return can be rephrased as: "by what constant factor would your
investment need to be multiplied by each year in order to achieve the same effect as multiplying by 1.10 one year, 1.60
the next, and 1.20 the third?" The answer is the
If you calculate this geometric mean you get approximately 1.283, so the average rate of return
which is what the arithmetic mean of 10%, 60%, and 20% would give you).
Measure of Variability (Average, Standard Deviation & Variance):
• The standard deviation is a kind of average of individual deviations from the mean of a distri
• The mean serves as an index of variability. It’s the reference point.
AD = ∑ |x| / N
• Average Deviation is the sum of individual deviations
• The standard deviation is a summation of the average individual
More formally, its:
Sample SD is:
where is the sample and
The geometric mean is useful to determine "average factors".
For example, if a stock rose 10% in the first year, 20% in the second year and fell 15% in the third year, then we compute
the geometric mean of the factors 1.10, 1.20 and 0.85 as (1.10 × 1.20 × 0.85)1/3 = 1.0391
... and we conclude that the stock rose 3.91 percent per year, on average.
The question about finding the average rate of return can be rephrased as: "by what constant factor would your
ied by each year in order to achieve the same effect as multiplying by 1.10 one year, 1.60
the next, and 1.20 the third?" The answer is the geometric mean .
If you calculate this geometric mean you get approximately 1.283, so the average rate of return
which is what the arithmetic mean of 10%, 60%, and 20% would give you).
(Average, Standard Deviation & Variance):
The standard deviation is a kind of average of individual deviations from the mean of a distri
The mean serves as an index of variability. It’s the reference point.
sum of individual deviations (distances) from the mean population.
summation of the average individual deviation from the mean of the distribution.
is the sample and is the mean of the sample.
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For example, if a stock rose 10% in the first year, 20% in the second year and fell 15% in the third year, then we compute
The question about finding the average rate of return can be rephrased as: "by what constant factor would your
ied by each year in order to achieve the same effect as multiplying by 1.10 one year, 1.60
If you calculate this geometric mean you get approximately 1.283, so the average rate of return is about 28% (not 30%
The standard deviation is a kind of average of individual deviations from the mean of a distribution.
(distances) from the mean population.
from the mean of the distribution.
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To Find Average Deviation (precision of measurement)
1. Find the average value of your measurements.
2. Find the difference between your first value and the average value. This is called the deviation.
3. Take the absolute value of this deviation.
4. Repeat steps 2 and 3 for your other values.
5. Find the average of the deviations. This is the average deviation
The average deviation is an estimate of how far off the actual values are from the average value, assuming that your measuring
device is accurate. You can use this as the estimated error. Sometimes it is given as a number (numerical form) or as a percentage.
To Find Percent Error
1. Divide the average deviation by the average value.
2. Multiply this value by 100.
3. Add the % symbol.
The mean average deviation is the average distance to the mean, i.e the average of |X-mean|. Absolute values have to be used
since the average of (X-mean) is zero.
The standard deviation uses square instead of absolute value to eliminate the sign. Since this leaves squares the units of
measurement, for example the spread of, say, heights in cm. would be an area, square root is applied at the end to recover the
original units of measurement.
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The z –score: Standard Score
Makes it possible for individuals (raw scores/votes/values) onto a single, common scale and compare it.
How?
First we need a unit that can be used to compensate for variability in the distribution. To do that, we use a measure of variability.
Divide the raw-score units by the standard deviation, so:
z = x /S
or more formally:
• X is a raw score to be standardized
• σ is the standard deviation of the population
• μ is the mean of the population
...basically you’re dividing each individual deviation by the standard deviation.
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Correlation: Finding Relationships
• When you want to know the relationships between a set of values (variables)
• Coefficient of correlation is an index of relationship that is high when there is a strong degree of relationship between
variable and low value when the relationship is weak.
• 1.0 is a perfect correlation.
• We have to know not only about the strength, but direction as well. Direct or Indirect (Inverse)
• Note: Any correlation coefficient carries information about two aspects of a relationship: its strength – measured on a scale
from zero to unity – and its direction – indicated by the presence of a minus sign.
Two types of correlation indices:
• Rank-Difference (Spearman)
• Product-Moment (Pearson)
Raw Data:
Example of Spearman:
The raw data used in this example is shown to left. There are 10 scores. n=10
The first step is to sort this data by the first column. Next, two more columns are
created. Both of these are for ranking the first two columns.
Notice how the rank of values that are the same is the mean of what their ranks
would otherwise be.
Then a column "d" is created to hold the differences between the two rank
columns. Finally another column "d2" should be created. This is just column d
squared.
After doing this process with the example data you should end up with something
like:
The values in the d2 column can now be added to find
The value of n is 10. So these values can now be substituted back into the equation.
Which evaluates to ρ = − 0.175758. In the case of ties in the original values, then
this formula should not be used.
Instead, the Pearson correlation coefficient should be calculated on the ranks
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Regression Analysis:
• Identifying the relationship between a dependent variable and one or more independent variables.
• A model of the relationship is hypothesized, and estimates of the parameter values are used to develop an estimated
regression equation
• .Its purpose is to take a series of independent variables and determine whether a particular dependent variable is related to
the independent variables.
• Regression analysis is very useful in that it does forecasts that do not rely on time. Dependent variables can rely on many
different independent variables.
• A Regression Line displays how well independent and dependent variables fit together. Points are plotted on a graph, and
after regression calculation, a line is drawn that is the "best fit" to the points on the graph.
• The closer the line is to each point, the stronger the relationship is between the independent and dependent variables.
Any future forecast on the dependent variable can be found on the regression line.
For Example:
A sales manager may want to determine whether product price, advertising budgets and competitor's prices have any effect on
total sales. A regression analysis determines, within a certain error bound, how well the variables "fit" together, and which ones
have the most effect on total sales.
A Dependent Variable is the variable that you wish to make forecasts on. In the above example, the dependent variable is Sales. An
Independent Variable is one or more variables that you wish to base the forecast of the dependent variable on. In the above
example, price, advertising budgets, and competitor price are independent variables.
22
Suppose you are the manager and you wish to forecast sales for a
particular brand of 3-D Video Card. You also wish to know how well the
Sales of this particular video card are related to the price of the card.
The 6 month snapshot is shown below:
Please note the Month is not a variable in the regression, it is in the
table to seperate the data, and not used in the actual analysis.
Regression Equation: Y = a + bX
where:
Y = Dependent Variable(Sales)
a = Y-axis Intercept
b = Slope of the regression line
X = Independent Variable(Price)
To calculate a & b we use the following formulas:
The formula for a is:
a = (µY) - b(µX)
The formula for b is:
ΣXY – n (µX) (µY)
ΣX2 - nX
2
In the Above formulas, n is the number of values to be analyses; in this case 6.
S stands for the sum, and µX & µµµµY stand for the Arithmetic Mean of all the X and Y values. The calculations are shown below.
First we calculate the arithmetic means of X & Y:
µY = SX
n =
102
6 = 17
µX = SY
n =
$1200
6 = $200
Next we plug these values into the formulas for a & b
b = 19910 - 6(200)(17)
245200 - 6(2002)
= -490
5200 = -0.1
Notice the value of b is negative, which means that the regression line has a decreasing slope
a = 17 - (-0.1)200 = 17 + 20 = 37
Now that we have the values of a and b, we can plug these values into our original regression equation. Therefore, our Regression
equation is:
Y = 37 - 0.1X
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You can now use this equation to forecast sales. For example, you would like to forecast how many video cards you will sell if you
put the card on sale in September for $145. Simply plug this into the equation to find the Y value.
Y = 37 - 0.1(145) = 22.5
Therefore, according to the regression analysis, the forecast for September is between 22 and 23 units sold.