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 Confidence
Tables
A common way to calculate confidence limits is with a table
of cumulative values for a "standard" Normal distribution. A standard Normal
distribution has a mean of 0 and a standard deviation of 1. Most statistics
books or books ... [full story]
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The "S"
Curve
Recall that the cumulative probability accumulates from 0 to
1 regardless of the actual distribution being summed or integrated. We can
easily equate the accumulating value as accumulating from 0 to 100%. For
example, if we accumulate all ... [full story]
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Confidence Intervals and Limits for Projects
The whole point of studying statistics in the context of
projects is to make it easier to forecast outcomes and put plans in place to
affect those outcomes if they are not acceptable or reinforce ... [full story]
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Central Limit
Theorem
Every bit as important as the Law of Large Numbers is to
sampling or diversification, the Central Limit Theorem helps to simplify matters
regarding probability distributions to the point of heuristics in many cases.
Here is what it ... [full story]
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Maximum
Likelihood and Unbiased Estimators
Hopefully, you can see that the Law of Large Numbers
simplifies matters greatly when it comes to estimating an expected value or the
mean of a distribution. Without knowledge of the distribution, or knowledge of
the ... [full story]
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The Law of Large
Numbers and Sample Average
The Law of Large Numbers deals with estimating expected
value from a large number of observations of values of events from the same
population. The Law of Large Numbers will be very valuable ... [full story]
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The
Central Limit Theorem and Law of Large Numbers
Two very important concepts for the project practitioner are
the Law of Large Numbers and the Central Limit Theorem because they integrate
much of what we have discussed and provide very useful ... [full story]
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Three-Point
Estimate Approximations
Quite useful results for project statistics are obtainable
by developing three-point estimates that can be used in equations to calculate
expected value, variance, and standard deviation. The three points commonly used
are:
Most pessimistic value that yet has some ... [full story]
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Probability Distribution Statistics
Most often we do not know every value and its probability.
Thus we cannot apply the equations we have discussed to calculate statistics
directly. However, if we know the probability distribution of values, or can
estimate what the ... [full story]
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The
Arithmetic of Operations on Statistics and Random Variables
When it comes to arithmetic, random variables are not much
different than deterministic variables. We can add, subtract, multiply, and
divide random variables. For instance, we can define a random variable Z ... [full story]
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Median
The median is the value that is half the distance between
the absolute value of the most pessimistic value and the most optimistic
value.
Median = 1/2 * | (optimistic value - pessimistic
value)
[13]Balsley, Howard, Introduction
to Statistical Method, Littlefield, Adams ... [full story]
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Mode
The mode of a random outcome is the most probable or most
likely outcome of any single occurrence of an event. If you look at the
distribution of outcome values versus their probabilities, the mode is the value
at the ... [full story]
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Variance and
Standard Deviation
Variance and standard deviation are measures of the spread
of values around the expected value. As a practical matter for project
practitioners, the larger the spread, the less meaningful is the expected value
per se.
Variance and standard ... [full story]
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Expected Value
and Average
The best-known statistic familiar to everyone is "average"
(more properly, arithmetic average), which is arithmetically equal to a specific
case of expected value. Expected value, E, is the most
important statistic for project managers. The idea of ... [full story]
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Key
Statistics Used in Projects
Strictly speaking, statistics are data. The data need not be
a result of analysis. Statistics are any collection of data. We often hear,
"What are the statistics on that event?" In other words, what are the ... [full story]
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Other
Distributions
There are many other distributions that are useful in
operations, sales, engineering, etc. They are amply described in the literature,
[12] and a brief listing is
given in Table 2-3.
Table 2-3: Other Distributions
Distribution
General Application
Poisson
The Poisson distribution ... [full story]
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The Normal
Distribution
The Normal distribution is a well-known shape, sometimes
referred to as the "bell curve" for its obvious similarity to a bell. In some
texts, it will be referred to as the Gaussian distribution after the
19th century mathematician ... [full story]
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The BETA
Distribution
The BETA distribution is a distribution with two parameters,
typically denoted "a" and "b" in its PDF, that influence its shape quite
dramatically. Depending on the values of "a" and "b", the BETA distribution can
be all the ... [full story]
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Triangular
Distribution
The Triangular distribution is applied to continuous random
variables. The Triangular distribution is usually shown with a skew to one side
or the other. The Triangular distribution portrays the situation that not all
outcomes are equally likely as was ... [full story]
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Uniform
Distribution
The discrete Uniform distribution is illustrated in Figure 2-4. The toss of the coin and
the roll of the single die are discrete Uniform distributions. The principal
attribute is that each value of the random variable has the same ... [full story]
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Probability Distributions for Project Managers
If we plot the probability (density) function (PDF) on a
graph with vertical axis as probability and horizontal axis as value of X, then
that plot is called a "distribution." The PDF is aptly named because ... [full story]
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Cumulative
Probability Functions
It is useful in many project situations to think of the
accumulating probability of an event happening. For instance, it might be useful
to convey to the project sponsor that "...there is a 0.6 probability that the
schedule ... [full story]
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Continuous
Random Variables
As the number of values of X increases in a given range of values, the spacing
between them becomes smaller, so small in the limit that one cannot distinguish
between one unique value and another. So also do ... [full story]
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Discrete Random
Variables
So far, our examples of random variables have been discrete
random variables. H or T could only take on discrete values on any specific
toss: 1 or 0. On any given toss, we have no way of knowing ... [full story]
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Probability
Functions
Random variables do not have deterministic values. In
advance of a random outcome, like the uncertain duration or cost of a work
package, the project team can only estimate the probable values, but the team
will not know for ... [full story]
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Random
Variables and Their Functions in Projects
Random
Variables
So far, we have discussed random events (tails coming up on
a coin toss) and probability spaces (a coin toss can only be heads or tails
because there is nothing else in the ... [full story]
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Subjective
Probability
What about statements like "there is a 20% chance of rain or
snow today in the higher elevations"? A statement of this type does not express
a relative frequency probability. We do not have an idea of the population ... [full story]
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The (1-p)
Space
To this point, we have been using a number of conventions
adopted for probability analysis:
All quantitative probabilities lie in the range between the
numbers 0 (absolute certainty that an outcome will not occur) and 1 (absolute
certainty that ... [full story]
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Conditional
Probabilities
When A and B are not independent, then one becomes a
condition on the outcome of the other. For example, the question might be: What
is the probability of A given the
condition that B has occurred? [3]
Consider ... [full story]
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AND and OR with
Overlap or Collisions
We can now go one step further and consider the situation
where events A and B are not mutually exclusive (that is, A and B might
occur together sometimes or perhaps overlap in some ... [full story]
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AND and OR
Let's begin with OR. Using the relative frequency
mathematics already developed, what is the probability of the event "5"? There
are only four outcomes, so the event "5" probability is 4/36 or 1/9.
To make it more interesting, let's ... [full story]
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Relative
Frequency Definitions
The exercise of flipping coins or rolling dice illustrates
the "relative frequency" view of probability. Any specific result is an
"outcome," like rolling the pair (1,6). The six pairs that total seven on the
faces are collectively an ... [full story]
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