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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

Calculating Probability Most people would say that the probability of an outcome of heads or tails in a fair coin toss is 50%; some might say it is half, 0.5, or one chance in two. They all would be correct. There ... [full story]

Coin Toss 101 As an experiment, toss a coin 100 times. Let heads represent one estimate of the duration of a project task, say writing a specification, of 10 days, and let tails represent an estimate of duration of 15 days ... [full story]