In both for-profit and not-for-profit organizations, statistics are critical pieces of information that allow decision makers to steer the organization in directions that are in the

Probability.html

Probability

Uncertainty is a threat in business because it puts scarce resources at risk. Accordingly, a considerable amount of effort is put into diminishing uncertainty across all aspects of business operations.

One of the best tools available to business managers to quantify the risk associated with the unknown is probability theory. Probability theory offers a means of reducing uncertainty by estimating the chances of different outcomes occurring. For example, knowing that in a particular process, Outcome A has a 5 percent chance of materializing while Outcome B has a 95 percent chance of materializing will greatly reduce the uncertainty associated with the outcome that can be expected from this process. However, uncertainty can rarely be eliminated altogether.

Analysis using probabilities for Outcome A and Outcome B can be achieved through the use of decision trees.  Organizations simply represent each possible outcome of a situation, the likelihood that each outcome will occur (5% for Outcome A and 95% for Outcome B), and the monetary value associated with each outcome.  Then, based on those probabilities, the organization can decide which alternative to choose, based on which outcome gives the likelihood of the highest profit. 

Probability estimation can be based on prior knowledge of the process involved, for example, the roll of a die, or on an analysis of past events such as the frequency with which an event occurred. Alternatively, probability estimation can be based on qualitative considerations, such as a survey of expert opinion. Probabilities based on known processes are known as classical probabilities. Those based on an analysis of recurring events are known as frequentist probabilities. The subjective estimation of probabilities is known as Bayesian probability (named after its inventor, Thomas Bayes). This course focuses on classical and frequentist approaches. Qualitative approaches (like Bayesian probability) are available for further study.

An example of probability based on known events is the probabilities associated with the throw of a die. In a business context, instances of known probabilities associated with a process are rare. However, frequentist probability determinations are common. Examples include frequency of product returns, the proportion of defective parts in a shipment, the length of time of telephone calls, and so on. Bayesian probabilities are used to estimate the probabilities of particular customers placing orders, the rate of output of a research program, or the risks associated with political or economic changes. Again, as Bayesian approaches address circumstances that are rare, unusual, and/or difficult to quantify, they will not be treated in this course.

A typical business application of probability theory might address, for example, the question of preferences for regular or decaffeinated coffee. To do this a survey is carried out at a selected coffee shop. Of 200 respondents, 125 are male and 75 are female. 120 prefer regular coffee and 80 prefer decaffeinated coffee. Of the males, only 45 prefer decaffeinated coffee.

Using a frequentist approach to estimating probabilities, it is apparent from this data that the percentage of people who prefer regular coffee can be estimated at 120/200 or 60%. The percentage of males who prefer decaffeinated coffee is given as 45/125 or 36%. The percentage of females who prefer decaffeinated coffee is, therefore, 35/75 or approximately 47%.

Measures of Central Tendency.html

Measures of Central Tendency

Summary data are generally communicated by representative measures. The "typical" age of a student in an MBA class might be represented by the average age of the students in the MBA class. The entire collection of all possible observations (in this case, all of the students in the MBA class) represents the population in which the statistician is interested. When a single measure is used, it generally represents the most common measurement of the population statistic. For this reason, such measures are commonly termed measures of central tendency, because they represent the measure that is more or less in the middle of the observations. Common measures of central tendency are the mean (also known as the simple average), the median, and the mode.

Measures of central tendency can vary in their appropriateness as a measure of central tendency. For example, the mean age of participants in a Boy Scouts camping trip may be 21, but that may not be a realistic reflection of the age of any of the participants in the trip (which might include young boys and older Scout leaders, but might not include any young adults that might be suggested by the calculated mean value of age). Thus, appropriate judgment must be exercised in deciding which measure of central tendency is appropriate in a given situation.

Most of the time, organizations choose between mean and median as the appropriate measure of central tendency.  If a measure is expected to take all data into consideration, then the mean is likely the appropriate measure to consider.  However, if an organization is looking for a “typical” value for a variable, then the median may be the more appropriate choice. 

Measures of Variability.html

Measures of Variability

Often, to gain better insights into the nature of a population, it is desirable to describe the population by more than its measure of central tendency. One may question, as discussed in the previous video, how good a representation of the population the mean (or average) really is. To answer this question, measures of variability are used in conjunction with measures of central tendency to more fully describe the data. For example, a measure of the mean (a measure of central tendency) of a population may be augmented by information about the range of observations (a measure of variability of the population, which is calculated by subtracting the smallest value of a population from the largest value). Knowing the ages of the youngest and oldest participants in the camping trip discussed in the previous video provides us with more information than simply knowing the mean age of all participants. 

Two other measures of variability are the variance of the population and the standard deviation of the population.  There is a relationship between these two measures of variability, as the standard deviation squared (multiplied by itself) is the variance.  Looking at it another way, standard deviation is the square root of the variance. 

The units of measurement of the standard deviation make it more useful than the variance in some circumstances.  For example, if the measure of central tendency that we are considering is measured in dollars, then standard deviation will also be in dollars.  Variance would be in units of “dollars squared,” which is not something that is easy for us to comprehend, in terms of exactly what that means. 

Descriptive vs. Inferential Statistics.html

Descriptive vs. Inferential Statistics

Descriptive statistics are used to describe the characteristics that can be known about a collection of data.  Inferential statistics, on the other hand, are generalizations that we are able to make about a larger collection of data (known as a population) from the smaller collection of data that we collected (referred to as a sample). 

In statistics, we are interested in collecting the values for certain characteristics about the subjects that we are studying.  For example, the characteristics that we might collect from a patient in a doctor’s office would be height, weight, and blood pressure.  Each of these values is a characteristic, and we refer to each of these characteristics as variables. 

To understand the different types of variables, we need to start with the basic question, what are data? Let's begin by learning about data and the different types of data.

In business, data are simply facts. There are different types of facts, such as about purchases that have been made, outcomes that have materialized, or events that have been observed. There is an infinite variety of facts; yet, in terms of how these facts are recorded (that is, made into what we call data), there are four basic, mutually exclusive types of data groupings: nominal, ordinal, interval, and ratio. These types of data capture how individual facts are encoded, rather than their intrinsic meaning. For instance, the term "hundred" might be a nominal measure if it refers to the observation being part of a group named hundred, or it might be an interval scale measure as in the number 100.

 In theory, a given variable can be measured on any of the aforementioned four scales. However, in practice, some variables presuppose a particular scale. For instance, variables that are used to label states or phenomena, such as gender, are usually measured using nominal scales, while measures that connote ordering or ranking, such as finishing position in a race (first, second, third, etc.), are measured using ordinal scales.

See the Supplemental Media entitled “Variable Classification” for more information about how data is classified. 

Additional Materials

Variable Classification Qualitative or Quantitative?