Without knowing something about how data is dispersed, measures of central tendency may be misleading. For example, a residential street with 20 homes on it having a mean value of $200,000 with little variation from the mean would be very different from a street with the same mean home value but with 3 homes having a value of $1 million and the other 17 clustered around $60,000. Measures of dispersion provide a more complete picture. Dispersion measures include the range, average deviation, variance, and standard deviation. Range The simplest measure of dispersion is the range. The range is calculated by simply taking the difference between the maximum and minimum values in the data set. However, the range only provides information about the maximum and minimum values and does not say anything about the values in between. Average Deviation Another method is to calculate the average difference between each data point and the mean value, and divide by the number of points to calcuate the average deviation (mean deviation). However, performing this calcuation will result in an average deviation of zero since the values above the mean will cancel the values below the mean. If this method is used, the absolute…

A commonly used measure of dispersion is the standard deviation, which is simply the square root of the variance. The variance of a data set is calculated by taking the arithmetic mean of the squared differences between each value and the mean value. Squaring the difference has at least three advantages: Squaring makes each term positive so that values above the mean do not cancel values below the mean. Squaring adds more weighting to the larger differences, and in many cases this extra weighting is appropriate since points further from the mean may be more significant. The mathematics are relatively manageable when using this measure in subsequent statisitical calculations. Because the differences are squared, the units of variance are not the same as the units of the data. Therefore, the standard deviation is reported as the square root of the variance and the units then correspond to those of the data set. The calculation and notation of the variance and standard deviation depends on whether we are considering the entire population or a sample set. Following the general convention of using Greek characters to express population parameters and Arabic characters to express sample statistics, the notation for standard deviation and…

 Three Different Concepts of Probability The classical interpretation of probability is a theoretical probability based on the physics of the experiment, but does not require the experiment to be performed. For example, we know that the probability of a balanced coin turning up heads is equal to 0.5 without ever performing trials of the experiment. Under the classical interpretation, the probability of an event is defined as the ratio of the number of outcomes favorable to the event divided by the total number of possible outcomes. Sometimes a situation may be too complex to understand the physical nature of it well enough to calculate probabilities. However, by running a large number of trials and observing the outcomes, we can estimate the probability. This is the empirical probability based on long-run relative frequencies and is defined as the ratio of the number of observed outcomes favorable to the event divided by the total number of observed outcomes. The larger the number of trials, the more accurate the estimate of probability. If the system can be modeled by computer, then simulations can be performed in place of physical trials. A manager frequently faces situations in which neither classical nor empirical probabilities are…

Certain types of probability calculations involve dividing the number of outcomes associated with an event by the total number of possible outcomes. For simple problems it is easy to count the outcomes, but in more complex situations manual counting can become laborious or impossible. Fortunately, there are formulas for determining the number of ways in which members of a set can be arranged. Such arrangements are referred to as permutations or combinations, depending on whether the order in which the members are arranged is a distinguishing factor. The number of different orders in which members of a group can be arranged for a group of r members taken r at a time is: (r)(r-1)(r-2)…(1)   This is more easily expressed as simply r!. When order is a distinguishing factor, a group of n members taken r at a time results in a number of permutations equal to the first r terms of the following multiplication: (n)(n-1)(n-2)… This can be expressed as: nPr  =  n! / (n – r)!   In combinations, order is not a distinguishing factor: nCr   =   nPr / (r!)   =   n! / (n – r)!r!   For the special case of possible pairs in a group of…