Discussion - Due tomorrow

QSO 510 Module Three 1

Module Two reviewed tabular, graphical, and numerical methods for displaying and summarizing qualitative and quantitative data. Module Three examines the empirical rule and the normal distribution. The empirical rule, also referred to as the 68-95-99.7 rule, applies to any symmetric distribution of data. Symmetric distributions are bell-shaped. The rule states that

68% of the observations fall within 1 standard deviation of the mean, 95% of the observations are within 2 standard deviations of the mean, and 99.7% of the observations are within 3 standard deviations of the mean.

Related to the empirical rule is the concept of z-scores. A z-score represents the number of standard deviations that an observation is above or below the mean and is calculated as

z = x – µ, σ,

where x is an observation, µ is a population mean, and σ is a population standard deviation.

For example, a z-score of +3.0 suggests that an observation is 3 standard deviations above the mean, while a z-score of –2.0 says that the observation is 2 standard deviations below the mean. The normal distribution is arguably the most important distribution in operations, quantitative analysis, and business statistics. The normal distribution can be used to model a wide variety of data, such as the following:

 Heights of teenagers

 Weights of newborn children

 IQ scores

 Test scores on a final examination

 Graduate Management Admission Test (GMAT) scores

 Salaries of employees

 Rainfall levels in Houston, Texas

2 QSO 510 Module Three

The normal distribution is not a single distribution, but a family of distributions. Each normal distribution is defined by a mean µ and standard deviation σ. For example, stating that salaries are N (50,000, 5000) suggests that salaries are normally distributed with a mean µ = $50,000 and standard deviation σ = $5000.

The process of converting normal observations to z-scores using the formula is

called standardizing, or normalizing, the values and is particularly useful since tables of probabilities are widely available for z-scores. Normal probabilities can today be found in tables in statistics textbooks, statistical software, Excel, and in online website calculators. The true utility of the normal distribution lies in its applications. An operations manager may use a normal distribution to determine the probability of a shortage or an overage in inventory. A quality assurance analyst may use a normal distribution to determine control limits on a control chart and the presence of an out-of-control signal. An auditor may use a normal distribution to determine the proportion of accounts payable accounts to be audited. Clearly, the normal distribution is frequently used and continues to be an important part of business and operations.