Applications of the Normal Distribution in Business and Statistics

The normal distribution is one of the most widely applied tools in business statistics precisely because so many real variables approximate it once observations are numerous. Typical Gujarat Std-12 Commerce applications include:

• Quality control: the life of manufactured items (bulbs, machine parts), the weight/volume of packaged goods, or the diameter of components produced on a machine typically cluster normally around a target mean, letting a manufacturer estimate what proportion of output will fail to meet a specification.
• Wages and income analysis: wages of a large body of workers in a factory or organisation are often modelled as approximately normal, letting management estimate what proportion of the workforce falls into a particular wage bracket.
• Examination and performance analysis: marks scored by a large batch of students in a public examination tend to follow a normal (or near-normal) pattern, which is the basis for percentile ranking, grading on a curve, and identifying how many students are likely to cross a given cut-off.
• Finding a cut-off value for a target percentage — the reverse problem: instead of being given $x$ and asked for a probability, we are given a target probability (e.g. "the top 10% of workers") and asked to find the corresponding $x$. This is solved by first finding the $z$ that gives the required area from the table, and then reversing the standardization formula:

The general 4-step method used for every applied numerical in this chapter (the method every worked example and exercise below follows):

1. Identify $\mu$ and $\sigma$ from the question, and write down exactly what probability, percentage, or number of items is being asked for.
2. Standardize: compute $z = \dfrac{x-\mu}{\sigma}$ for every raw value given (or, in a reverse problem, look up the $z$ for the given probability first).
3. Apply the correct rule from Section 4 (less-than / more-than / between / symmetry) to get the area (probability). …