Applications of Normal Distribution

The normal distribution models human height variation to design doors that accommodate most people safely. Given a mean height of 1.7 meters and standard deviation of 0.06 meters, a 1.9-meter door allows 99.96% of people to pass through without striking their heads. This practical application uses z scores and area under the curve to calculate the probability that individuals can safely use the doorway.

A z score standardizes individual measurements by converting them to a common scale relative to the mean and standard deviation. For example, a height of 1.9 meters with mean 1.7 and standard deviation 0.06 yields a z score of 3.33. This conversion allows researchers to use standard normal distribution tables to find probabilities and make comparisons across different datasets.

Identify the z score corresponding to a cumulative probability of 0.85 from the z score table. Then use the z score formula in reverse: height equals mean plus z score multiplied by standard deviation. This calculation determines the specific door height needed to accommodate at least 85% of the population without requiring them to bend.

A z score table provides cumulative probabilities showing the proportion of a population falling below any given z score value. For instance, a z score of -1.5 corresponds to a probability of 0.0668, meaning 6.68% of the population falls below that point. This allows statisticians to quantify how common or rare specific measurements are within a normally distributed population.

Heights of 15 to 18-year-old males from Chile (1984-1985) followed a normal distribution with mean 172.36 cm and standard deviation 6.34 cm. Using z scores, researchers calculated that 6.68% of males had heights below 162.85 cm. This demonstrates how normal distribution analysis quantifies the probability of specific height ranges in real populations.

Converting measurements to z scores standardizes data to a common scale, enabling use of universal z score tables for probability lookup. Without this conversion, each dataset would require its own probability table. Z scores allow researchers to compare measurements across different populations and quickly determine cumulative probabilities for any normally distributed variable.

Normal distribution solves practical problems involving population measurements, such as designing infrastructure (door heights), assessing health risks, or setting manufacturing tolerances. By analyzing how measurements like height, weight, or test scores distribute across populations, organizations can make evidence-based decisions that accommodate the majority while identifying unusual cases.