6.12: Applications of Normal Distribution

The normal distribution is a useful statistical tool. One of its practical applications is determining the door height after considering the normal distribution of heights of persons, such that many can pass through it easily without striking their heads. The normal distribution can also determine the probability of a person having a height less than a specific height.

The heights of 15 to 18-year-old males from Chile from 1984 to 1985 followed a normal distribution. The mean height is 172.36 cm, and the standard deviation of 6.34 cm. This information can be used to find the probability of males from Chile having a height of less than 162.85 cm.

Begin by finding the z score for the height of 162.85 cm. After using the formula for the z score, the value is found to be -1.5. From the table for negative z scores, the cumulative area under the curve (from the left of the standard normal distribution) or the probability is found to be 0.0668. Converting this value to a percentage gives 6.68%. It can be concluded that there is a 6.68% probability of males among 15 to 18-year-old males that have a height below 162.85 cm.

The normal distribution is widely applicable to many problems in real life.

For example, the statistics of human height are used to decide the door height that allows the majority of the people to walk through without striking their heads.

Let's assume that humans have a mean height of 1.7 meters with a standard deviation of 0.06 meters.

The shaded region in the normal distribution represents humans who are 1.9 meters or less.

First, convert the random variable in the X axis into z scores to obtain a standard normal distribution.

A height of 1.9 meters corresponds to a z score of 3.33. The corresponding probability is looked up in the z score table.

The probability is 0.9996, which tells us that 99.96 percent of people can walk through a door 1.9 meters tall.

Similarly, we can calculate the door height that would allow at least 85% of people to pass through without bending.

From the z table, note the value of the z score for a probability of 0.85.

With this z score, the required door height is calculated.