Solution - Cumulative probability in the standard normal distribution

The normal distribution is important because we see it often in nature. Suppose we gather many unrelated measures, like human heights, blood pressure readings, or IQ scores. They will follow the normal distribution.

We see many normally distributed variables in psychology. For example, reading ability, introversion or job satisfaction. In investing, the normal distribution shows asset class returns. Although these distributions are only roughly normal, they are pretty close, and we can treat them as normal.

The normal distribution is easy to work with. Many statistical tests rely on it. Moreover, these tests work well even when the distribution is only approximately normal. For example, if a set's mean and standard deviation are known, and the set follows the normal distribution, we can easily convert between percentiles and raw scores.

Any normal distribution can be standardized to a standard normal distribution. That way, we can compare two or more separate data sets. Using standard normal distribution, we can estimate probabilities of events involving normal distribution. This way, we can estimate how tall a person is likely to grow, for instance.