A normal distribution is a type of probability distribution where the mean, the median, and the mode are all equal to each other. The graph that represents a normal distribution is called a normal curve and it is a continuous, bell-shaped curve that is symmetric with respect to the mean $\mu$ of the data set.

This type of distribution is the most common continuous probability distribution that can be observed in real life. When a normal distribution has a mean of $0$ and standard deviation of $1,$ it is called a standard normal distribution. A normal distribution can be standardized by transforming each of its values into their corresponding $z\text{-}$scores.

The total area under the normal curve is $100\%,$ or $1.$ Because of this, the area under the normal curve in a certain interval represents the percentage of data within that interval or the probability of randomly selecting a value that belongs to that interval. The Empirical Rule can be used to determine the area under the normal curve at specific intervals.

Example

Consider the weights of oranges as an example of normally distributed data. The mean weight of an orange is about $310$ grams and the standard deviation is approximately $15$ grams. The distribution of a sample of weights of $1000$ randomly chosen oranges is described by the following histogram.

The histogram will look more and more like a normal curve as more and more observations are made. In addition to this example, topics such as human height or income also tend to be normally distributed. It is also worth noting that not all data sets are normally distributed. If the mean and median are not equal, then the data set is skewed.