In order to qualify as a standard normal probability distribution, the distribution must meet certain requirements, such as having a mean of zero. Any random variable that has a normal distribution will follow certain rules which can be used to calculate the standard deviation or probability of a specific outcome. Normal probability distribution is an incredibly useful statistical tool.
Bell Shaped Curve
When random variables with a normal distribution are graphed, they result in a bell shaped curve. The bell shaped curve shows that the farther a value is from the mean, the less likely it is to occur. The lowest values of x have a low probability, but that probability increases as the value of x increases. The probability peaks at the mean, and descends steadily, so that the highest values of x are incredibly unlikely as well. The bell shaped curve is a mirror image, so that the right side of the mean looks identical to the left.
Everything is Normal
In statistics, there is the assumption that if the sample is large enough, it will end up with a normal distribution. How large the sample should be depends on the demographics of the group and the random variable being tested. The value of the mean may change, but as long as there are a few people who are extreme, and the majority cluster around the average, the probability distribution will have a bell shaped curve. In other words, they naturally create the bell shaped curve.
An Example of Normality
College students, as a general rule, have an above average IQ. If the IQs of the average population were graphed, it would result in a typical bell shaped curve, with most college students clustered above the mean. However, if college students were placed in their own category, their IQ scores would create a normal distribution as well. The mean IQ for college students would be higher, and the standard deviation would be smaller, but overall there would be some incredibly smart students, some students well below average, and most students would fall right in the middle.
Area of a Bell Shaped Curve
When working with a standard normal probability distribution, parts of the bell shaped curve are shaded in to indicate that they meet the requirements for a successful trial. In other words, the shaded area indicates p. The total area under the curve is said to equal one. The area also equals the combined probability of all possible results (p +q). Because of this, the probability of an event is simply equal to the shaded area. If the shaded part under the bell curve has an area equal to 0.95, the desired event has a probability of 95%.
Using Z Scores
The z score simply indicates how many standard deviations away from the mean something is. A positive z score refers to a value that is so many standard deviations above the mean, whereas a negative z score will lie below the mean. Z scores with the same number but opposite sign will be equidistant from the mean. Z scores are helpful since each z score represents a specific probability. There are tables available which compile all z scores and their corresponding probability, but many people find it easier to use computer programs such as STAQTDISK, Minitab, or Excel.
Itís assumed that a large enough sample will produce a normal probability distribution. This assumption is helpful because properties of a normal distribution help statisticians make a number of calculations. The distribution makes probability very easy to calculate and also helps to determine the standard deviation thanks to the z score.