Standard Normal Distribution
The standard normal distribution is a probability density function with a mean of and a standard deviation of . The input values to the standard normal distribution correspond to z-score values. The area under the curve of the function is equal to .
The standard normal distribution function is given in the equation above. This standardized form of the normal distribution allows for probabilities to be easily calculated or looked-up.
|The circle constant appears in the scaling factor that ensures the area under the distribution is equal to .|
|Euler’s number is chosen as the base of the exponential expression to give the function helpful properties and to make the values of the other variables more meaningful.|
|The input variable , commonly referred to as a “z-score”.|
The standard normal distribution “standardizes” values using z-score values, which is why the input to the PDF and CDF are represented using the variable . A z-score value represents a value in terms of its distance from the mean in terms of standard deviations. For example, to calculate a z-score for the value on a normal distribution with a mean equal to and a standard deviation equal to the formula would be:
This value represents that the value is standard deviations to the right of the mean of the distribution. Since the standard normal deviation and another normal deviation share the same properties, the standard normal distribution can be used to calculate probabilities
The probability of an event occuring on a probability density function between two values, and , is equal to the area under the curve from to . For example, the probability of an event occuring within standard deviation of the mean of a normal distribution is equal to . The general integral forms for calculating probability for PDFs are given below:
|The probability of an event occuring below a threshold .|
|The probability of an event occuring above a threshold .|
|The probability of an event occuring between and .|
There are three commons strategies, discussed below, for calculating these probabilities using the standard normal distribution: 1) Use a statistical function such as
NORM.S.DIST as implemented in Excel and Google Sheets. 2) Use the normal cumulative distribution function (CDF) defined with the error function. 3) Look up the probability corresponding to a z-value in a table.
= NORM.S.DIST(1.5) = .9332
The cumulative distribution function is described in the equation above. The output of the standard normal CDF is equal to the output of the
NORM.S.DIST function. For example, is equal to .
The output of the standard normal CDF corresponds area under the curve to the left of a value as shown in the graph below.
The plot of is given below.
The popularity of the standard normal distribution can likely be attributed to the difficulty of computing the area under the curve of the normal distribution. By standardizing the values of all normal distributions, different probabilities can be conveniented looked up in a table.
All three strategies for finding the area under the curve, discussed above, find the area to the left of a threshold. These examples below demonstrate how that information is sufficient to find the probability of an event below a threshold, above a threshold and between thresholds.
The probability of an event occuring below a threshold on the standard normal distribution corresponds to the area under the curve to the left of the threshold.
The probability of an event occuring above a threshold on the standard normal distribution corresponds to the area under the curve to the right of the threshold. This probability is equal to minus the probability of the event occuring below the threshold.
The probability of an event occuring in between thresholds defined by the values and , where is equal to .
The normal distribution is a continuous probability distribution that appears naturally in applications of statistics and probability. The shape of the function forms a "bell-curve" which is symmetric around the mean and whose shape is described by the standard deviation.