A normal distribution is a statistical graph used to help interpret data. The mean of the graph is the middle value, or the average of the set of data. The standard deviation is the value describing the spread of the data, or how far the data falls from the mean value.
The Normal Distribution
The normal distribution is a family of distributions. The distribution is commonly known as a bell curve. There are several properties that define a distribution as a normal distribution. Among then, the shape is symmetric, with the mean falling in the middle and the area under the curve equal to 1. Similarly, the area under the curve is equivalent to the probability. The mean defines the location of the graph on the number line, while the standard deviation defines the width of the graph. The larger the standard deviation, the wider the graph becomes.
Standardizing a Normal Distribution
A normal distribution can be standardized. This means that all data values in the distribution are converted to z-values. When a graph is standardized, the values create what is known as the standard normal distribution. The process to convert a normal distribution to a standard normal distribution is to convert all values in the data set to z-values by using the following equation:
(X - mean) / standard deviation
To compute all of the standardized values, you will need to find the mean and standard deviation. The value X stands for the data set of values; each value is separately insert into this equation in the place of X to find a new set of data referred to as the Z data set.
Standard Normal Distribution
Once a normal distribution has been standardized, there are a few additional properties that now hold true. The first is that for every standard normal curve the mean is equal to zero. This will always be true; if it is not true then the process of standardizing was not done properly. Additionally, the standard deviation is equal to one.
Finding the Mean and Standard Deviation
To find the mean of any distribution, if it is not given, simply find the average. You add all of the values in the data set and divide by the number of values in the set. For example, if the data is 1, 2, and 3 the average is (1 + 2 + 3) / 3 or 2.
To find the standard deviation takes a few steps.
Take all of the values in the set and subtract the mean from it. For the values above, this would yield (1-2), (2-2), and (3-2) or -1, 0, 1.
Take the square of each value found in step 1; this means take the value times itself. This would yield (-1 x -1), (0 x 0), and (1 x 1) or 1, 0, 1.
Add all of the values in the resulting set from step 2. This would mean taking 1 + 0 + 1, or 2.
Divide the resulting value from Step 3 by n-1, where n stands for the number of data in the set. Our set had three pieces of data, so n-1 is 2. This results in 3 / 2, or 1.5.
- The last step is to take the square root of the value from step 4. This would be the square root of 1.5 or 1.225.