The normal distribution is the most commonly used probability distribution in statistics.
It has the following properties:
- Mean and median are equal; both located at the center of the distribution
The mean of the normal distribution determines its location and the standard deviation determines its spread.
For example, the following plot shows three normal distributions with different means and standard deviations:
The standard normal distribution is a specific type of normal distribution where the mean is equal to 0 and the standard deviation is equal to 1.
The following plot shows a standard normal distribution:
How to Convert a Normal Distribution to Standard Normal Distribution
Any normal distribution can be converted into a standard normal distribution by converting the data values into z-scores, using the following formula:
z = (x – μ) / σ
- x: Individual data value
- μ: Mean of the distribution
- σ: Standard deviation of the distribution
For example, suppose we have the following dataset with a mean of 6 and a standard deviation of 2.152:
We can convert each individual data value into a z-score by subtracting 6 from each value and dividing by 2.152:
The z-score tells us how many standard deviations each data point lies from the mean. For example, the first data value of “3” lies 1.39 standard deviations below the mean.
The mean of this distribution of z-scores has a mean of zero and a standard deviation of one.
How to Use the Standard Normal Distribution
A standard normal distribution has the following properties:
- About 68% of data falls within one standard deviation of the mean
- About 95% of data falls within two standard deviations of the mean
- About 99.7% of data falls within three standard deviations of the mean
This is known as the Empirical Rule and is used to understand the distribution of values in a dataset.
For example, suppose the height of plants in a certain garden are normally distributed with a mean of 47.4 inches and a standard deviation of 2.4 inches.
According to the Empirical Rule, what percentage of plants are less than 54.6 inches tall?
The Empirical Rule states that for a given dataset with a normal distribution, 99.7% of data values fall within three standard deviations of the mean. This means that 49.85% of values fall between the mean and three standard deviations above the mean.
In this example, 54.6 is located three standard deviations above the mean. Since we know that 50% of data values fall below the mean in a normal distribution, a total of 50% + 49.85% = 99.85% of values fall below 54.6.
Thus, 99.85% of plants are less than 54.6 inches tall.
Empirical Rule Practice Problems
Empirical Rule Calculator
How to Apply the Empirical Rule in Excel