The normal distribution is the most commonly used probability distribution in statistics.
It has the following properties:
- Symmetrical
- Bell-shaped
- 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
where:
- 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.
Additional Resources
Empirical Rule Practice Problems
Empirical Rule Calculator
How to Apply the Empirical Rule in Excel
Thank you so much for your simple, clear and easy to understand article.
Fantastic and clearly written
It is really helpful. Could you please help me understand how are you considering three standard deviation i.e 99.7. Why not other standard deviations. Is it so because it is mentioned as 2.4 inches standard deviation in the problem ?
Hi Ritesh…It sounds like you’re referring to the application of the empirical rule, also known as the 68-95-99.7 rule, which is used in statistics to describe how data is distributed in a normal distribution.
### The Empirical Rule
The empirical rule states:
– **68%** of the data falls within **1 standard deviation** of the mean.
– **95%** of the data falls within **2 standard deviations** of the mean.
– **99.7%** of the data falls within **3 standard deviations** of the mean.
### Why 3 Standard Deviations (99.7%)?
The reason 3 standard deviations are often used is that it encompasses nearly all of the data in a normal distribution—99.7% to be exact. This makes it a common choice when you want to understand the “extremes” of your data or when you want to include almost all the possible outcomes.
However, the choice of how many standard deviations to consider depends on the specific context or problem at hand:
– **1 standard deviation** (68%) is used if you want to focus on the bulk of your data but aren’t concerned with more extreme values.
– **2 standard deviations** (95%) is common when you want a more comprehensive understanding that includes most of the data but excludes some outliers.
– **3 standard deviations** (99.7%) is used when you want to be almost certain that you are covering all possible outcomes.
### Specific to Your Problem
If your problem mentions a standard deviation of 2.4 inches, that number is simply describing the spread of your data. The choice of whether to consider 1, 2, or 3 standard deviations depends on how much of the data you want to include in your analysis.
For instance, if you’re working with a problem where you need to calculate a range that covers nearly all potential outcomes, using 3 standard deviations (99.7%) is appropriate. However, if you only need to cover the most typical outcomes, 1 or 2 standard deviations might be sufficient.
In summary, the choice of using 1, 2, or 3 standard deviations isn’t directly linked to the specific value of the standard deviation (like 2.4 inches); it’s about how much of the data you want to include in your analysis, based on the empirical rule.