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Daily Dose 003 | Probability and Statistics
How do you determine the proportion of range within a Normal Distribution?
Normal Distribution.
A problem type you are guaranteed to see on the exam. Question is –
Will you be ready for it.
A Normal Distribution problem can take one of many terms, but at the end of the day, the process can be broken down a simplified.
When you are done working through this lesson, I am confident that you will be ready to put this problem type in your bank of points.
Key Definition
What is the NORMAL DISTRIBUTION?
NORMAL DISTRIBUTION refers to a family of continuous probability distributions that graphically lays out as a BELL CURVE.
Sometimes referred to as Gaussian distribution, the majority of the data points in a NORMAL DISTRIBUTION are clustered near the center of the bell curve.
The standard NORMAL DISTRIBUTION has an AREA under the CURVE of 1 with a mean of zero and standard deviation of 1.
The NORMAL DISTRIBUTION (or BELL CURVE) is centered at zero and the degree to which a given measurement deviates from the mean is given by the standard deviation.
The NORMAL DISTRIBUTION follows the 68-95-99.7 rule, which says that for a the standard NORMAL DISTRIBUTION, 68% of the observations lie within 1 standard deviation of the mean; 95% lie within two standard deviation of the mean; and 99.9% lie within 3 standard deviations of the mean.
The Process
How can you determine the proportion of range within a Normal Distribution?
Determining the proportion within a range of a normal distribution requires us to use the the principles of continuous probability distributions found within the FE Reference Handbook.
Because it requires us to define a number of variables that are then cross referenced with tables, the process may first appear complicated. But it doesn’t need to be, there’s a general step-by-step process that can be used to simplify your work no matter how complicated it appears.
Here’s the process:
- Step 1: Identify the parameters of the Normal Distribution
- Mean (μ): The average value around which the distribution is centered.
- Standard deviation (σ): A measure of the dispersion or spread of the data points.
- Step 2: Define the range of interest
- Step 3: Calculate the Z-scores for the lower and upper bounds
- Z = (X – μ) / σ
- X is the value within the range you want to evaluate
- μ = Mean of the normal distribution
- σ = Standard deviation of the normal distribution
- Step 4: Look up the cumulative probability (proportion) associated with the Z-scores:
- Step 5: Calculate the proportion within the range:
As you are reading through the problem statement, note the data that is presented, ideally the:
It’s important to note that these variables may be presented in many different forms, including just raw experimental data that must then be converted into the appropriate parameters using each respective general formula.
Once you have read through the problem statement and know specifically what is being asked, determine the lower and upper bounds of the range you want to evaluate.
The Z-score represents the number of standard deviations a data point is from the mean and helps standardize the normal distribution.
Each Z-score can be defined using the formula:
Where:
Referencing the standard normal distribution table in the FE Reference Handbook, find the cumulative probability (proportion) for the calculated Z-scores.
Subtract the cumulative probability (proportion) for the lower bound from the cumulative probability (proportion) for the upper bound to determine the proportion within the specified range.
This step by step procedure will allow you to determine proportions of even the most complex Normal Distribution problems, but to keep it simple, here is a quick example.
Let’s say you are given a problem that illustrates a normal distribution with a mean (μ) of 50 and a standard deviation (σ) of 5, and you are asked to find the proportion of data points that fall between 45 and 55.
Our range is between 45 and 55, and are being asked to determine the proporation of the entire Normal Distribution that falls within that range – let’s go step by step.
- Step 1: Identify the parameters of the Normal Distribution
- μ = 50
- σ = 5
- Step 2: Define the range of interest
- Step 3: Calculate the Z-scores for the lower and upper bounds
- Z = (X – μ) / σ
- Z_lower = (45 – 50) / 5 = -1
- Z_upper = (55 – 50) / 5 = 1
- Step 4: Look up the cumulative probability (proportion) associated with the Z-scores:
- Cumulative probability for Z_lower ≈ 0.1587
- Cumulative probability for Z_upper ≈ 0.8413
- Step 5: Calculate the proportion within the range:
- 0.8413 – 0.1587 ≈ 0.6826
Reading through the problem statement, we are given:
As already determined, our range is from 45 to 55.
Each Z-score can be defined using the formula:
Which gives us:
Referencing the standard normal distribution table in the FE Reference Handbook, we determine that the cumulative probability (proportion) for each Z-score is:
Subtracting the cumulative probability (proportion) for the lower bound from the cumulative probability (proportion) for the upper bound to determine the proportion within the specified range, we get:
Therefore, approximately 68.26% of the data points are within the range of 45 to 55 in the given normal distribution.
Check out the video to see an FE Exam Probability and Statistics Practice Problem that deals with a much more complex scenario and how we can go about solving it in the most efficient manner.
As always, we are here to help, Prepineer
