The cumulative probability for a value is the probability less than or equal to that value. In notation, this is \(P(X\leq x)\). The proportion at or below a given value is also known as a percentile.
Minitab® – Proportion Less Than a z Value
Question: What proportion of the standard normal distribution is less than a z score of -2?
Recall that the standard normal distribution (i.e., z distribution) has a mean of 0 and standard deviation of 1. This is the default normal distribution in Minitab.
- From the tool bar select Graph > Probability Distribution Plot > One Curve > View Probability
- Check that the Mean is 0 and the Standard deviation is 1
- Select Options
- Select A specified x value
- Select Left tail
- For X value enter -2
- Click Ok
- Click Ok
This should result in the following output:
The proportion of the standard normal distribution that is less than a z score of -2 is 0.02275.
This could also be written as P(z < -2) = 0.02275.
Minitab® – Proportion Less Than a Value on a Normal Distribution
Scenario: Vehicle speeds at a highway location have a normal distribution with a mean of 65 mph and a standard deviation of 5 mph. What is the probability that a randomly selected vehicle will be going 73 mph or slower?
Let's construct a normal distribution with a mean of 65 and standard deviation of 5 to find the area less than 73.
- From the tool bar select Graph > Probability Distribution Plot > One Curve > View Probability
- Change the Mean to 65 and the Standard deviation to 5
- Select Options
- Select A specified x value
- Select Left tail
- For X value enter 73
- Click Ok
- Click Ok
This should result in the following output:
On a normal distribution with a mean of 65 mph and standard deviation of 5 mph, the proportion less than 73 mph is 0.9452.
In other words, 94.52% of vehicles will be going 73 mph or slower.