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Course: Statistics and probability > Unit 4
Lesson 5: Normal distributions and the empirical ruleNormal distributions review
Normal distributions come up time and time again in statistics. A normal distribution has some interesting properties: it has a bell shape, the mean and median are equal, and 68% of the data falls within 1 standard deviation.
What is a normal distribution?
Early statisticians noticed the same shape coming up over and over again in different distributions—so they named it the normal distribution.
Normal distributions have the following features:
- symmetric bell shape
- mean and median are equal; both located at the center of the distribution
of the data falls within standard deviation of the mean of the data falls within standard deviations of the mean of the data falls within standard deviations of the mean
Want to learn more about what normal distributions are? Check out this video.
Drawing a normal distribution example
The trunk diameter of a certain variety of pine tree is normally distributed with a mean of and a standard deviation of .
Sketch a normal curve that describes this distribution.
Solution:
Step 1: Sketch a normal curve.
Step 2: The mean of goes in the middle.
Step 3: Each standard deviation is a distance of .
Finding percentages example
A certain variety of pine tree has a mean trunk diameter of and a standard deviation of .
Approximately what percent of these trees have a diameter greater than ?
Solution:
Step 1: Sketch a normal distribution with a mean of and a standard deviation of .
Step 2: The diameter of is two standard deviations above the mean. Shade above that point.
Step 3: Add the percentages in the shaded area:
About of these trees have a diameter greater than
Want to see another example like this? Check out this video.
Want to practice more problems like this? Check out this exercise on the empirical rule.
Finding a whole count example
A certain variety of pine tree has a mean trunk diameter of and a standard deviation of .
A certain section of a forest has of these trees.
Approximately how many of these trees have a diameter smaller than ?
Solution:
Step 1: Sketch a normal distribution with a mean of and a standard deviation of .
Step 2: The diameter of is one standard deviation below the mean. Shade below that point.
Step 3: Add the percentages in the shaded area:
About of these trees have a diameter smaller than
Step 4: Find how many trees in the forest that percent represents.
We need to find how many trees of is.
About trees have a diameter smaller than .
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