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## Statistics and probability

### Unit 4: Lesson 5

Normal distributions and the empirical rule

# Normal 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
• approximately equals, 68, percent of the data falls within 1 standard deviation of the mean
• approximately equals, 95, percent of the data falls within 2 standard deviations of the mean
• approximately equals, 99, point, 7, percent of the data falls within 3 standard deviations of the mean

### Drawing a normal distribution example

The trunk diameter of a certain variety of pine tree is normally distributed with a mean of mu, equals, 150, start text, c, m, end text and a standard deviation of sigma, equals, 30, start text, c, m, end text.
Sketch a normal curve that describes this distribution.
Solution:
Step 1: Sketch a normal curve.
Step 2: The mean of 150, start text, c, m, end text goes in the middle.
Step 3: Each standard deviation is a distance of 30, start text, c, m, end text.
Practice problem 1
The heights of the same variety of pine tree are also normally distributed. The mean height is mu, equals, 33, start text, m, end text and the standard deviation is sigma, equals, 3, start text, m, end text.
Which normal distribution below best summarizes the data?

### Finding percentages example

A certain variety of pine tree has a mean trunk diameter of mu, equals, 150, start text, c, m, end text and a standard deviation of sigma, equals, 30, start text, c, m, end text.
Approximately what percent of these trees have a diameter greater than 210, start text, c, m, end text?
Solution:
Step 1: Sketch a normal distribution with a mean of mu, equals, 150, start text, c, m, end text and a standard deviation of sigma, equals, 30, start text, c, m, end text.
Step 2: The diameter of 210, start text, c, m, end text is two standard deviations above the mean. Shade above that point.
2, point, 35, percent, plus, 0, point, 15, percent, equals, 2, point, 5, percent
About 2, point, 5, percent of these trees have a diameter greater than 210, start text, c, m, end text, point
Want to see another example like this? Check out this video.
practice problem 2
Approximately what percent of these trees have a diameter between 90 and 210 centimeters?
percent

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 mu, equals, 150, start text, c, m, end text and a standard deviation of sigma, equals, 30, start text, c, m, end text.
A certain section of a forest has 500 of these trees.
Approximately how many of these trees have a diameter smaller than 120, start text, c, m, end text?
Solution:
Step 1: Sketch a normal distribution with a mean of mu, equals, 150, start text, c, m, end text and a standard deviation of sigma, equals, 30, start text, c, m, end text.
Step 2: The diameter of 120, start text, c, m, end text is one standard deviation below the mean. Shade below that point.
0, point, 15, percent, plus, 2, point, 35, percent, plus, 13, point, 5, percent, equals, 16, percent
About 16, percent of these trees have a diameter smaller than 120, start text, c, m, end text, point
Step 4: Find how many trees in the forest that percent represents.
We need to find how many trees 16, percent of 500 is.
16, percent, start text, space, o, f, space, end text, 500, equals, 0, point, 16, dot, 500, equals, 80
About 80 trees have a diameter smaller than 120, start text, c, m, end text.
practice problem 3
A certain variety of pine tree has a mean trunk diameter of mu, equals, 150, start text, c, m, end text and a standard deviation of sigma, equals, 30, start text, c, m, end text.
A certain section of a forest has 500 of these trees.
Approximately how many of these trees have a diameter between 120 and 180 centimeters?
approximately equals
trees