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## Statistics and probability

### Course: 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
• $\approx 68\mathrm{%}$ of the data falls within $1$ standard deviation of the mean
• $\approx 95\mathrm{%}$ of the data falls within $2$ standard deviations of the mean
• $\approx 99.7\mathrm{%}$ 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 $\mu =150\phantom{\rule{0.167em}{0ex}}\text{cm}$ and a standard deviation of $\sigma =30\phantom{\rule{0.167em}{0ex}}\text{cm}$.
Sketch a normal curve that describes this distribution.
Solution:
Step 1: Sketch a normal curve.
Step 2: The mean of $150\phantom{\rule{0.167em}{0ex}}\text{cm}$ goes in the middle.
Step 3: Each standard deviation is a distance of $30\phantom{\rule{0.167em}{0ex}}\text{cm}$.
Practice problem 1
The heights of the same variety of pine tree are also normally distributed. The mean height is $\mu =33\phantom{\rule{0.167em}{0ex}}\text{m}$ and the standard deviation is $\sigma =3\phantom{\rule{0.167em}{0ex}}\text{m}$.
Which normal distribution below best summarizes the data?

### Finding percentages example

A certain variety of pine tree has a mean trunk diameter of $\mu =150\phantom{\rule{0.167em}{0ex}}\text{cm}$ and a standard deviation of $\sigma =30\phantom{\rule{0.167em}{0ex}}\text{cm}$.
Approximately what percent of these trees have a diameter greater than $210\phantom{\rule{0.167em}{0ex}}\text{cm}$?
Solution:
Step 1: Sketch a normal distribution with a mean of $\mu =150\phantom{\rule{0.167em}{0ex}}\text{cm}$ and a standard deviation of $\sigma =30\phantom{\rule{0.167em}{0ex}}\text{cm}$.
Step 2: The diameter of $210\phantom{\rule{0.167em}{0ex}}\text{cm}$ is two standard deviations above the mean. Shade above that point.
$2.35\mathrm{%}+0.15\mathrm{%}=2.5\mathrm{%}$
About $2.5\mathrm{%}$ of these trees have a diameter greater than $210\phantom{\rule{0.167em}{0ex}}\text{cm}.$
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?
$\mathrm{%}$

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 $\mu =150\phantom{\rule{0.167em}{0ex}}\text{cm}$ and a standard deviation of $\sigma =30\phantom{\rule{0.167em}{0ex}}\text{cm}$.
A certain section of a forest has $500$ of these trees.
Approximately how many of these trees have a diameter smaller than $120\phantom{\rule{0.167em}{0ex}}\text{cm}$?
Solution:
Step 1: Sketch a normal distribution with a mean of $\mu =150\phantom{\rule{0.167em}{0ex}}\text{cm}$ and a standard deviation of $\sigma =30\phantom{\rule{0.167em}{0ex}}\text{cm}$.
Step 2: The diameter of $120\phantom{\rule{0.167em}{0ex}}\text{cm}$ is one standard deviation below the mean. Shade below that point.
$0.15\mathrm{%}+2.35\mathrm{%}+13.5\mathrm{%}=16\mathrm{%}$
About $16\mathrm{%}$ of these trees have a diameter smaller than $120\phantom{\rule{0.167em}{0ex}}\text{cm}.$
We need to find how many trees $16\mathrm{%}$ of $500$ is.
About $80$ trees have a diameter smaller than $120\phantom{\rule{0.167em}{0ex}}\text{cm}$.
A certain variety of pine tree has a mean trunk diameter of $\mu =150\phantom{\rule{0.167em}{0ex}}\text{cm}$ and a standard deviation of $\sigma =30\phantom{\rule{0.167em}{0ex}}\text{cm}$.
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?
$\approx$