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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
$\approx68\%$ of the data falls within $1$ standard deviation of the mean
$\approx95\%$ of the data falls within $2$ standard deviations of the mean
$\approx99.7\%$ of the data falls within $3$ 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

\mu=150\,\text{cm}

and a standard deviation of

\sigma=30\,\text{cm}

Sketch a normal curve that describes this distribution.

Solution:

Step 1: Sketch a normal curve.

Step 2: The mean of

150\,\text{cm}

goes in the middle.

Step 3: Each standard deviation is a distance of

30\,\text{cm}

Practice problem 1

The heights of the same variety of pine tree are also normally distributed. The mean height is

\mu=33\,\text{m}

and the standard deviation is

\sigma=3\,\text{m}

Which normal distribution below best summarizes the data?

Explain

Finding percentages example

A certain variety of pine tree has a mean trunk diameter of

\mu=150\,\text{cm}

and a standard deviation of

\sigma=30\,\text{cm}

Approximately what percent of these trees have a diameter greater than $210\,\text{cm}$ ?

Solution:

Step 1: Sketch a normal distribution with a mean of

\mu=150\,\text{cm}

and a standard deviation of

\sigma=30\,\text{cm}

Step 2: The diameter of

210\,\text{cm}

is two standard deviations above the mean. Shade above that point.

Step 3: Add the percentages in the shaded area:

2.35\%+0.15\%=2.5\%

About $2.5\%$ of these trees have a diameter greater than $210\,\text{cm}.$

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?

\%

Explain

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\,\text{cm}

and a standard deviation of

\sigma=30\,\text{cm}

A certain section of a forest has

500

of these trees.

Approximately how many of these trees have a diameter smaller than $120\,\text{cm}$ ?

Solution:

Step 1: Sketch a normal distribution with a mean of

\mu=150\,\text{cm}

and a standard deviation of

\sigma=30\,\text{cm}

Step 2: The diameter of

120\,\text{cm}

is one standard deviation below the mean. Shade below that point.

Step 3: Add the percentages in the shaded area:

0.15\%+2.35\%+13.5\%=16\%

About

16\%

of these trees have a diameter smaller than

120\,\text{cm}.

Step 4: Find how many trees in the forest that percent represents.

We need to find how many trees

16\%

500

is.

16\%\text{ of } 500=0.16\cdot500=80

About $80$ trees have a diameter smaller than $120\,\text{cm}$ .

practice problem 3

A certain variety of pine tree has a mean trunk diameter of

\mu=150\,\text{cm}

and a standard deviation of

\sigma=30\,\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

trees

Explain

Empirical rule