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
  • 68%\approx68\% of the data falls within 11 standard deviation of the mean
  • 95%\approx95\% of the data falls within 22 standard deviations of the mean
  • 99.7%\approx99.7\% of the data falls within 33 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 μ=150cm\mu=150\,\text{cm} and a standard deviation of σ=30cm\sigma=30\,\text{cm}.
Sketch a normal curve that describes this distribution.
Solution:
Step 1: Sketch a normal curve.
Step 2: The mean of 150cm150\,\text{cm} goes in the middle.
Step 3: Each standard deviation is a distance of 30cm30\,\text{cm}.
Practice problem 1
The heights of the same variety of pine tree are also normally distributed. The mean height is μ=33m\mu=33\,\text{m} and the standard deviation is σ=3m\sigma=3\,\text{m}.
Which normal distribution below best summarizes the data?
Choose 1 answer:
Choose 1 answer:

Finding percentages example

A certain variety of pine tree has a mean trunk diameter of μ=150cm\mu=150\,\text{cm} and a standard deviation of σ=30cm\sigma=30\,\text{cm}.
Approximately what percent of these trees have a diameter greater than 210cm210\,\text{cm}?
Solution:
Step 1: Sketch a normal distribution with a mean of μ=150cm\mu=150\,\text{cm} and a standard deviation of σ=30cm\sigma=30\,\text{cm}.
Step 2: The diameter of 210cm210\,\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%2.35\%+0.15\%=2.5\%
About 2.5%2.5\% of these trees have a diameter greater than 210cm.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 9090 and 210210 centimeters?
%\%
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 μ=150cm\mu=150\,\text{cm} and a standard deviation of σ=30cm\sigma=30\,\text{cm}.
A certain section of a forest has 500500 of these trees.
Approximately how many of these trees have a diameter smaller than 120cm120\,\text{cm}?
Solution:
Step 1: Sketch a normal distribution with a mean of μ=150cm\mu=150\,\text{cm} and a standard deviation of σ=30cm\sigma=30\,\text{cm}.
Step 2: The diameter of 120cm120\,\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%0.15\%+2.35\%+13.5\%=16\%
About 16%16\% of these trees have a diameter smaller than 120cm.120\,\text{cm}.
Step 4: Find how many trees in the forest that percent represents.
We need to find how many trees 16%16\% of 500500 is.
16% of 500=0.16500=8016\%\text{ of } 500=0.16\cdot500=80
About 8080 trees have a diameter smaller than 120cm120\,\text{cm}.
practice problem 3
A certain variety of pine tree has a mean trunk diameter of μ=150cm\mu=150\,\text{cm} and a standard deviation of σ=30cm\sigma=30\,\text{cm}.
A certain section of a forest has 500500 of these trees.
Approximately how many of these trees have a diameter between 120120 and 180180 centimeters?
\approx
trees
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