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Statistics and probability

Course: Statistics and probability > Unit 4

Lesson 5: Normal distributions and the empirical rule
Math
Statistics and probability
Modeling data distributions
Normal distributions and the empirical rule
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Normal distributions review

Google Classroom
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.
A normal distribution curve is plotted along a horizontal axis labeled, Mean, which ranges from negative 3 to 3 in increments of 1 The curve rises from the horizontal axis at negative 3 with increasing steepness to its peak at 0, before falling with decreasing steepness through 3, then appearing to plateau along the horizontal axis. All values estimated. The area under the curve to the left of negative 3 and right of 3 are each labeled 0.15%. The area between negative 3 and negatve 2, and 2 and 3, are each labeled 2.35%. The area between negative 2 and negative 1, and 1 and 2, are each labeled 13.5%. The area between negative 1 and 0, and 0 and 1, are each labeled 34%.
Normal distributions have the following features:
  • symmetric bell shape
  • mean and median are equal; both located at the center of the distribution
  • ≈68% of the data falls within 1 standard deviation of the mean
  • ≈95% of the data falls within 2 standard deviations of the mean
  • ≈99.7% 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 μ=150cm and a standard deviation of σ=30cm.
Sketch a normal curve that describes this distribution.
Solution:
Step 1: Sketch a normal curve.
Step 2: The mean of 150cm goes in the middle.
Step 3: Each standard deviation is a distance of 30cm.
A normal distribution curve is plotted along a horizontal axis labeled, Trunk Diameter in centimeters, which ranges from 60 to 240 in increments of 30. The curve rises from the horizontal axis at 60 with increasing steepness to its peak at 150, before falling with decreasing steepness through 240, then appearing to plateau along the horizontal axis. All values estimated.
Practice problem 1
The heights of the same variety of pine tree are also normally distributed. The mean height is μ=33m and the standard deviation is σ=3m.
Which normal distribution below best summarizes the data?
Choose 1 answer:

Finding percentages example

A certain variety of pine tree has a mean trunk diameter of μ=150cm and a standard deviation of σ=30cm.
Approximately what percent of these trees have a diameter greater than 210cm?
Solution:
Step 1: Sketch a normal distribution with a mean of μ=150cm and a standard deviation of σ=30cm.
Step 2: The diameter of 210cm 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 210cm.
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?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
%

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 and a standard deviation of σ=30cm.
A certain section of a forest has 500 of these trees.
Approximately how many of these trees have a diameter smaller than 120cm?
Solution:
Step 1: Sketch a normal distribution with a mean of μ=150cm and a standard deviation of σ=30cm.
Step 2: The diameter of 120cm is one standard deviation below the mean. Shade below that point.
Step 3: Add the percentages in the shaded area:
A normal distribution curve is plotted along a horizontal axis labeled, Trunk Diameter in centimeters, which ranges from 60 to 240 in increments of 30. The curve rises from the horizontal axis at 60 with increasing steepness to its peak at 150, before falling with decreasing steepness through 240, then appearing to plateau along the horizontal axis. All values estimated. The area under the curve to the left of 60 and right of 240 are each labeled 0.15%. The area between 60 and 90, and 210 and 240, are each labeled 2.35%. The area between 90 and 120, and 180 and 210, are each labeled 13.5%. The area between 120 and 150, and 150 and 180. The regions at 120 and less are all shaded.
0.15%+2.35%+13.5%=16%
About 16% of these trees have a diameter smaller than 120cm.
Step 4: Find how many trees in the forest that percent represents.
We need to find how many trees 16% of 500 is.
16% of 500=0.16⋅500=80
About 80 trees have a diameter smaller than 120cm.
practice problem 3
A certain variety of pine tree has a mean trunk diameter of μ=150cm and a standard deviation of σ=30cm.
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?
≈
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
trees

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