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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

## Want to join the conversation?

• Hello folks, For your finding percentages practice problem, the part of the explanation "the upper boundary of 210 is one standard deviation above the mean" probably should be two standard deviations. You do a great public service. Thanks.
• Nice one Richard, we can all trust you to keep the streets of Khan academy safe from errors
• Anyone else doing khan academy work at home because of corona?
• Yea I just don't understand the point of this it makes no sense and how do I need this to be able to throw a football, I don't.
• Why do the mean, median and mode of the normal distribution coincide?
• 16% percent of 500, what does the 500 represent here? and where it was given in the shape
(1 vote)
• 500 represent the number of total population of the trees. And the question is asking the NUMBER OF TREES rather than the percentage. So we need to figure out the number of trees that is 16 percent of the 500 trees, which would be 0.16*500.
Hope it helps.
• What is the mode of a normal distribution? The way I understand, the probability of a given point(exact location) in the normal curve is 0. So,is it possible to infer the mode from the distribution curve?
(1 vote)
• A normal distribution has a mean of 80 and a standard deviation of 20. Refer to the table in Appendix B.1.

Determine the value above which 80 percent of the values will occur. (Round z-score computation to 2 decimal places and the final answer to 2 decimal places.)

X
(1 vote)
• So, my teacher wants us to graph bell curves, but I was slightly confused about how to graph them. Do you just make up the curve and write the deviations or whatever underneath?
(1 vote)
• hello, I am really stuck with the below question, and unable to understand on text. I'd be really appreciated if someone can help to explain this quesion
6) The total home-game attendance for major-league baseball is the sum of all attendees for all stadiums during the entire season. The home attendance ( in millions) for a number of years is shown in the table below

Year Home Attendance (millions)

1978 40.6

1979 43.5

1980 43.0

1981 26.6

1982 44.6

1983 46.3

1984 48.7

1985 49.0

1986 50.5

1987 51.8

1988 53.2

a) Make a scatterplot showing the trend in home attendance. Describe what you see.

Inline image 2

a) Make a scatterplot showing the trend in home attendance. Describe what you see.

b) Determine the correlation, and comment on its significance.

c) Find the equation of the line of regression. Interpret the slope of the equation.

d) Use your model to predict the home attendance for 1998. How much confidence do you have in this

prediction? Explain.

e) Use the internet or other resource to find reasons for any outliers you observe in the scatterplot.
(1 vote)
• These questions include a few different subjects. Basically you try to approximate a (linear) line of regression by minimizing the distances between all the data points and their predictions. When you have modeled the line of regression, you can make predictions with the equation you get. Correlation tells if there's a connection between the variables to begin with etc. That's a very short summary, but suggest studying a lot more on the subject.
• The mean of a normal probability distribution is 490; the standard deviation is 145.

a. μ ± 1σ of the observations lie between what two values?

Lower Value

Upper Value