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Page 1: The Normal Distribution and the 68-95-99.7 Rule p.html p.html

The Normal Distribution The Normal Distribution and the 68-95-99.7 Ruleand the 68-95-99.7 Rule

http://www.mathsnet.net/js/balldrop.html

Page 2: The Normal Distribution and the 68-95-99.7 Rule p.html p.html

Normal means typicalNormal means typical If the average woman is 5 feet 4 inches tall If the average woman is 5 feet 4 inches tall

(64 inches) would you expect to see many (64 inches) would you expect to see many women who are around that height?women who are around that height?

Is it common to see women who are 6 feet Is it common to see women who are 6 feet tall?tall?

Is it common to see women who are 4 feet Is it common to see women who are 4 feet tall?tall?

Page 3: The Normal Distribution and the 68-95-99.7 Rule p.html p.html

Describing what is NormalDescribing what is Normal

MeanMean (EVERY ONE KNOWS THIS (EVERY ONE KNOWS THIS ONE)ONE) This is the average, the center. Here it is 64. This is the average, the center. Here it is 64. ItIt’’s located right in the middle of the normal s located right in the middle of the normal

curve.curve.

Page 4: The Normal Distribution and the 68-95-99.7 Rule p.html p.html

Describing what is NormalDescribing what is NormalStandard Deviation Standard Deviation

(NOT MANY KNOW THIS ONE)(NOT MANY KNOW THIS ONE) This tells us This tells us how spread out how spread out the distribution the distribution

is.is. Here it is 3.Here it is 3.

Page 5: The Normal Distribution and the 68-95-99.7 Rule p.html p.html

What is a typical womanWhat is a typical woman’’s s height?height?

68% of all women are within 1 standard 68% of all women are within 1 standard deviation of the mean.deviation of the mean.

Here 68% of all women are within 3 inches Here 68% of all women are within 3 inches of 64.of 64.

Page 6: The Normal Distribution and the 68-95-99.7 Rule p.html p.html

Going out farther…Going out farther…

95% of all women are within 2 standard 95% of all women are within 2 standard deviations of the mean.deviations of the mean.

Here 95% of all women are within 6 inches Here 95% of all women are within 6 inches of 64.of 64.

Page 7: The Normal Distribution and the 68-95-99.7 Rule p.html p.html

Almost all women are…Almost all women are…

99.7% of all women are within 3 standard 99.7% of all women are within 3 standard deviations of the mean.deviations of the mean.

Here 99.7% of all women are within 9 in. of Here 99.7% of all women are within 9 in. of 64.64.

Page 8: The Normal Distribution and the 68-95-99.7 Rule p.html p.html

The average grade on the quiz The average grade on the quiz was 30 points.was 30 points.

Could all of the kids gotten a 30? Could all of the kids gotten a 30? Did half the kids get a 20 and the other Did half the kids get a 20 and the other

half get a 40?half get a 40? What would be a good score on the quiz?What would be a good score on the quiz? What would be a bad score?What would be a bad score?

This is not enough information.This is not enough information.It leaves a lot of questions…It leaves a lot of questions…

Page 9: The Normal Distribution and the 68-95-99.7 Rule p.html p.html

What if we were only given the What if we were only given the standard deviation?standard deviation?

The standard deviation on the quiz was The standard deviation on the quiz was 5 points.5 points.

So we know the spread of the grades, but So we know the spread of the grades, but from where was the quiz centered?from where was the quiz centered?

How did the class do in general?How did the class do in general? Was my grade good or bad?Was my grade good or bad?

This would still leave us with This would still leave us with plenty of questions…plenty of questions…

Page 10: The Normal Distribution and the 68-95-99.7 Rule p.html p.html

With both Mean and St. Dev.With both Mean and St. Dev.

Consider the following statement:Consider the following statement: The mean was 30 points and the standard The mean was 30 points and the standard

deviation on the quiz was 5 points.deviation on the quiz was 5 points. Now we know that about 68% of the class Now we know that about 68% of the class

scored between…scored between… 25 and 35 points25 and 35 points Now we can say that it was rare to get a score Now we can say that it was rare to get a score

that was above 40 points.that was above 40 points. It was extremely rare to score above 45 points.It was extremely rare to score above 45 points.

Page 11: The Normal Distribution and the 68-95-99.7 Rule p.html p.html

The Normal Distribution needs The Normal Distribution needs both statistics to survive.both statistics to survive.

We can also graph the quiz scores.We can also graph the quiz scores.

Page 12: The Normal Distribution and the 68-95-99.7 Rule p.html p.html

A typical score…A typical score…was between 25 and 35 pointswas between 25 and 35 points

Page 13: The Normal Distribution and the 68-95-99.7 Rule p.html p.html

An extremely rare score…An extremely rare score…

What percent of the students did this What percent of the students did this student beat?student beat?

Page 14: The Normal Distribution and the 68-95-99.7 Rule p.html p.html

An extremely rare score…An extremely rare score…

About 99.7 percent scored between a About 99.7 percent scored between a

15 to 45. That does not include about 15 to 45. That does not include about 0.3% 0.3%

Page 15: The Normal Distribution and the 68-95-99.7 Rule p.html p.html

An extremely rare score…An extremely rare score…

You would have to split this in half to get about You would have to split this in half to get about 0.15%.0.15%.

So out of 2000 students, only 3 would score that So out of 2000 students, only 3 would score that well! well!

Page 16: The Normal Distribution and the 68-95-99.7 Rule p.html p.html

Practice with 68-95-99.7 RulePractice with 68-95-99.7 Rule Suppose the average height of boys at EHS is Suppose the average height of boys at EHS is

66 inches, with a standard deviation of 4.5 66 inches, with a standard deviation of 4.5 inches.inches.

Draw the normal curve representing this information.Draw the normal curve representing this information. Answer:Answer:

Page 17: The Normal Distribution and the 68-95-99.7 Rule p.html p.html

Practice with 68-95-99.7 RulePractice with 68-95-99.7 Rule

Suppose the average height of boys at EHS is Suppose the average height of boys at EHS is 66 inches, with a standard deviation of 4.5 66 inches, with a standard deviation of 4.5 inches.inches.

How tall is someone 2 standard deviations How tall is someone 2 standard deviations aboveabove the the mean?mean?

Answer: 66 + 4.5 + 4.5 = 75 inchesAnswer: 66 + 4.5 + 4.5 = 75 inches

Page 18: The Normal Distribution and the 68-95-99.7 Rule p.html p.html

Practice with 68-95-99.7 RulePractice with 68-95-99.7 Rule

Suppose the average height of boys at EHS is Suppose the average height of boys at EHS is 66 inches, with a standard deviation of 4.5 66 inches, with a standard deviation of 4.5 inches.inches.

What percent of boys are between 61.5 and 70.5 inches What percent of boys are between 61.5 and 70.5 inches tall ?tall ?

Answer: 68%Answer: 68%

Page 19: The Normal Distribution and the 68-95-99.7 Rule p.html p.html

Practice with 68-95-99.7 RulePractice with 68-95-99.7 Rule

Suppose the average height of boys at EHS is Suppose the average height of boys at EHS is 66 inches, with a standard deviation of 4.5 66 inches, with a standard deviation of 4.5 inches.inches.

How many SDHow many SD’’s above the mean is someone who is s above the mean is someone who is 79.5 inches tall?79.5 inches tall?

Answer: 3Answer: 3

Page 20: The Normal Distribution and the 68-95-99.7 Rule p.html p.html

Practice with 68-95-99.7 RulePractice with 68-95-99.7 Rule

Suppose the average height of boys at Sweet Suppose the average height of boys at Sweet Home is 66 inches, with a standard deviation of Home is 66 inches, with a standard deviation of 4.5 inches.4.5 inches.

What percent of boys are between 57 and 70.5 inches What percent of boys are between 57 and 70.5 inches tall?tall?

Answer:Answer:

68%+ the bit between 57 68%+ the bit between 57

and 61.5and 61.5

68%+13.5% = 81.5%68%+13.5% = 81.5%


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